# Existence and instability of standing waves for the biharmonic nonlinear
Schrödinger equation with combined nonlinearities

Xiaojun Chang<EMAIL_ADDRESS>School of Mathematics and Statistics &
Center for Mathematics and Interdisciplinary Sciences,
Northeast Normal University, Changchun 130024, Jilin, China  Hichem Hajaiej
<EMAIL_ADDRESS>Department of Mathematics, Cal State LA, Los Angeles CA
90032, USA  Zhouji Ma<EMAIL_ADDRESS>School of Mathematics and
Statistics & Center for Mathematics and Interdisciplinary Sciences,
Northeast Normal University, Changchun 130024, Jilin, China  Linjie Song
<EMAIL_ADDRESS>Institute of Mathematics, AMSS, Chinese Academy
of Science, Beijing 100190, China
University of Chinese Academy of Science, Beijing 100049, China

###### Abstract

We consider the following biharmonic nonlinear Schrödinger equation with
combined nonlinearities

$\displaystyle
i\partial_{t}\psi-\Delta^{2}\psi+\mu|\psi|^{q-2}\psi+|\psi|^{p-2}\psi=0,\qquad(t,x)\in\mathbb{R}\times\mathbb{R}^{N},$

where $N\geq 5,\ \psi\in\mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb{C},\
\mu>0,\ 2+\frac{8}{N}\leq q<p\leq 4^{*}$ and $4^{*}=\frac{2N}{N-4}$. We prove
the existence of normalized ground state solutions for the biharmonic
Schrödinger equation with combined nonlinearities and show that all ground
states correspond to the local minima of the associated energy functional
restricted to the Pohozaev set. Moreover, we prove that the standing waves are
strongly unstable by blowup. In particular, our results cover the Sobolev case
$p=4^{*}=\frac{2N}{N-4}$. Our method is novel and innovative as previous ideas
cannot be used in many cases under this study.

###### keywords:

Biharmonic Schrödinger equation; Normalized ground state solutions; Combined
nonlinearities; Instability.

##  1 Introduction and main results

In the last decades, there has been a growing interest in the study of the
biharmonic Schrödinger equation

$i\partial_{t}\psi+\gamma\Delta^{2}\psi+\epsilon\Delta\psi+f(|\psi|)\psi=0,$
(1.1)

where $N\geq 5,\gamma\not=0,\ \epsilon=\pm 1$ or $\epsilon=0$,
$\psi:\mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb{C}$. In [27, 28], the
biharmonic Schrödinger equation with mixed-dispersion term was introduced by
Karpman and Shagalov to capture the effects of small fourth-order dispersion
terms in the propagation of intense laser beams in a bulk medium with Kerr
nonlinearity. Ben-Artzi et al. [5] obtained necessary Strichartz estimates for
the biharmonic operator $\Delta^{2}$. Here, we are particularly interested in
the case $\epsilon=0$. This kind of biharmonic Schrödinger equations was first
introduced in [22, 43] to study the stability of solitons in magnetic
materials once the effective quasi particle mass becomes infinite. Fibich,
Ilan and Papanicolaou [17] described some properties of the biharmonic
Schrödinger equation in the subcritical case. Pausader [38] proved the global
well-posedness and scattering theory under some conditions for the mass
critical biharmonic Schrödinger equation. The mass supercritical and Sobolev
subcritical regime were considered in [20]. For the global well-posedness and
scattering of the focusing Sobolev critical biharmonic Schrödinger equations,
one can refer to [35] and [37]. For more related studies, the reader can see
[19, 34, 39, 47, 48] for the biharmonic Schrödinger equation, and [8, 9, 10,
11, 13, 16, 17] for the biharmonic Schrödinger equation with mixed-dispersion
term.

In this paper, we are concerned with the existence and orbital instability of
ground state solutions having prescribed mass, also called normalized ground
state solutions for the following biharmonic nonlinear Schrödinger equation
with combined nonlinearities

$i\partial_{t}\psi-\Delta^{2}\psi+\mu|\psi|^{q-2}\psi+|\psi|^{p-2}\psi=0,\quad\psi(0,x)=\psi_{0}\in
H^{2}(\mathbb{R}^{N}),\quad(t,x)\in\mathbb{R}\times\mathbb{R}^{N},$ (1.2)

where $N\geq 5,\ \psi:\mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb{C},\
\mu>0,\ 2+\frac{8}{N}\leq q<p\leq 4^{*}$ and $4^{*}=\frac{2N}{N-4}$.

Firstly, we recall that the standing waves of (1.2), are solutions of the form
$\psi(t,x)=e^{i\omega t}u(x),\ \omega\in\mathbb{R}$. Then the ansatz yields
the elliptic equation

$\Delta^{2}u+\omega u-\mu|u|^{q-2}u-|u|^{p-2}u=0,\quad\mathrm{in}\
\mathbb{R}^{N}.$ (1.3)

While looking for solutions to (1.3), one reasonable approach is to fix
$\omega\in\mathbb{R}$ and search for the critical points of the following
action functional

$A_{p,q}(u)=\frac{1}{2}\|\Delta
u\|_{2}^{2}+\frac{\omega}{2}\|u\|_{2}^{2}-\frac{\mu}{q}\|u\|_{q}^{q}-\frac{1}{p}\|u\|_{p}^{p}.$

Alternatively, we can search for the solutions to (1.3) with prescribed mass,
which is a meaningful problem from both mathematical and physical
perspectives. Indeed, the mass of a quantum system is a fundamental property
that is related to the probability density of the system. In quantum
mechanics, the squared modulus of the wave function represents the probability
density of finding the system in a particular state. For normalized solutions,
the total probability of finding the system in any state is conserved over
time, as the mass is conserved. By considering solutions to the biharmonic
Schrödinger equation with prescribed mass, we can gain insights into the
properties of the system in a consistent and meaningful way. Furthermore,
studying the solutions to the biharmonic Schrödinger equation with prescribed
mass can lead to interesting mathematical challenges and the discovery of new
tools and insights.

We define on $H^{2}=H^{2}(\mathbb{R}^{N},\mathbb{C})$ the energy functional

$E_{p,q}(u):=\frac{1}{2}\|\Delta
u\|_{2}^{2}-\frac{\mu}{q}\|u\|_{q}^{q}-\frac{1}{p}\|u\|_{p}^{p}.$

It is standard to check that $E_{p,q}$ is of class $C^{1}$ and that a critical
point of $E_{p,q}$ restricted to the mass constraint

$S(c)=\\{u\in H^{2}:\|u\|_{2}^{2}=c\\}$

corresponds to a solution of (1.3) with prescribed mass $\|u\|_{2}^{2}=c$.
Here, the parameter $\omega\in\mathbb{R}$ is unknown and it appears as a
Lagrange multiplier. In this paper, we are interested in the solutions that
minimize $E_{p,q}(u)$ among all non-trivial solutions, namely ground state
solutions. We define it in the following way:

######  Definition 1.1.

We say that a solution $u_{c}\in S(c)$ to (1.3) is a ground state solution to
(1.3) if it is a solution with the least energy among all the solutions that
belong to $S(c)$. Namely, it satisfies

$E_{p,q}(u_{c})=\inf\\{E_{p,q}(u),u\in
S(c),(E_{p,q}|_{S(c)})^{{}^{\prime}}(u)=0\\}.$

In recent years, the study of the normalized ground state solutions of
nonlinear Schrödinger type equations has been a hot topic. Particularly, there
have been several results for the mixed-dispersion fourth-order Schrödinger
equations. However, there are few references to the corresponding study of
biharmonic Schrödinger equation with mixed nonlinearities in the literature.
Recently, in [34], the authors proved the existence of the normalized ground
state solutions to (1.2) for $2<q<2+\frac{8}{N}<p=4^{*}$. In this paper, we
will study the normalized ground state solutions to (1.2) in the remaining
regimes.

Under the assumption $2+\frac{8}{N}\leq q<p\leq 4^{*}$, $E_{p,q}$ restricted
to $S(c)$ is unbounded from below. To overcome this difficulty, we construct a
subset of $S(c)$ on which $E_{p,q}$ is bounded from below and coercive. We
exploit a natural constraint called Pohozaev set of $E_{p,q}$ restricted to
$S(c)$, in which all the critical points of $E_{p,q}|_{S(c)}$ belong to this
Pohozaev set. The set is defined as follows

$\mathcal{Q}_{p,q}(c)=\\{u\in S(c):Q_{p,q}(u)=0\\},$

where

$Q_{p,q}(u)=\|\Delta
u\|_{2}^{2}-\mu\gamma_{q}\|u\|_{q}^{q}-\gamma_{p}\|u\|_{p}^{p}.$

with

$\gamma_{r}=\frac{N(r-2)}{4r}=\frac{N}{2}(\frac{1}{2}-\frac{1}{r}),\forall
r\in(2,4^{*}].$

Hence it is natural to consider the minimization problem on this Pohozaev set

$m_{p,q}(c)=\inf\limits_{u\in\mathcal{Q}_{p,q}(c)}E_{p,q}(u).$

Then we state the following existence results.

######  Theorem 1.2.

Let $N\geq 5$, $c>0$, $\mu>0$, $2+\frac{8}{N}\leq q<p<4^{*}$. If
$q=2+\frac{8}{N}$, we further assume that $\mu
c^{\frac{4}{N}}<\frac{N+4}{NC^{q}_{N,q}}$. Then $E_{p,q}|_{S(c)}$ has a
critical point $u$ at a positive level $E_{p,q}(u)>0$ with the following
properties: $u$ solves (1.3) for some $\omega>0$, and is a normalized ground
state of (1.3) on $S(c)$.

######  Theorem 1.3.

Let $N\geq 5$, $c>0$, $\mu>0$ and $2+\frac{8}{N}\leq q<4^{*}$. If
$q=2+\frac{8}{N}$, we further assume that $\mu
c^{\frac{4}{N}}<\frac{N+4}{NC^{q}_{N,q}}$. If $N=5$, we further assume that
$q>\frac{22}{3}$. Then $E_{4^{*},q}|_{S(c)}$ has a critical point $u$ at a
level $E_{4^{*},q}(u)\in(0,\frac{2}{N}\mathcal{S}^{\frac{N}{4}})$ with the
following properties: $u$ solves (1.3) for some $\omega>0$, and is a
normalized ground state of (1.3) on $S(c)$ for $p=4^{*}$.

To search for the normalized ground state solutions of (1.2), the first
challenging thing is to construct a bounded Palais-Smale sequence
$\\{u_{n}\\}$ of $E_{p,q}|_{S(c)}$. To overcome this difficulty, we prove the
existence of a Palais-Smale sequence $\\{{u_{n}}\\}$ of $E_{p,q}|_{S(c)}$ at
level $m_{p,q}(c)$ by constructing it within the subspace
$\mathcal{Q}_{p,q}(c)$, where the boundedness of the sequence can be obtained,
see Lemma 2.5. In fact, motivated by the work of Bartsch and Soave [1, 3], we
first show that $t_{u}$ defined in Lemma 2.3 is unique, and the map $u\mapsto
t_{u}$ is of $C^{1}$ as stated in Lemma 2.4. This implies that the functional
$\Psi(u)$ defined in Section 3 is also $C^{1}$, as mentioned in Lemma 3.1. By
adapting the arguments used in [1] to the $C^{1}$ constrained functional
$I:=\Psi(u)|_{S(c)}$, we can obtain the desired Palais-Smale sequence.

In addition, another major obstruction is faced because of the lack of the
compactness for the obtained sequence $\\{u_{n}\\}$. For the Sobolev
subcritical case, the key ingredient is to show that the map $c\rightarrow
m_{p,q}(c)$ is strictly decreasing, as stated in Lemma 2.10. In fact, up to a
translation and up to a subsequence, we can see that $\\{u_{n}\\}$ has a
nontrivial weak limit $u$ with $\|u\|_{2}^{2}=c_{1}\in(0,c]$, which satisfies

$\Delta^{2}u+\omega u-\mu_{n}|u|^{q-2}u-|u|^{p-2}u=0,$

and $u\in\mathcal{Q}_{p,q}(c_{1})$. By using some arguments, we can show that
$\lim\limits_{n\rightarrow\infty}E_{p,q}(u_{n}-u)\geq 0$ and it follows that

$m_{p,q}(c)=\lim\limits_{n\rightarrow\infty}E_{p,q}(u_{n})=E_{p,q}(u)+\lim\limits_{n\rightarrow\infty}E_{p,q}(u_{n}-u)\geq
m_{p,q}(c_{1}).$

This allows us to get the strong convergence of the sequence $\\{u_{n}\\}$ if
$c\rightarrow m_{p,q}(c)$ is strictly decreasing for $c>0$. However, in the
Sobolev critical case, the problem becomes more complicated. We need to
construct an appropriate testing function to derive an upper bound for
$m_{4^{,}q}(c)$, which is denoted as
$m_{4^{,}q}(c)<\frac{2}{N}\mathcal{S}^{\frac{N}{4}}$ and is obtained in Lemma
4.1. This upper bound provides an estimate on the energy level
$m_{4^{*},q}(c)$, which is crucial for further analysis and convergence
results.

Together with above analysis, we obtain a non-existence result as follows.

######  Theorem 1.4.

Let $N\geq 5$, $c>0$, $\mu>0$, $2+\frac{8}{N}=q<p\leq 4^{*}$. We further
assume that $\mu c^{\frac{4}{N}}\geq\frac{N+4}{NC^{q}_{N,q}}$. Then
$E_{p,q}|_{S(c)}$ has no critical point $u$ at the level
$E_{p,q}(u)=m_{p,q}(c)$ (including the case
$\mathcal{Q}_{p,q}^{\mu}(c)=\emptyset$).

Finally in this section, we will discuss the strong instability of standing
waves to (1.2). First we recall the definition of instability below.

######  Definition 1.5.

We say that a solution $u\in H^{2}({\mathbb{R}^{N}})$ to (1.3) is unstable by
blowup in finite time, if for any $\epsilon>0$, there exists a $v\in
H^{2}({\mathbb{R}^{N}})$ such that $\|v-u\|_{H^{2}}<\epsilon$ and the solution
$\psi(t)$ to (1.2) with initial data $\psi_{0}=v$ blows up in finite time.

The instability of standing waves to nonlinear Schrödinger equations has been
extensively studied in the literature, we refer the reader to [21, 29, 40, 41]
and the references therein. The strategy for proving instability is often
based on the variational characterization of the ground states on the Pohozaev
or Nehari manifold, see [12] for example. The key point is to show the
following inequality

$Q_{p,q}(\psi)\leq E_{p,q}(\psi_{0})-E_{p,q}(u),$

where $u$ is the corresponding ground state. Then by using the virial
identity, we can obtain a differential inequality as follows

$\frac{d^{2}}{dt^{2}}\int_{\mathbb{R}^{N}}|x\psi(t)|^{2}dx=8Q_{p,q}(\psi(t))\leq-(E_{p,q}(u)-E_{p,q}(\psi_{0}))<0.$

This implies that the solution $\psi(t)$ blows up in finite time.

While for the biharmonic nonlinear Schrödinger equation with mixed dispersion

$i\partial\psi-\gamma\Delta^{2}\psi+\epsilon\Delta\psi+|\psi|^{2\sigma}\psi=0,\qquad\gamma>0,\
\epsilon\in\mathbb{R},\ 4\leq\sigma\leq\frac{4}{N-4},$ (1.4)

the virial identity is not available due to the appearance of the operator
$\Delta^{2}$. Boulenger and Lenzmann [12] used a localized version of the
virial identity from [35, 37] to prove blowup results for radial solutions in
$H^{2}(\mathbb{R}^{N})$ of (1.4) with $\gamma=1$ under various criteria in the
mass-critical, mass-supercritical and Sobolev critical cases. In particular
when $\epsilon=0$, they assumed that the energy of the initial data was either
negative or non-negative with some additional conditions on the initial data.
Bonheure et al. [9] proved the instability by blowup of the radial ground
state solutions for (1.4) when $\epsilon=0$ in both the mass-critical and
mass-supercritical regimes. Later, in [8], they showed that the instability of
the radial normalized ground state solutions can be proved by blowup with
$\epsilon=1$ when $4\leq\sigma<\frac{4}{N-4}$.

In the present paper, we aim to study the instability by blowup for ground
states of (1.2). To the best of our knowledge, so far, there has been no
existing instability results for the biharmonic nonlinear Schrödinger
equations with combined nonlinearities. We will borrow an idea from Boulenger
and Lenzmann [12] to study this issue. However, we cannot directly use the
arguments from Boulenger and Lenzmann’s work due to the fact that the energy
of the initial data satisfies $E_{p,q}(\psi_{0})>0$ without any additional
assumptions on $\psi_{0}$. Therefore, we need to come up with a different
approach to establish instability results for (1.2).

More precisely, in order to study the instability of (1.2), one of the main
challenges is to obtain the following estimate

$\frac{d}{dt}M_{\varphi_{R}}[\psi(t)]\leq-\delta\|\Delta\psi(t)\|^{2}_{2},\qquad\delta>0,$
(1.5)

where the modified energy $M_{\varphi_{R}}[\psi(t)]$ is defined in (6.2). This
estimate is crucial for establishing the instability result. However, there
are several terms in the inequality that make it difficult to estimate (1.5),
including the term $\mu|u|_{q}^{q}$ or $|u|_{p}^{p}$ that remains in the
inequality after applying variational arguments. Another challenge is that the
term $|\Delta\psi(t)|^{2}_{2}$ in (1.5) may be unbounded, which further
complicates the estimation process. To overcome these challenges, a key point
is to carefully choose a suitable bound for $|\Delta\psi(t)|^{2}_{2}$, which
can be used to obtain the desired estimate for
$\frac{d}{dt}M_{\varphi_{R}}[u(t)]$.

In what follows, we state our main instability results in Sobolev subcritical
case and Sobolev critical case respectively.

######  Theorem 1.6.

Let $2+\frac{8}{N}\leq q<p<4^{*}$. Then standing waves associated with the
ground state solutions to (1.3) with prescribed mass are strongly unstable.

######  Theorem 1.7.

Let $2+\frac{8}{N}\leq q<p=4^{*}$. Then standing waves associated with the
ground state solutions to (1.2) with prescribed mass are strongly unstable.

  * Remark:

When considering the instability of normalized ground states solutions to
(1.2), we work in the space $H^{2}_{r}(\mathbb{R}^{N})$. It is still not clear
for us how to control the error terms without the radial symmetry, even though
the local virial identities hold true without radial assumptions. If we set
$p,q\in\mathbb{N}$, then we may use the symmetric-decreasing rearrangement in
Fourier space to prove a radial symmetry result for ground states. See [12,
30, 31].

This paper is organized as follows: In Section 2, some preliminary results are
presented. In Section 3, we give the proof of Theorem 1.2. In Section 4, the
proof of Theorem 1.3 is established. In Section 5, we prove Theorem 1.4.
Finally in Section 6, we give the proofs of Theorem 1.6 and Theorem1.7 which
are used to study the strong instability of standing waves corresponding to
the normalized ground states. $\|\cdot\|_{q}$ will denote the standard norm in
$L^{q}(\mathbb{R}^{N})$.

##  2 Preliminaries

In this section, we present some useful lemmas to be often used in this paper.

Take $\|u\|_{H^{2}}=(\int_{\mathbb{R}^{N}}|\Delta
u|^{2}+|u|^{2}dx)^{\frac{1}{2}}$ to be the norm of $H^{2}$, which is
equivalent to the standard one
$\|u\|_{H^{2}}=(\int_{\mathbb{R}^{N}}|D^{2}u|^{2}+|\nabla
u|^{2}+|u|^{2}dx)^{\frac{1}{2}}.$ We first recall the well-known Gagliardo-
Nirenberg inequality for any $u\in H^{2}$.

######  Lemma 2.1.

Let $N\geq 5$ and $2<r<4^{*}$. Then the following sharp Gagliardo-Nirenberg
inequality

$\|u\|_{r}^{r}\leq C^{r}_{N,r}\|\Delta
u\|_{2}^{r\gamma_{r}}\|u\|_{2}^{r(1-\gamma_{r})},$ (2.1)

holds for any $u\in H^{2}(\mathbb{R}^{N})$, where $C_{N,r}$ denotes the sharp
constant.

Then, by similar arguments as in [46], we have the following Lions’ type lemma
in $H^{2}(\mathbb{R}^{N})$.

######  Lemma 2.2.

Let $R>0$. If $\\{u_{n}\\}$ is bounded in $H^{2}$ and if

$\sup\limits_{y\in\mathbb{R}^{N}}\int_{B(y,R)}|u_{n}|^{2}dx\rightarrow
0,\quad\mathrm{as}\quad n\rightarrow\infty,$

then $u_{n}\rightarrow 0$ in $L^{r}(\mathbb{R}^{N})$ for $2<p<4^{*}$.

For any $u\in S(c)$, we define

$u_{t}(x)=t^{\frac{N}{4}}u(\sqrt{t}x).$ (2.2)

Clearly, $u_{t}\in S(c)$ for any $t>0$. Furthermore,

$E_{p,q}(u_{t})=\frac{1}{2}t^{2}\|\Delta
u\|_{2}^{2}-\frac{\mu}{q}t^{\frac{N(q-2)}{4}}\|u\|_{q}^{q}-\frac{1}{p}t^{\frac{N(p-2)}{4}}\|u\|_{p}^{p}$
(2.3)

and

$Q_{p,q}(u_{t})=t^{2}\|\Delta
u\|_{2}^{2}-\mu\gamma_{q}t^{\frac{N(q-2)}{4}}\|u\|_{q}^{q}-\gamma_{p}t^{\frac{N(p-2)}{4}}\|u\|_{p}^{p}.$

Then one has the following properties for $E_{p,q}(u_{t})$ and
$Q_{p,q}(u_{t})$.

######  Lemma 2.3.

Let $N\geq 5$, $c>0$, $\mu>0$ and $2+\frac{8}{N}\leq q<p\leq 4^{*}$. If
$q=2+\frac{8}{N}$, we further assume that $\mu
c^{\frac{4}{N}}<\frac{N+4}{NC^{q}_{N,q}}$. Then for any $u\in S(c)$, there
exists a unique $t_{u}\in(0,+\infty)$ for which
$u_{t_{u}}\in\mathcal{Q}_{p,q}(c)$. Moreover, $t_{u}$ is the unique critical
point of $E_{p,q}(u_{t})$ such that
$E_{p,q}(u_{t_{u}})=\max_{t\in(0,+\infty)}E_{p,q}(u_{t})$ and the function
$u\mapsto E_{p,q}(u_{t_{u}})$ is concave on $[t_{u},+\infty)$. In particular,
if $Q_{p,q}(u)\leq 0$, then $t_{u}\in(0,1]$.

###### Proof.

First we let

$Q_{p,q}(u_{t})=t^{2}(\|\Delta
u\|_{2}^{2}-\mu\gamma_{q}t^{\frac{N(q-2)}{4}-2}\|u\|_{q}^{q}-\gamma_{p}t^{\frac{N(p-2)}{4}-2}\|u\|_{p}^{p})=t^{2}h(t).$

When $2+\frac{8}{N}<q<p\leq 4^{*}$, we have
$\frac{N(p-2)}{4}-2>\frac{N(q-2)}{4}-2>0$. When $2+\frac{8}{N}=q<p\leq 4^{*}$
and $\mu c^{\frac{4}{N}}<\frac{N+4}{NC^{q}_{N,q}}$, we have
$\frac{N(p-2)}{4}-2>\frac{N(q-2)}{4}-2=0$. Using the Gagliardo-Nirenberg
inequality, we get

$\mu\gamma_{q}\|u\|_{q}^{q}\leq\mu\gamma_{q}C^{q}_{N,q}\|\Delta
u\|_{2}^{q\gamma_{q}}\|u\|_{2}^{q(1-\gamma_{q})}<\|\Delta u\|_{2}^{2}.$

Hence, for $2+\frac{8}{N}\leq q<p\leq 4^{*}$, it always holds that $h(t)>0$
for $t$ small enough, $h(t)<0$ for $t$ large enough and $h^{\prime}(t)<0$ for
all $t\in(0,+\infty)$. So $Q_{p,q}(u_{t})$ has a unique zero $t_{u}$.

By direct calculations, we have
$\frac{d}{dt}E_{p,q}(u_{t})=t^{-1}Q_{p,q}(u_{t})$, $E_{p,q}(u_{t})>0$ for
$t>0$ small enough and
$\lim\limits_{t\rightarrow\infty}E_{p,q}(u_{t})=-\infty$. Thus, $t_{u}$ is the
unique critical point of $E_{p,q}(u_{t})$ and
$E_{p,q}(u_{t_{u}})=\max_{t\in(0,+\infty)}E_{p,q}(u_{t})$. Now writing
$t=st_{u}$, we have

$\displaystyle\frac{d^{2}}{d^{2}t}E_{p,q}(u_{t})$ $\displaystyle=\|\Delta
u\|_{2}^{2}-\mu\gamma_{q}(\frac{N(q-2)}{4}-1)t^{\frac{N(q-2)}{4}-2}\|u\|_{q}^{q}-\gamma_{p}(\frac{N(p-2)}{4}-1)t^{\frac{N(p-2)}{4}-2}\|u\|_{p}^{p}$
$\displaystyle=\|\Delta
u\|_{2}^{2}-\mu\gamma_{q}(\frac{N(q-2)}{4}-1)s^{\frac{N(q-2)}{4}-2}t_{u}^{\frac{N(q-2)}{4}-2}\|u\|_{q}^{q}$
$\displaystyle-\gamma_{p}(\frac{N(p-2)}{4}-1)s^{\frac{N(p-2)}{4}-2}t_{u}^{\frac{N(p-2)}{4}-2}\|u\|_{p}^{p}$
$\displaystyle=\frac{1}{t_{u}^{2}}\big{[}t_{u}^{2}\|\Delta
u\|_{2}^{2}-\mu\gamma_{q}(\frac{N(q-2)}{4}-1)s^{\frac{N(q-2)}{4}-2}t_{u}^{\frac{N(q-2)}{4}}\|u\|_{q}^{q}$
$\displaystyle-\gamma_{p}(\frac{N(p-2)}{4}-1)s^{\frac{N(p-2)}{4}-2}t_{u}^{\frac{N(p-2)}{4}}\|u\|_{p}^{p}\big{]}.$

Since

$0=Q_{p,q}(u_{t_{u}})=t_{u}^{2}\|\Delta
u\|_{2}^{2}-\mu\gamma_{q}t_{u}^{\frac{N(q-2)}{4}}\|u\|_{q}^{q}-\gamma_{p}t_{u}^{\frac{N(p-2)}{4}}\|u\|_{p}^{p},$

we infer that $\frac{d^{2}}{d^{2}t}E_{p,q}(u_{t})<0$ for any $s\geq 1$. In
addition, if $Q_{p,q}(u)\leq 0$, it follows that $0<t_{u}\leq 1$. This
completes the proof. ∎

######  Lemma 2.4.

Let $2+\frac{8}{N}\leq q<p\leq 4^{*}$. For any $u\in S(c)$, the map $u\mapsto
t_{u}$ is of class $C^{1}$, where $t_{u}$ is defined in Lemma 2.3.

###### Proof.

For $2+\frac{8}{N}\leq q<p\leq 4^{*}$ and $u\in S(c)$, we consider the $C^{1}$
function $\Psi:\mathbb{R}\times S(c)\mapsto\mathbb{R}$ defined by
$\Psi(t,u)=\psi^{\prime}_{u}(t)$, where $\psi_{u}(t)=E_{p,q}(u_{t})$. By
direct computation and the definition of $t_{u}$, we have

$\Psi(t_{u},u)=\frac{1}{t_{u}}Q_{p,q}(u_{t_{u}})=0\quad\mathrm{and}\quad\partial_{t}\Psi(t_{u},u)=\psi^{\prime\prime}_{u}(t_{u})<0.$

By the Implicit Function Theorem, we can get that the map $u\mapsto t_{u}$ is
of class $C^{1}$. ∎

######  Lemma 2.5.

Let $N\geq 5$, $c>0$, $\mu>0$ and $2+\frac{8}{N}\leq q<p\leq 4^{*}$. If
$q=2+\frac{8}{N}$, we further assume that $\mu
c^{\frac{4}{N}}<\frac{N+4}{NC^{q}_{N,q}}$. Then,

$(i)\ \mathcal{Q}_{p,q}(c)\not=\emptyset;$

$(ii)\ \inf\limits_{u\in\mathcal{Q}_{p,q}(c)}\|\Delta u\|_{2}^{2}>0;$

$(iii)\ m_{p,q}(c)>0;$

$(iv)\ E_{p,q}$ is coercive on $\mathcal{Q}_{p,q}(c)$, that is
$E_{p,q}(u_{n})\rightarrow+\infty$ for any
$\\{u_{n}\\}\subset\mathcal{Q}_{p,q}(c)$ with
$\|u_{n}\|_{H^{2}}\rightarrow\infty$.

###### Proof.

$(i)$ By Lemma 2.3, for any $u\in S(c)$, there always exists $t_{u}>0$ such
that $u_{t_{u}}\in\mathcal{Q}_{p,q}(c)$, it follows that
$\mathcal{Q}_{p,q}(c)\neq\emptyset$.

$(ii)$ We divide the proof into two cases.

Case $1:$ $(p<4^{*})\ $For any $u\in\mathcal{Q}_{p,q}(c)$, by the Gagliardo-
Nirenberg inequality, we have

$\displaystyle\|\Delta u\|_{2}^{2}$
$\displaystyle=\mu\gamma_{q}\|u\|_{q}^{q}+\gamma_{p}\|u\|_{p}^{p}$
$\displaystyle\leq\mu\gamma_{q}C^{q}_{N,q}\|u\|_{2}^{q(1-\gamma_{q})}\|\Delta
u\|_{2}^{q\gamma_{q}}+\gamma_{p}C^{p}_{N,p}\|u\|_{2}^{p(1-\gamma_{p})}\|\Delta
u\|_{2}^{p\gamma_{p}}$
$\displaystyle=\mu\gamma_{q}C^{q}_{N,q}(\sqrt{c})^{q(1-\gamma_{q})}\|\Delta
u\|_{2}^{q\gamma_{q}}+\gamma_{p}C^{p}_{N,p}(\sqrt{c})^{p(1-\gamma_{p})}\|\Delta
u\|_{2}^{p\gamma_{p}}.$

If $2+\frac{8}{N}<q<p$, then $p\gamma_{p}>q\gamma_{q}>2$. It follows that
there exists a constant $C>0$ such that $\|\Delta u\|_{2}^{2}\geq C$. If
$2+\frac{8}{N}=q<p$ and $\mu c^{\frac{4}{N}}<\frac{N+4}{NC^{q}_{N,q}}$, then
$p\gamma_{p}>q\gamma_{q}=2$ and $\frac{\mu
NC^{q}_{N,q}}{N+4}c^{\frac{4}{N}}<1$. This also implies that there exists a
constant $C>0$ such that $\|\Delta u\|_{2}^{2}\geq C$.

Case $2:$ $(p=4^{*})\ $The proof here is similar to Case 1. The only
difference is that we estimate the term $\int_{\mathbb{R}^{N}}|u|^{4^{*}}dx$
by using

$\mathcal{S}\|u\|^{2}_{4^{*}}\leq\|\Delta u\|^{2}_{2}.$

$(iii)$ If $2+\frac{8}{N}<q<p$, we can easily get
$\mu\gamma_{q}\|u\|_{q}^{q}+\gamma_{p}\|u\|_{p}^{p}=\|\Delta u\|_{2}^{2}\geq
C$. If $2+\frac{8}{N}=q<p$ and $\mu c^{\frac{4}{N}}<\frac{N+4}{NC^{q}_{N,q}}$,
it follows that

$\gamma_{p}\|u\|_{p}^{p}\geq(1-\frac{\mu
NC^{q}_{N,q}}{N+4}c^{\frac{4}{N}})\|\Delta u\|_{2}^{2}\geq C(1-\frac{\mu
NC^{q}_{N,q}}{N+4}c^{\frac{4}{N}}).$ (2.4)

Then, for any $u\in\mathcal{Q}_{p,q}(c)$, we have

$\displaystyle E_{p,q}(u)$
$\displaystyle=(\frac{1}{2}-\frac{1}{q\gamma_{q}})\|\Delta
u\|_{2}^{2}+(\frac{\gamma_{p}}{q\gamma_{q}}-\frac{1}{p})\|u\|_{p}^{p}$
$\displaystyle=(\frac{1}{2}-\frac{1}{p\gamma_{p}})\|\Delta
u\|_{2}^{2}+(\frac{\gamma_{q}}{p\gamma_{p}}-\frac{1}{q})\mu\|u\|_{q}^{q}$
$\displaystyle=(\frac{\gamma_{p}}{2}-\frac{1}{p})\|u\|_{p}^{p}+(\frac{\gamma_{q}}{2}-\frac{1}{q})\mu\|u\|_{q}^{q}\geq
C_{1}$ (2.5)

for some $C_{1}>0$, which implies that $m_{p,q}(c)>0$.

$(iv)$ By (2) it is easily seen that the conclusion holds. ∎

Since for given $c>0$, one can see from Lemma 2.5 that the infimum

$m_{p,q}(c)=\inf\limits_{u\in\mathcal{Q}_{p,q}(c)}E_{p,q}(u)$

is well-defined and strictly positive. We will characterize further the
behavior of $m_{p,q}(c)$ when $c>0$ varies.

######  Lemma 2.6.

The function $c\mapsto m_{p,q}(c)$ is non-increasing on $(0,+\infty)$.

###### Proof.

First we set $\gamma(c)=\inf\limits_{u\in
S(c)}\max\limits_{t>0}E_{p,q}(u_{t})$. We claim that $\gamma(c)=m_{p,q}(c)$.
Indeed, for any $u\in S(c)$, by Lemma 2.3, there exists a $t_{u}>0$ such that
$u_{t_{u}}\in\mathcal{Q}_{p,q}(c)$ and
$\max\limits_{t>0}E_{p,q}(u_{t})=E_{p,q}(u_{t_{u}})\geq m_{p,q}(c)$, thus we
get $\gamma(c)\geq m_{p,q}(c)$. On the other hand, for any
$u\in\mathcal{Q}_{p,q}(c),\ \max\limits_{t>0}E_{p,q}(u_{t})=E_{p,q}(u)$, this
immediately implies that $\gamma(c)\leq m_{p,q}(c)$. Then we conclude that
$\gamma(c)=m_{p,q}(c)$.

Next we assume that $0<c_{1}<c_{2}<+\infty$. For any $u\in S(c_{1})$, define
$v(x)=(\frac{c_{1}}{c_{2}})^{\frac{N-4}{8}}u((\frac{c_{1}}{c_{2}})^{\frac{1}{4}}x),\
\forall x\in\mathbb{R}^{N}$. then we have the followings:

$\|v\|_{2}^{2}=c_{2},\quad\|\Delta v\|_{2}^{2}=\|\Delta u\|_{2}^{2},$
$\|v\|_{q}^{q}=(\frac{c_{1}}{c_{2}})^{\frac{q(N-4)-2N}{8}}\|u\|_{q}^{q},\quad\|v\|_{p}^{p}=(\frac{c_{1}}{c_{2}})^{\frac{p(N-4)-2N}{8}}\|u\|_{p}^{p}.$

By the definition of $m_{p,q}(c)$ and the fact $\gamma(c)=m_{p,q}(c)$, we have

$\displaystyle m_{p,q}(c_{2})$
$\displaystyle\leq\max_{t\in(0,+\infty)}E_{p,q}(v_{t})$
$\displaystyle=E_{p,q}(v_{t_{v}})=E_{p,q}(t_{v}^{\frac{N}{4}}v(\sqrt{t_{v}}x))$
$\displaystyle=E_{p,q}(t_{v}^{\frac{N}{4}}(\frac{c_{1}}{c_{2}})^{\frac{N-4}{8}}u((\frac{c_{1}}{c_{2}})^{\frac{1}{4}}\sqrt{t_{v}}x))$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{N}}|\Delta
u_{t_{v}}|^{2}dx-(\frac{c_{1}}{c_{2}})^{\frac{q(N-4)-2N}{8}}\frac{\mu}{q}\int_{\mathbb{R}^{N}}|u_{t_{v}}|^{q}dx-(\frac{c_{1}}{c_{2}})^{\frac{p(N-4)-2N}{8}}\frac{1}{p}\int_{\mathbb{R}^{N}}|u_{t_{v}}|^{p}dx$
$\displaystyle<\frac{1}{2}\int_{\mathbb{R}^{N}}|\Delta
u_{t_{v}}|^{2}dx-\frac{\mu}{q}\int_{\mathbb{R}^{N}}|u_{t_{v}}|^{q}dx-\frac{1}{p}\int_{\mathbb{R}^{N}}|u_{t_{v}}|^{p}dx$
$\displaystyle=E_{p,q}(u_{t_{v}})\leq\max_{t\in(0,+\infty)}E_{p,q}(u_{t}),$

where $2+\frac{8}{N}\leq q<p\leq 4^{*}$ and $N\geq 5$. Since $u\in S(c_{1})$
is arbitrary, we have

$m_{p,q}(c_{2})\leq\inf_{u\in
S(c_{1})}\max_{t\in(0,+\infty)}E_{p,q}(u_{t})=m_{p,q}(c_{1}).$ (2.6)

∎

Next we will prove that $m_{p,q}(c)$ is continuous. Before that, we will show
some lemmas which will be used in the proof for the continuity of
$m_{p,q}(c)$.

######  Lemma 2.7.

Let $2+\frac{8}{N}\leq q<p\leq 4^{*}$. Let $\\{u_{n}\\}_{n\in\mathbb{N}^{+}}$
be a bounded sequence in $H^{2}(\mathbb{R}^{N})$ and $a_{n}$ be a sequence
satisfying $\lim\limits_{n\rightarrow\infty}a_{n}=1$. Then

$\lim\limits_{n\rightarrow\infty}\big{|}E_{p,q}(a_{n}u_{n})-E_{p,q}(u_{n})\big{|}=0.$

###### Proof.

It follows from direct calculations that

$\big{|}E_{p,q}(a_{n}u_{n})-E_{p,q}(u_{n})\big{|}\leq\big{|}\frac{(a_{n}^{2}-1)}{2}\|\Delta
u_{n}\|_{2}^{2}\big{|}+\big{|}\frac{\mu}{q}(a_{n}^{q}-1)\|u_{n}\|_{q}^{q}\big{|}+\big{|}\frac{1}{p}(a_{n}^{p}-1)\|u_{n}\|_{p}^{p}\big{|}$

Since$\\{u_{n}\\}_{n\in\mathbb{R}}$ is bounded in $H^{2}(\mathbb{R}^{N})$, we
can directly get the conclusion. ∎

######  Lemma 2.8.

Let $2+\frac{8}{N}\leq q<p\leq 4^{*}$. If $q=2+\frac{8}{N}$, we further assume
that $\mu c^{\frac{4}{N}}<\frac{N+4}{NC^{q}_{N,q}}$. For any $u\in S(c)$, let
$t_{u}$ be be given as in Lemma 2.3. Then $t_{u}$ satisfies

$t_{u}^{\frac{N(p-2)}{4}-2}<C\frac{\|\Delta u\|_{2}^{2}}{\|u\|_{p}^{p}}.$

where $C$ is a constant independent of $u$.

###### Proof.

For convenience, in this part, we set $f(t)=\mu|t|^{q-2}t+|t|^{p-2}t$, then
$F(t)=\mu\frac{1}{q}|t|^{q}+\frac{1}{p}|t|^{p}$. We claim that

$f(t)t>(2+\frac{8}{N})F(t)>0,\quad\mathrm{for}\ \mathrm{all}\ t\not=0.$

Indeed, when $2+\frac{8}{N}<q<p\leq 4^{*}$, we obviously have

$f(t)t-(2+\frac{8}{N})F(t)=\mu(1-\frac{2+8/N}{q})|t|^{q}+(1-\frac{2+8/N}{p})|t|^{p}>0,\quad\mathrm{for}\
\mathrm{all}\ t\not=0.$

When $2+\frac{8}{N}=q<p\leq 4^{*}$, we still find that

$f(t)t-(2+\frac{8}{N})F(t)=(1-\frac{2+8/N}{p})|t|^{p}>0,\quad\mathrm{for}\
\mathrm{all}\ t\not=0.$

Therefore, we have the following inequality,

$f(t)t-2F(t)>\frac{8}{N}F(t),\quad\mathrm{for}\ \mathrm{all}\ t\not=0.$ (2.7)

According to Lemma 2.3, we have $u_{t_{u}}\in\mathcal{Q}_{p,q}(c)$. Hence
$Q_{p,q}(u_{t_{u}})=0,\ i.e.$,

$\|\Delta
u_{t_{u}}\|_{2}^{2}=\mu\gamma_{q}\|u_{t_{u}}\|_{q}^{q}+\gamma_{p}\|u_{t_{u}}\|_{p}^{p}.$

Then by the inequality in (2.7), we have

$\displaystyle t_{u}^{2}\|\Delta u\|_{2}^{2}$
$\displaystyle=\mu\gamma_{q}\|u_{t_{u}}\|_{q}^{q}+\gamma_{p}\|u_{t_{u}}\|_{p}^{p}$
$\displaystyle=\frac{N}{4}(\mu\frac{q-2}{q}\|u_{t_{u}}\|_{q}^{q}+\frac{p-2}{p}\|u_{t_{u}}\|_{p}^{p})$
$\displaystyle=\frac{N}{4}\int_{\mathbb{R}^{N}}(f(u_{t_{u}})u_{t_{u}}-2F(u_{t_{u}}))dx$
$\displaystyle>\frac{N}{4}\cdot\frac{8}{N}\int_{\mathbb{R}^{N}}F(u_{t_{u}})dx=2\int_{\mathbb{R}^{N}}F(u_{t_{u}})dx$
$\displaystyle=2(\mu\frac{1}{q}\|u_{t_{u}}\|_{q}^{q}+\frac{1}{p}\|u_{t_{u}}\|_{p}^{p})$
$\displaystyle=2(t_{u}^{\frac{N(q-2)}{4}}\mu\frac{1}{q}\|u\|_{q}^{q}+t_{u}^{\frac{N(p-2)}{4}}\frac{1}{p}\|u\|_{p}^{p})$
$\displaystyle>2t_{u}^{\frac{N(p-2)}{4}}\frac{1}{p}\|u\|_{p}^{p}.$

Hence, we deduce that

$t_{u}^{\frac{N(p-2)}{4}-2}<\frac{p}{2}\frac{\|\Delta
u\|_{2}^{2}}{\|u\|_{p}^{p}}.$

∎

######  Lemma 2.9.

Let $N\geq 5$, $\mu>0$ and $2+\frac{8}{N}\leq q<p\leq 4^{*}$. If
$q=2+\frac{8}{N}$, we further assume that $\mu
c^{\frac{4}{N}}<\frac{N+4}{NC^{q}_{N,q}}$. It holds that

$(i)$

$c\mapsto m_{p,q}(c)$ is a continuous mapping.

$(ii)$

Let $c\in(0,+\infty)$. We have $m_{p,q}(c)\leq
m_{p,q}(\alpha)+m_{p,q}(c-\alpha)$ for all $\alpha\in(0,c)$. The inequality is
strict if $m_{p,q}(\alpha)$ or $m_{p,q}(c-\alpha)$ is reached.

###### Proof.

$(i)\ $ Based on the above lemma, for any fixed $c>0$, $m_{p,q}(c-h)$ and
$m_{p,q}(c+h)$ are monotonic and bounded as $h\rightarrow 0^{+}$, so they have
limits. Moreover, $m_{p,q}(c-h)\geq m_{p,q}(c)\geq m_{p,q}(c+h)$, thus we
obtain

$\lim\limits_{h\rightarrow 0^{+}}m_{p,q}(c-h)\geq
m_{p,q}(c)\geq\lim\limits_{h\rightarrow 0^{+}}m_{p,q}(c+h).$

It remains to show the other direction. We finish it by proving two claims.

Claim $1$: $\lim\limits_{h\rightarrow 0^{+}}m_{p,q}(c-h)\leq m_{p,q}(c)$.

Take $u\in\mathcal{Q}_{p,q}(c)$, for any $h>0$, we set
$u^{h}(x)=\sqrt{1-\frac{h}{c}}u(x)$. Noticing that $\|u^{h}\|_{2}^{2}=c-h$, by
Lemma 2.3, there exists $t_{h}>0$ such that
$u^{h}_{t_{h}}\in\mathcal{Q}_{p,q}(c-h)$. By Lemma 2.8, we have

$t_{h}^{\frac{N(p-2)}{4}-2}<\frac{p}{2}\frac{\|\Delta
u^{h}\|_{2}^{2}}{\|u^{h}\|_{p}^{p}}=\frac{p}{2}\frac{(1-\frac{h}{c})}{(1-\frac{h}{c})^{\frac{p}{2}}}\frac{\|\Delta
u\|_{2}^{2}}{\|u\|_{p}^{p}},$

which implies that $t_{h}$ is bounded as $h\rightarrow 0^{+}$. Hence
$u_{t_{h}}$ is bounded in $H^{2}(\mathbb{R}^{N})$. So according to Lemma 2.7,
we have

$m_{p,q}(c-h)\leq
E_{p,q}(u^{h}_{t_{h}})=E_{p,q}(\sqrt{1-\frac{h}{c}}u_{t_{h}})\leq
E_{p,q}(u_{t_{h}})+o(h)\leq E_{p,q}(u)+o(h).$

Then we prove the claim.

Claim $2$: $\lim\limits_{h\rightarrow 0^{+}}m_{p,q}(c+h)\geq m_{p,q}(c)$.

Choose $u\in\mathcal{Q}_{p,q}(c+h)$ for $h>0$ and let
$v^{h}(x)=\frac{1}{\sqrt{1+\frac{h}{c}}}u(x)$. Since $\|v^{h}\|_{2}^{2}=c$,
similar to Claim $1$, there exists $t_{h}>0$ such that
$v^{h}_{t_{h}}\in\mathcal{Q}_{p,q}(c)$ and

$t_{h}^{\frac{N(p-2)}{4}-2}<\frac{p}{2}\frac{\|\Delta
v^{h}\|_{2}^{2}}{\|v^{h}\|_{p}^{p}}=\frac{p}{2}\frac{(1+\frac{h}{c})^{\frac{p}{2}}}{(1+\frac{h}{c})}\frac{\|\Delta
u\|_{2}^{2}}{\|u\|_{p}^{p}}.$

So $u_{t_{h}}$ is bounded in $H^{2}(\mathbb{R}^{N})$. Using Lemma 2.7, it
follows that

$m_{p,q}(c)\leq
E_{p,q}(v^{h}_{t_{h}})=E_{p,q}(\frac{1}{\sqrt{1+\frac{h}{c}}}u_{t_{h}})\leq
E_{p,q}(u_{t_{h}})+o(h)\leq E_{p,q}(u)+o(h).$

Hence, we prove that $c\mapsto m_{p,q}(c)$ is continuous.

$(ii)\ $ We observe that for fixed $\alpha\in(0,c)$, it is sufficient to prove
that the following holds

$\forall\theta\in(1,\frac{c}{\alpha}]:m_{p,q}(\theta\alpha)\leq\theta
m_{p,q}(\alpha)$ (2.8)

and that, if $m_{p,q}(\alpha)$ is reached, the inequality is strict. In fact,
if (2.8) holds then we have

$\displaystyle m_{p,q}(c)$
$\displaystyle=\frac{c-\alpha}{c}m_{p,q}(c)+\frac{\alpha}{c}_{p,q}m_{p,q}(c)$
$\displaystyle=\frac{c-\alpha}{c}m_{p,q}(\frac{c}{c-\alpha}(c-\alpha))+\frac{\alpha}{c}m_{p,q}(\frac{c}{\alpha}\alpha)$
$\displaystyle\leq m_{p,q}(c-\alpha)+m_{p,q}(\alpha),$

with a strict inequality if $m_{p,q}(\alpha)$ is reached. By the definition of
$m_{p,q}(c)$, we notice that, for any $\epsilon>0$ sufficiently small, there
exists a $u\in\mathcal{Q}_{p,q}(\alpha)$ such that

$E_{p,q}(u)\leq m_{p,q}(\alpha)+\epsilon.$ (2.9)

Since $Q_{p,q}(u)=0$, we define
$\widetilde{u}(x)=(\frac{1}{\theta})^{\frac{1}{p-2}}u((\frac{1}{\theta})^{\frac{p}{N(p-2)}}x)$.
Then

$\int_{\mathbb{R}^{N}}|\widetilde{u}|^{2}dx=\theta\alpha,\quad\int_{\mathbb{R}^{N}}|\widetilde{u}|^{p}dx=\int_{\mathbb{R}^{N}}|u|^{p}dx,$
$\int_{\mathbb{R}^{N}}|\widetilde{u}|^{q}dx=\int_{\mathbb{R}^{N}}|\widetilde{u}|^{q}dx=(\frac{1}{\theta})^{\frac{q-p}{p-2}}\int_{\mathbb{R}^{N}}|u|^{q}dx\geq\int_{\mathbb{R}^{N}}|u|^{q}dx,$
$\int_{\mathbb{R}^{N}}|\Delta\widetilde{u}|^{2}dx=(\frac{1}{\theta})^{\frac{2N+p(4-N)}{N(p-2)}}\int_{\mathbb{R}^{N}}|\Delta
u|^{2}dx\leq\int_{\mathbb{R}^{N}}|\Delta u|^{2}dx{\color[rgb]{0,0,1}.}$

Hence $Q_{p,q}(\widetilde{u})\leq 0$. So there exists $t_{0}\in(0,1]$ such
that $\widetilde{u}_{t_{0}}\in\mathcal{Q}_{p,q}(\theta\alpha)$ and for any
$\theta>1$,

$\displaystyle m_{p,q}(\theta\alpha)$ $\displaystyle\leq
E_{p,q}(\widetilde{u}_{t_{0}})=(\frac{1}{2}-\frac{1}{q\gamma_{q}})\int_{\mathbb{R}^{N}}|\Delta\widetilde{u}_{t_{0}}|^{2}dx+(\frac{\gamma_{p}}{q\gamma_{q}}-\frac{1}{p})\int_{\mathbb{R}^{N}}|\widetilde{u}_{t_{0}}|^{p}dx$
$\displaystyle=(\frac{1}{2}-\frac{1}{q\gamma_{q}})t_{0}^{2}\int_{\mathbb{R}^{N}}|\Delta\widetilde{u}|^{2}dx+(\frac{\gamma_{p}}{q\gamma_{q}}-\frac{1}{p})t_{0}^{\frac{Np}{4}-\frac{N}{2}}\int_{\mathbb{R}^{N}}|\widetilde{u}|^{p}dx$
$\displaystyle=(\frac{1}{2}-\frac{1}{q\gamma_{q}})t_{0}^{2}(\frac{1}{\theta})^{\frac{2N+p(4-N)}{N(p-2)}}\int_{\mathbb{R}^{N}}|\Delta
u|^{2}dx$
$\displaystyle+(\frac{\gamma_{p}}{q\gamma_{q}}-\frac{1}{p})t_{0}^{\frac{Np}{4}-\frac{N}{2}}\int_{\mathbb{R}^{N}}|u|^{p}dx$
$\displaystyle\leq\theta(\frac{1}{2}-\frac{1}{q\gamma_{q}})\int_{\mathbb{R}^{N}}|\Delta
u|^{2}dx+\theta(\frac{\gamma_{p}}{q\gamma_{q}}-\frac{1}{p})\int_{\mathbb{R}^{N}}|u|^{p}dx$
$\displaystyle\leq\theta E_{p,q}(u)<\theta(m_{p,q}(\alpha)+\epsilon).$

Since $\epsilon$ is arbitrary, we have that $m_{p,q}(\theta\alpha)\leq\theta
m_{p,q}(\alpha)$. If $m_{p,q}(\alpha)$ is reached, we can choose $\epsilon=0$.
Then the strict inequality follows. ∎

  * Remark:

For Lemma 2.9 $(ii)$, it is not used in the rest of the paper but is useful
for future application. Although the key point for us to show the strong
convergence is Lemma 2.10, it is possible to prove the strong convergence by
this property.

######  Lemma 2.10.

Let $2+\frac{8}{N}\leq q<p\leq 4^{*}$, $\mu>0$ and $c>0$. If
$q=2+\frac{8}{N}$, we further assume that $\mu
c^{\frac{4}{N}}<\frac{N+4}{NC^{q}_{N,q}}$. If there exists $u\in S(c)$ such
that $E_{p,q}(u)=m_{p,q}(c)$ and

$\Delta^{2}u+\omega u=\mu|u|^{q-2}u+u|^{p-2}u,$ (2.10)

then $Q_{p,q}(u)=0$ and $\omega>0$. Moreover, the function $c\mapsto
m_{p,q}(c)$ is strictly decreasing in a right neighborhood of $c$.

###### Proof.

The fact that any solution to (2.10) satisfies $Q_{p,q}(u)=0$ is a direct
consequence of the Pohozaev identity. Now we deduce from (2.10)

$\|\Delta u\|_{2}^{2}+\omega\|u\|_{2}^{2}-\mu\|u\|_{q}^{q}-\|u\|_{p}^{p}=0.$
(2.11)

Combining (2.11) with $Q_{p,q}(u)=0$, we have

$\omega\|u\|_{2}^{2}=\mu(1-\gamma_{q})\|u\|_{q}^{q}+(1-\gamma_{p})\|u\|_{p}^{p}.$

Since $0<\gamma_{q}<\gamma_{p}\leq 1$, it follows that $\omega>0$. Then we let
$u_{\lambda,t}(x)=t^{\frac{N}{4}}\sqrt{\lambda}u(\sqrt{t}x)$ for $\lambda,\
t>0$. We define

$\mathcal{K}(\lambda,t)=E_{p,q}(u_{\lambda,t})=\frac{1}{2}t^{2}\lambda\|\Delta
u\|_{2}^{2}-\frac{\mu}{q}t^{\frac{N(q-2)}{4}}\lambda^{\frac{q}{2}}\|u\|_{q}^{q}-\frac{1}{p}t^{\frac{N(p-2)}{4}}\lambda^{\frac{p}{2}}\|u\|_{p}^{p}$

and

$\mathcal{M}(\lambda,t)=Q_{p,q}(u_{\lambda,t})=t^{2}\lambda\|\Delta
u\|_{2}^{2}-\mu\gamma_{q}t^{\frac{N(q-2)}{4}}\lambda^{\frac{q}{2}}\|u\|_{q}^{q}-\gamma_{p}t^{\frac{N(p-2)}{4}}\lambda^{\frac{p}{2}}\|u\|_{p}^{p}.$

By suitable calculations, we have

$\frac{\partial\mathcal{K}}{\partial\lambda}(1,1)=\frac{1}{2}\|\Delta
u\|_{2}^{2}-\frac{\mu}{2}\|u\|_{q}^{q}-\frac{1}{2}\|u\|_{p}^{p}=-\frac{1}{2}\omega
c,$ $\frac{\partial\mathcal{K}}{\partial t}(1,1)=\|\Delta
u\|_{2}^{2}-\mu\gamma_{q}\|u\|_{q}^{q}-\gamma_{p}\|u\|_{p}^{p}=0,$

and

$\frac{\partial^{2}\mathcal{K}}{\partial t^{2}}(1,1)=\|\Delta
u\|_{2}^{2}-\mu\gamma_{q}(\frac{N(q-2)}{4}-1)\|u\|_{q}^{q}-\gamma_{p}(\frac{N(p-2)}{4}-1)\|u\|_{p}^{p}<0,$

which yields for $\delta_{t}$ small enough and $\delta_{\lambda}>0$,

$\mathcal{K}(1+\delta_{\lambda},1+\delta_{t})<\mathcal{K}(1,1)\quad\mathrm{for}\
\omega>0.$ (2.12)

In addition, we can also observe that

$\mathcal{M}(1,1)=Q_{p,q}(u)=\|\Delta
u\|_{2}^{2}-\mu\gamma_{q}\|u\|_{q}^{q}-\gamma_{p}\|u\|_{p}^{p}=0$

and

$\frac{\partial\mathcal{M}}{\partial t}(1,1)=2\|\Delta
u\|_{2}^{2}-\mu\gamma_{q}\frac{N(q-2)}{4}\|u\|_{q}^{q}-\gamma_{p}\frac{N(p-2)}{4}\|u\|_{p}^{p}\not=0.$

From the Implicit Function Theorem, we deduce that there exists $\epsilon>0$
and a continuous function $g:[1-\epsilon,\ 1+\epsilon]\mapsto\mathbb{R}$
satisfying $g(1)=1$ such that $\mathcal{M}(\lambda,g(\lambda))=0$ for
$\lambda\in[1-\epsilon,\ 1+\epsilon]$. Therefore, we inter from (2.12) that

$m_{p,q}((1+\epsilon)c)=\inf\limits_{u\in\mathcal{Q}_{p,q}((1+\epsilon)c)}E_{p,q}(u)\leq
E_{p,q}(u_{1+\epsilon,g(1+\epsilon)})<E(u)=m_{p,q}(c).$

This proves the last assertion of the lemma. ∎

##  3 Ground States In Sobolev Subcritical Case

In this section we will establish the existence of normalized ground states to
(1.3) in Sobolev subcritical case. Our prime aim is to prove the existence of
a Palais-Smale sequence $\\{u_{n}\\}\subset\mathcal{Q}_{p,q}(c)$ at level
$m_{p,q}(c)$ for $E_{p,q}$ restricted on $S(c)$.

First, let us introduce the functional
$\Psi(u):H^{2}(\mathbb{R}^{N})\backslash\\{0\\}\rightarrow\mathbb{R}$ defined
by

$\Psi(u):=E_{p,q}(u_{t_{u}})=\frac{1}{2}t_{u}^{2}\|\Delta
u\|_{2}^{2}-\frac{\mu}{q}t_{u}^{\frac{N(q-2)}{4}}\|u\|_{q}^{q}-\frac{1}{p}t_{u}^{\frac{N(p-2)}{4}}\|u\|_{p}^{p},$

where $t_{u}$ is the unique number determined in Lemma 2.3. Then we get

######  Lemma 3.1.

The functional
$\Psi(u):H^{2}(\mathbb{R}^{N})\backslash\\{0\\}\rightarrow\mathbb{R}$ is of
class $C^{1}$ and

$\displaystyle d\Psi(u)[\varphi]$
$\displaystyle=t_{u}^{2}\int_{\mathbb{R}^{N}}\Delta u\cdot\Delta\varphi dx-
t_{u}^{\frac{N(q-2)}{4}}\mu\int_{\mathbb{R}^{N}}|u|^{q-2}u\cdot\varphi dx-
t_{u}^{\frac{N(p-2)}{4}}\int_{\mathbb{R}^{N}}|u|^{p-2}u\cdot\varphi dx$
$\displaystyle=dE_{p,q}(u_{s_{u}})[\varphi_{s_{u}}]$

for any $u\in H^{2}(\mathbb{R}^{N})\backslash\\{0\\}$ and $\varphi\in
H^{2}(\mathbb{R}^{N})$.

###### Proof.

Let $u\in H^{2}(\mathbb{R}^{N})\backslash\\{0\\}$ and $\varphi\in
H^{2}(\mathbb{R}^{N})$. By the definition of $\Psi$, we have

$\Psi(u+s\varphi)-\Psi(u)=E_{p,q}((u+s\varphi)_{t_{s}})-E_{p,q}(u_{t_{0}}),$

where $|s|$ is small enough, $t_{s}=t_{u+s\varphi}$ and $t_{0}=t_{u}$ is the
unique maximum point of the functional $E_{p,q}(u)$. By the mean value theorem
we get

$\displaystyle E_{p,q}((u+s\varphi)_{t_{s}})-E_{p,q}(u_{t_{0}})$
$\displaystyle\leq E_{p,q}((u+s\varphi)_{t_{s}})-E_{p,q}(u_{t_{s}})$
$\displaystyle=\frac{t_{s}^{2}}{2}(\|\Delta(u+s\varphi)\|_{2}^{2}-\|\Delta
u\|_{2}^{2})-\frac{t_{s}^{\frac{N(p-2)}{4}}}{p}(\|u+s\varphi\|_{p}^{p}-\|u\|_{p}^{p})$
$\displaystyle-\mu\frac{t_{s}^{\frac{N(q-2)}{4}}}{q}(\|u+s\varphi\|_{q}^{q}-\|u\|_{q}^{q})$
$\displaystyle=\frac{t_{s}^{2}}{2}(\int_{\mathbb{R}^{N}}2s\Delta
u\cdot\Delta\varphi+s^{2}|\Delta\varphi|^{2}dx)$ $\displaystyle-\mu
t_{s}^{\frac{N(q-2)}{4}}\int_{\mathbb{R}^{N}}(\int_{0}^{1}|u+s\eta_{s}\varphi|^{q-2}(u+s\eta_{s}\varphi)s\varphi
d\eta_{s})dx$ $\displaystyle-
t_{s}^{\frac{N(p-2)}{4}}\int_{\mathbb{R}^{N}}(\int_{0}^{1}|u+s\eta_{s}\varphi|^{p-2}(u+s\eta_{s}\varphi)s\varphi
d\eta_{s})dx,$

where $\eta_{s}\in(0,1)$. Similarly,

$\displaystyle E_{p,q}((u+s\varphi)_{t_{s}})-E_{p,q}(u_{t_{0}})$
$\displaystyle\geq E_{p,q}((u+s\varphi)_{t_{0}})-E_{p,q}(u_{t_{0}})$
$\displaystyle=\frac{t_{0}^{2}}{2}(\int_{\mathbb{R}^{N}}2s\Delta
u\cdot\Delta\varphi+s^{2}|\Delta\varphi|^{2}dx)$ $\displaystyle-\mu
t_{0}^{\frac{N(q-2)}{4}}\int_{\mathbb{R}^{N}}(\int_{0}^{1}|u+s\tau_{s}\varphi|^{q-2}(u+s\tau_{s}\varphi)s\varphi
d\tau_{s})dx$ $\displaystyle-
t_{0}^{\frac{N(p-2)}{4}}\int_{\mathbb{R}^{N}}(\int_{0}^{1}|u+s\tau_{s}\varphi|^{p-2}(u+s\tau_{s}\varphi)s\varphi
d\tau_{s})dx,$

where $\tau_{s}\in(0,1)$. Since the map $u\mapsto t_{u}$ is of class $C^{1}$,
it follows from the two inequalities above that

$\lim\limits_{s\rightarrow
0}\frac{\Psi(u+s\varphi)-\Psi(u)}{s}=t_{u}^{2}\int_{\mathbb{R}^{N}}\Delta
u\cdot\Delta\varphi dx-
t_{u}^{\frac{N(q-2)}{4}}\mu\int_{\mathbb{R}^{N}}|u|^{q-2}u\cdot\varphi dx-
t_{u}^{\frac{N(p-2)}{4}}\mu\int_{\mathbb{R}^{N}}|u|^{p-2}u\cdot\varphi dx.$

Hence, we see that the G$\hat{a}$teaux derivative of $\Psi$ is bounded linear
in $\varphi$ and continuous in $u$. Therefore, $\Psi$ is of class $C^{1}$. In
particular, by changing variables in the integrals, we have

$\displaystyle d\Psi(u)[\varphi]$
$\displaystyle=t_{u}^{2}\int_{\mathbb{R}^{N}}\Delta u\cdot\Delta\varphi dx-
t_{u}^{\frac{N(q-2)}{4}}\mu\int_{\mathbb{R}^{N}}|u|^{q-2}u\cdot\varphi dx-
t_{u}^{\frac{N(p-2)}{4}}\int_{\mathbb{R}^{N}}|u|^{p-2}u\cdot\varphi dx$
$\displaystyle=\int_{\mathbb{R}^{N}}\Delta
u_{t_{u}}\cdot\Delta\varphi_{t_{u}}dx-\mu\int_{\mathbb{R}^{N}}|u_{t_{u}}|^{q-2}u_{t_{u}}\cdot\varphi_{t_{u}}dx-\int_{\mathbb{R}^{N}}|u_{t_{u}}|^{p-2}u_{t_{u}}\cdot\varphi_{t_{u}}dx.$

This completes the proof. ∎

For any given $c>0$ and $2+\frac{8}{N}\leq q<p<4^{*}$, If $q=2+\frac{8}{N}$,
we further assume that $\mu c^{\frac{4}{N}}<\frac{N+4}{NC^{q}_{N,q}}$. We
consider the constrained functional
$I:=\Psi|_{S(c)}:S(c)\rightarrow\mathbb{R}$.

######  Lemma 3.2.

The functional $I$ is of class $C^{1}$ and

$dI(u)[\varphi]=d\Psi(u)[\varphi]=dE_{p,q}(u_{t_{u}})[\varphi_{t_{u}}]$

for any $u\in S(c)$ and $\varphi\in T_{u}S(c)$.

Then we turn to search the existence of a Palais-Smale sequence
$\\{u_{n}\\}\subset\mathcal{Q}_{p,q}(c)$ at level $m_{p,q}(c)$ for $E_{p,q}$
restricted on $S(c)$. The following arguments are inspired from [1, 2, 3]. We
recall the below definition from [18] first.

######  Definition 3.3.

Let $B$ be a closed subset of a metric space $X$. We say that a class
$\mathcal{G}$ of compact subsets of $Y$ is a homotopy stable family with
closed boundary $B$ provided

$(i)$ every set in $\mathcal{G}$ contains $B$.

$(ii)$ for any $A\in\mathcal{G}$ and any $\eta\in C([0,1]\times Y,Y)$
satisfying $\eta(t,x)=x$ for all $(t,x)\in(\\{0\\}\times Y)\bigcup([0,1],B)$,
we have $\eta(\\{1\\}\times A)\in\mathcal{G}$.

We remark that $B=\emptyset$ is admissible.

######  Lemma 3.4.

Let $2+\frac{8}{N}\leq q<p<4^{*}$. If $q=2+\frac{8}{N}$, we further assume
that $\mu c^{\frac{4}{N}}<\frac{N+4}{NC^{q}_{N,q}}$. Let $\mathcal{G}$ be a
homotopy stable family of compact subsets of $S(c)$ (with $B=\emptyset$) and
set

$c_{\mathcal{G}}:=\inf\limits_{A\in\mathcal{G}}\max\limits_{u\in A}I(u).$

If $c_{\mathcal{G}}>0$, then there exists a Palais-Smale sequence
$\\{u_{n}\\}\subset\mathcal{Q}_{p,q}(c)$ for the constrained functional
$E_{p,q}|_{S(c)}$ at level $c_{\mathcal{G}}$.

###### Proof.

Let $\\{D_{n}\\}\subset\mathcal{G}$ be an arbitrary minimizing sequence of
$c_{\mathcal{G}}$. We define the mapping

$\eta:[0,1]\times S(c)\rightarrow S(c),\qquad\eta(s,u)=u_{1-s+st_{u}},$

which is continuous by Lemma 2.4. Since $t_{u}=1$ for any
$u\in\mathcal{Q}_{p,q}(c)$ and $\emptyset=B\subset\mathcal{Q}_{p,q}(c)$, we
have $\eta(s,u)=u$ for any $(s,u)\in(\\{0\\}\times S(c))$. Thus, by the
definition of $\mathcal{G}$, one has

$A_{n}:=\eta(1,D_{n})=\\{u_{t_{u}}:u\in D_{n}\\}\in\mathcal{G}.$ (3.1)

In particular, $A_{n}\subset\mathcal{G}$ for all $n\in\mathbb{N}^{+}$. Let
$v\in A_{n}$, $i.e.\ v=u_{t_{u}}$ for some $u\in D_{n}$ and
$I(v)=E_{p,q}(v_{t_{v}})=E_{p,q}(v)=E_{p,q}(u_{t_{u}})=I(u)$. It is clear that
$\max_{A_{n}}I=\max_{D_{n}}I\rightarrow c_{\mathcal{G}}$. Therefore
$\\{A_{n}\\}\subset\mathcal{Q}_{p,q}(c)$ is another minimizing sequence of
$c_{\mathcal{G}}$. Now, using the equivalent minimax principle [18], we obtain
a Palais-Smale sequence $\\{v_{n}\\}\subset S(c)$ for $I$ at the level
$c_{\mathcal{G}}$ such that
$dist_{H^{2}(\mathbb{R}^{N})}(v_{n},A_{n})\rightarrow 0$ as
$n\rightarrow\infty$. Let

$t_{n}=t_{v_{n}}\quad\mathrm{and}\quad
u_{n}=(v_{n})_{t_{n}}=(v_{n})_{t_{v_{n}}}.$

Then we claim that there exists a $C>0$ such that,

$t_{n}^{-2}\leq C$ (3.2)

for $n\in\mathbb{N}$ large enough. First, we notice that

$t_{n}^{-2}=\frac{\int_{H^{2}(\mathbb{R}^{N})}|\Delta
v_{n}|^{2}dx}{\int_{H^{2}(\mathbb{R}^{N})}|\Delta u_{n}|^{2}dx}.$

Since $E_{p,q}(u_{n})=I(v_{n})\rightarrow c_{\mathcal{G}}$, by Lemma 2.3 and
Lemma 2.5, we can deduce that $|\Delta u_{n}\|_{2}$ is bounded from below by a
positive constant. On the other hand, since
$\\{A_{n}\\}\subset\mathcal{Q}_{p,q}(c)$ is a minimizing sequence for
$c_{\mathcal{G}}$ and $E_{p,q}$ is coercive on $\mathcal{Q}_{p,q}(c)$, we
conclude that $\\{A_{n}\\}$ is uniformly bounded in $H^{2}(\mathbb{R}^{N})$
and thus from $dist_{H^{2}(\mathbb{R}^{N})}(v_{n},A_{n})\rightarrow 0$ as
$n\rightarrow\infty$, it follows that $\sup_{n}\|v_{n}\|<\infty$. Clearly,
this proves the claim.

Now, for $\\{u_{n}\\}\subset\mathcal{Q}_{p,q}(c)$, it follows that

$E_{p,q}(u_{n})=I(u_{n})=I(v_{n})\rightarrow c_{\mathcal{G}}.$

We then only need to show that $\\{u_{n}\\}$ is a Palais-Smale sequence for
$E_{p,q}$ on $S(c)$ at level $c_{\mathcal{G}}$. For any $\phi\in
T_{u_{n}}S(c)$, we have

$\int_{\mathbb{R}^{N}}v_{n}\phi_{t_{n}^{-1}}dx=\int_{\mathbb{R}^{N}}(v_{n})_{t_{n}}\phi
dx=\int_{\mathbb{R}^{N}}u_{n}\phi dx,$

which means $\phi_{t_{n}^{-1}}\in T_{v_{n}}S(c)$. Also, by the claim it
follows that

$\|\phi_{t_{n}^{-1}}\|_{H^{2}(\mathbb{R}^{N})}\leq\max\\{\sqrt{C},1\\}\|\phi\|_{H^{2}(\mathbb{R}^{N})}.$

Denoting by $\|\cdot\|_{\ast}$ the dual norm of $(T_{u}S(c))^{\ast}$ and using
Lemma 3.2, we deduce that

$\displaystyle\|dE_{p,q}(u_{n})\|_{\ast}$ $\displaystyle=\sup\limits_{\phi\in
T_{u_{n}}S(c),\|\phi\|_{H^{2}(\mathbb{R}^{N})}\leq 1}|dE_{p,q}(u_{n})[\phi]|$
$\displaystyle=\sup\limits_{\phi\in
T_{u_{n}}S(c),\|\phi\|_{H^{2}(\mathbb{R}^{N})}\leq
1}|dE_{p,q}((v_{n})_{t_{n}})[(\phi_{t_{n}^{-1}})_{t_{n}}]|$
$\displaystyle=\sup\limits_{\phi\in
T_{u_{n}}S(c),\|\phi\|_{H^{2}(\mathbb{R}^{N})}\leq
1}|dI(v_{n})[\phi_{t_{n}^{-1}}]|$
$\displaystyle\leq\|dI(v_{n})\|_{\ast}\cdot\sup\limits_{\phi\in
T_{u_{n}}S(c),\|\phi\|_{H^{2}(\mathbb{R}^{N})}\leq
1}\|\phi_{t_{n}^{-1}}\|_{H^{2}(\mathbb{R}^{N})}$
$\displaystyle\leq\max\\{\sqrt{C},1\\}\|dI(v_{n})\|_{\ast}.$

Since $v_{n}\subset S(c)$ is a Palais-Smale sequence of $I$, it follows that
$\|dI(v_{n})\|_{\ast}\rightarrow 0$ as $n\rightarrow\infty$. Hence, we deduce
that $\\{u_{n}\\}\subset\mathcal{Q}_{p,q}(c)$ is a Palais-Smale sequence for
$E_{p,q}$ on $S(c)$ at the level $c_{\mathcal{G}}$. ∎

Finally, we obtain the following lemma, which is crucial in the proof of
Theorem 1.2.

######  Lemma 3.5.

Let $2+\frac{8}{N}\leq q<p<4^{*}$, If $q=2+\frac{8}{N}$, we further assume
that $\mu c^{\frac{4}{N}}<\frac{N+4}{NC^{q}_{N,q}}$. There exists a Palais-
Smale sequence $\\{u_{n}\\}\subset\mathcal{Q}_{p,q}(c)$ at level $m_{p,q}(c)$
for $E_{p,q}$ restricted on $S(c)$.

###### Proof.

We use Lemma 3.4 in the particular case where $\mathcal{G}$ is the class of
all singletons belonging to $S(c)$ and then we denote it by
$\mathcal{\overline{G}}$. Since $m_{p,q}(c)>0$, we only need to prove that
$m_{p,q}(c)=c_{\mathcal{\overline{G}}}$. We notice that

$c_{\mathcal{\overline{G}}}=\inf\limits_{A\in\mathcal{\overline{G}}}\max\limits_{u\in
A}I(u)=\inf\limits_{u\in
S(c)}\sup\limits_{t>0}E_{p,q}(u_{t})=\inf\limits_{u\in
S(c)}E_{p,q}(u_{t_{u}}).$

For any $u\in S(c)$, we deduce from $u_{t_{u}}\in\mathcal{Q}_{p,q}(c)$ that
$E_{p,q}(u_{t_{u}})\geq m_{p,q}(c)$. Therefore $c_{\mathcal{\overline{G}}}\geq
m_{p,q}(c)$. On the other hand, for any $u\in\mathcal{Q}_{p,q}(c)$, we have
$t_{u}=1$ and thus $E_{p,q}(u)=E_{p,q}(u_{1})\geq c_{\mathcal{\overline{G}}}$.
This clearly implies that $m_{p,q}(c)\geq c_{\mathcal{\overline{G}}}$. The
proof is complete. ∎

######  Lemma 3.6.

Let $2+\frac{8}{N}\leq q<p<4^{*}$. If $q=2+\frac{8}{N}$, we further assume
that $\mu c^{\frac{4}{N}}<\frac{N+4}{NC^{q}_{N,q}}$. Let
$\\{u_{n}\\}\subset\mathcal{Q}_{p,q}(c)$ be a Palais-Smale sequence for
$E_{p,q}$ restricted on $S(c)$ at level $m_{p,q}(c)$. Then there exists $u\in
S(c)$ and $\omega>0$ such that, up to translation and up to the extraction of
a subsequence, $u_{n}\rightarrow u$ strongly in $H^{2}(\mathbb{R}^{N})$ and
$\Delta^{2}u+\omega u-\mu|u|^{q-2}u-|u|^{p-2}u=0$.

###### Proof.

By Lemma 2.5 $(iv)$, we can assume without loss of generality that
$\\{u_{n}\\}\subset\mathcal{Q}_{p,q}(c)$ is a bounded Palais-Smale sequence in
$H^{2}(\mathbb{R}^{N})$. After a suitable translation in $\mathbb{R}^{N}$ and
up to the extraction of a subsequence, we claim that $\\{u_{n}\\}$ is non-
vanishing. Indeed, if not, then applying Lemma 2.2, we infer that

$0=\lim\limits_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}|u_{n}|^{p}dx=\lim\limits_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}|u_{n}|^{q}dx.$

It follows that

$\|\Delta
u_{n}\|_{2}^{2}=\mu\gamma_{q}\|u_{n}\|_{q}^{q}+\gamma_{p}\|u_{n}\|_{p}^{p}\rightarrow
0\quad\mathrm{as}\quad n\rightarrow\infty,$

which contradicts the fact $\inf\limits_{u_{n}\in\mathcal{Q}_{p,q}(c)}\|\Delta
u_{n}\|_{2}^{2}>0$ in Lemma 2.5 $(ii)$. Since
$\\{u_{n}\\}\subset\mathcal{Q}_{p,q}(c)$ is bounded in
$H^{2}(\mathbb{R}^{N})$, from the condition that
$\|dE_{p,q}(u_{n})\|_{\ast}\rightarrow 0$ and Lemma $3$ in [6], it follows
that

$\Delta^{2}u_{n}+\omega_{n}u_{n}-\mu_{n}|u_{n}|^{q-2}u_{n}-|u_{n}|^{p-2}u_{n}\rightarrow
0\quad\mathrm{in}\quad(H^{2}(\mathbb{R}^{N}))^{*},$

where

$-c\omega_{n}=(\int_{\mathbb{R}^{N}}|\Delta
u_{n}|^{2}dx-\mu\int_{\mathbb{R}^{N}}|u_{n}|^{q}dx-\int_{\mathbb{R}^{N}}|u_{n}|^{p}dx).$

Noting that $\omega_{n}\rightarrow\omega$ for some $\omega\in\mathbb{R}$, we
have

$\Delta^{2}u_{n}(\cdot+y_{n})+\omega
u_{n}(\cdot+y_{n})-\mu_{n}|u_{n}(\cdot+y_{n})|^{q-2}u_{n}(\cdot+y_{n})-|u_{n}(\cdot+y_{n})|^{p-2}u_{n}(\cdot+y_{n})\rightarrow
0\quad\mathrm{in}\quad(H^{2}(\mathbb{R}^{N}))^{*}$ (3.3)

for some $\\{y_{n}\\}\subset\mathbb{R}^{N}$. Since $\\{u_{n}\\}$ is non-
vanishing, up to a subsequence, there exists
$\\{y_{n}^{1}\\}\subset\mathbb{R}^{N}$ and $z^{1}$ such that
$u_{n}(\cdot+y_{n}^{1})\rightharpoonup z^{1}$ in $H^{2}(\mathbb{R}^{N})$,
$u_{n}(\cdot+y_{n}^{1})\rightarrow z^{1}$ in $L_{loc}^{r}(\mathbb{R}^{N})$ for
all $r\in(2,4^{*})$ and $u_{n}(\cdot+y_{n}^{1})\rightarrow z^{1}$ almost
everywhere in $\mathbb{R}^{N}$, where $z^{1}\in B(c)=\\{u\in
H^{2}(\mathbb{R}^{N}),\|u\|_{2}^{2}\leq c\\}$. In view of (3.3), we obtain

$\Delta^{2}z^{1}+\omega z^{1}-\mu_{n}|z^{1}|^{q-2}z^{1}-|z^{1}|^{p-2}z^{1}=0.$
(3.4)

By the Nehari and Pohozaev identity corresponding to (3.4), we get
$Q_{p,q}(z^{1})=0$. Let $v_{n}^{1}:=u_{n}-z^{1}(\cdot-y_{n}^{1})$ for every
$n\in\mathbb{N}^{+}$. Clearly, $v_{n}^{1}(\cdot+y_{n})\rightharpoonup 0$ in
$H^{2}(\mathbb{R}^{N})$. Thus it follows that

$\|u_{n}\|_{2}^{2}=\|v_{n}^{1}(\cdot+y_{n})+z^{1}\|_{2}^{2}=\|v_{n}^{1}\|_{2}^{2}+\|z^{1}\|_{2}^{2}+o_{n}(1){\color[rgb]{0.7,0.04,0.8},}$
(3.5)

$\|\Delta u_{n}\|_{2}^{2}=\|\Delta v_{n}^{1}(\cdot+y_{n})+\Delta
z^{1}\|_{2}^{2}=\|\Delta v_{n}^{1}\|_{2}^{2}+\|\Delta
z^{1}\|_{2}^{2}+o_{n}(1).$ (3.6)

By Brezis-Lieb lemma, we get

$\|u_{n}\|_{p}^{p}=\|v_{n}^{1}\|_{p}^{p}+\|z^{1}\|_{p}^{p}+o_{n}(1)\quad\mathrm{and}\quad\|u_{n}\|_{q}^{q}=\|v_{n}^{1}\|_{q}^{q}+\|z^{1}\|_{q}^{q}+o_{n}(1).$

It follows that

$m_{p,q}(c)=\lim\limits_{n\rightarrow\infty}E_{p,q}(u_{n})=\lim\limits_{n\rightarrow\infty}E_{p,q}(u_{n}(\cdot+y_{n}))=\lim\limits_{n\rightarrow\infty}E_{p,q}(v_{n}^{1})+E_{p,q}(z^{1}).$
(3.7)

We claim that $\lim\limits_{n\rightarrow\infty}E_{p,q}(v_{n}^{1})\geq 0$.
Indeed, since $Q_{p,q}(u_{n})=0$ and $Q_{p,q}(z^{1})=0$, it then follows that
$\lim\limits_{n\rightarrow\infty}Q_{p,q}(v_{n}^{1})=0$. Then, for any
$2+\frac{8}{N}\leq q<p<4^{*}$, we have

$\lim\limits_{n\rightarrow\infty}E_{p,q}(v_{n}^{1})=\lim\limits_{n\rightarrow\infty}(E_{p,q}(v_{n}^{1})-\frac{1}{2}Q_{p,q}(v_{n}^{1}))=\lim\limits_{n\rightarrow\infty}\big{[}\mu(\frac{\gamma_{q}}{2}-\frac{1}{q})\|v_{n}^{1}\|_{q}^{q}+(\frac{\gamma_{p}}{2}-\frac{1}{p})\|v_{n}^{1}\|_{p}^{p}\big{]}\geq
0.$

Now we set $c_{1}:=\|z^{1}\|_{2}^{2}\in(0,c]$. Since
$\lim\limits_{n\rightarrow\infty}E_{p,q}(v_{n}^{1})\geq 0$ and
$z^{1}\in\mathcal{Q}_{p,q}(c_{1})$, it follows from (3.7) that

$m_{p,q}(c)=\lim\limits_{n\rightarrow\infty}E_{p,q}(v_{n}^{1})+E_{p,q}(z^{1})\geq
E_{p,q}(z^{1})\geq m_{p,q}(c_{1}).$

Recalling that $m_{p,q}(c)$ is non-increasing in $c>0$, then we have

$m_{p,q}(c)=E_{p,q}(z^{1})=m_{p,q}(c_{1})$ (3.8)

and

$\lim\limits_{n\rightarrow\infty}E_{p,q}(v_{n}^{1})=0.$ (3.9)

At last, we need to show that $c_{1}=c$. First similar in Lemma 2.10, by (3.4)
and the fact $Q_{p,q}(z^{1})=0$, we get $\omega>0$. If $c_{1}<c$, taking into
account (3.4), (3.8) and Lemma 2.10, we would have

$E_{p,q}(z^{1})=m_{p,q}(c_{1})>m_{p,q}(c),$

which contradicts (3.8). Therefore, $c_{1}=\|z^{1}\|_{2}^{2}=c$ and
$\|v_{n}^{1}\|_{2}^{2}\rightarrow 0$. By Lemma 2.2, we have

$0=\lim\limits_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}|v_{n}^{1}|^{p}dx=\lim\limits_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}|v_{n}^{1}|^{q}dx,$

it follows that

$\lim\limits_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}|\Delta
v_{n}^{1}|^{2}dx=0.$

Thus, $u_{n}(\cdot+y_{n}^{1})\rightarrow z^{1}$ strongly in
$H^{2}(\mathbb{R}^{N})$. At this point, this lemma is proved. ∎

######  Proof of Theorem 1.2 1.

By Lemma 2.5 and Lemma 3.5, we have a bounded Palais-Smale sequence
$\\{u_{n}\\}\subset\mathcal{Q}_{p,q}(c)$ for the constrained functional
$E_{p,q}|_{S(c)}$ at the level $m_{p,q}(c)>0$. And then Lemma 3.6 applies and
provides the existence of a ground state $u\in S(c)$ at the level
$m_{p,q}(c)$.

##  4 Ground States In Sobolev Critical Case

In this section we will establish the existence of normalized ground states in
Sobolev critical case. The main difficuly and challenge is to recover the
compactness. To do this, we need to give a upper bound for $m_{4^{*},q}(c)$.

######  Lemma 4.1.

Let $N\geq 5$, $c>0$, $\mu>0$ and $2+\frac{8}{N}\leq q<4^{*}$. If
$q=2+\frac{8}{N}$, we further assume that $\mu
c^{\frac{4}{N}}<\frac{N+4}{NC^{q}_{N,q}}$. If $N=5$, we further assume that
$q>\frac{22}{3}$. Then $m_{4^{*},q}(c)<\frac{2}{N}\mathcal{S}^{\frac{N}{4}}$,
where $\mathcal{S}$ is the best constant in the Sobolev embedding
$H^{2}(\mathbb{R}^{N})\hookrightarrow L^{4^{*}}(\mathbb{R}^{N})$.

###### Proof.

Define
$v_{\epsilon}(x)=\left(\sqrt[4]{c^{-1}\|u_{\epsilon}\|_{2}^{2}}\right)^{\frac{N-4}{2}}u_{\epsilon}(\sqrt[4]{c^{-1}\|u_{\epsilon}\|_{2}^{2}}x)$,
where $u_{\epsilon}$ is given by Appendix. Then

$\int_{\mathbb{R}^{N}}|v_{\epsilon}|^{2}dx=c,\int_{\mathbb{R}^{N}}|\Delta
v_{\epsilon}|^{2}dx=\int_{\mathbb{R}^{N}}|\Delta
u_{\epsilon}|^{2}dx,\int_{\mathbb{R}^{N}}|v_{\epsilon}|^{4^{*}}dx=\int_{\mathbb{R}^{N}}|u_{\epsilon}|^{4^{*}}dx,$

and for $q\in[2+\frac{8}{N},4^{*})$,

$\displaystyle\int_{\mathbb{R}^{N}}|v_{\epsilon}|^{q}dx$ $\displaystyle=$
$\displaystyle\left(\sqrt[4]{c^{-1}\|u_{\epsilon}\|_{2}^{2}}\right)^{\frac{N-4}{2}q-N}\int_{\mathbb{R}^{N}}|u_{\epsilon}|^{q}dx$
(4.7) $\displaystyle\geq$ $\displaystyle
K_{1}c^{\frac{N}{4}-\frac{N-4}{8}q}\left\\{\begin{array}[]{cc}1,&N>8,\\\
|\log\epsilon|^{\frac{q-4}{2}},&N=8,\\\
\epsilon^{\frac{8-N}{4}(N-\frac{N-4}{2}q)},&N=6,7,\\\
\epsilon^{\frac{3}{4}(N-\frac{N-4}{2}q)},&N=5,q>5,\\\
\epsilon^{\frac{15}{8}}|\log\epsilon|,&N=5,q=5,\\\
\epsilon^{\frac{5}{8}q-\frac{5}{4}},&N=5,\frac{18}{5}\leq
q<5,\end{array}\right.$

for some $K_{1}>0$ and $\epsilon$ small enough, where the last inequality
follows by Lemma 7.1. Next we use $v_{\epsilon}$ to estimate $m_{4^{*},q}(c)$.
By direct calculations, one has

$\displaystyle E_{4^{*},q}((v_{\epsilon})_{t_{\epsilon}})$ (4.14)
$\displaystyle=$ $\displaystyle\frac{1}{2}t_{\epsilon}^{2}\|\Delta
v_{\epsilon}\|_{2}^{2}-\frac{\mu}{q}t_{\epsilon}^{\frac{N(q-2)}{4}}\|v_{\epsilon}\|_{q}^{q}-\frac{1}{4^{*}}t_{\epsilon}^{\frac{N(4^{*}-2)}{4}}\|v_{\epsilon}\|_{4^{*}}^{4^{*}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}t_{\epsilon}^{2}\left(\mathcal{S}^{\frac{N}{4}}+O(\epsilon^{N-4})\right)-\frac{1}{4^{*}}t_{\epsilon}^{4^{*}}\left(\mathcal{S}^{\frac{N}{4}}+O(\epsilon^{N})\right)$
$\displaystyle-$
$\displaystyle\frac{\mu}{q}t_{\epsilon}^{\frac{N(q-2)}{4}}K_{1}c^{\frac{N}{4}-\frac{N-4}{8}q}\left\\{\begin{array}[]{cc}1,&N>8,\\\
|\log\epsilon|^{\frac{q-4}{2}},&N=8,\\\
\epsilon^{\frac{8-N}{4}(N-\frac{N-4}{2}q)},&N=6,7,\\\
\epsilon^{\frac{3}{4}(N-\frac{N-4}{2}q)},&N=5,q>5,\\\
\epsilon^{\frac{15}{8}}|\log\epsilon|,&N=5,q=5,\\\
\epsilon^{\frac{5}{8}q-\frac{5}{4}},&N=5,\frac{18}{5}\leq
q<5,\end{array}\right.$

where $t_{\epsilon}=t_{v_{\epsilon}}$ is given by Lemma 2.3 and
$(v_{\epsilon})_{t_{\epsilon}}\in\mathcal{Q}_{4^{*},q}(c)$. The definition of
$m_{4^{*},q}(c)$ yields that $m_{4^{*},q}(c)\leq
E_{4^{*},q}((v_{\epsilon})_{t_{\epsilon}})$.

We claim that there exist $t_{0},t_{1}>0$ independent of $\epsilon$ such that
$t_{\epsilon}\in[t_{0},t_{1}]$ for $\epsilon>0$ small. Suppose by
contradiction that $t_{\epsilon}\rightarrow 0$ or
$t_{\epsilon}\rightarrow\infty$ as $\epsilon\rightarrow 0$. The fact that
$(v_{\epsilon})_{t_{\epsilon}}\in\mathcal{Q}_{4^{*},q}(c)$ yields that

$\|\Delta
v_{\epsilon}\|_{2}^{2}=\mu\gamma_{q}t_{\epsilon}^{\frac{N(q-2)}{4}-2}\|v_{\epsilon}\|_{q}^{q}+\gamma_{4^{*}}t_{\epsilon}^{\frac{N(4^{*}-2)}{4}-2}\|v_{\epsilon}\|_{4^{*}}^{4^{*}}.$
(4.15)

When $q>2+\frac{8}{N}$, the left side of (4.15) is
$\mathcal{S}^{\frac{N}{4}}+O(\epsilon^{N-4})>0$ for small $\epsilon$ while the
right side of (4.15) tends to $0$ as $t_{\epsilon}\rightarrow 0$ and tends to
infinity as $t_{\epsilon}\rightarrow\infty$, which is impossible. When
$q=2+\frac{8}{N}$ and $t_{\epsilon}\rightarrow\infty$, we find a similar
contradiction since the right side of (4.15) tends to infinity. Furthermore,
when $q=2+\frac{8}{N}$, Gagliardo-Nirenberg inequality shows that

$\|v_{\epsilon}\|_{q}^{q}\leq C^{q}_{N,q}\|\Delta
v_{\epsilon}\|_{2}^{2}c^{\frac{4}{N}}.$ (4.16)

We further assume that $\mu c^{\frac{4}{N}}<\frac{N+4}{NC^{q}_{N,q}}$.
Therefore,

$\frac{N}{N+4}\mu\|v_{\epsilon}\|_{q}^{q}<\|\Delta
v_{\epsilon}\|_{2}^{2}=\mathcal{S}^{\frac{N}{4}}+O(\epsilon^{N-4}).$ (4.17)

The above inequality, together with (4.15), enables us to show that it is
impossible that $t_{\epsilon}\rightarrow 0$. Thus, the claim holds.

In (4.14), when $N=5$, $q>\frac{22}{3}$, or $N\geq 6$, $O(\epsilon^{N-4})$ can
be controlled by the last term for $\epsilon>0$ small enough. Hence,

$m_{4^{*},q}(c)\leq E_{4^{*},q}((v_{\epsilon})_{t_{\epsilon}})<\sup_{t\geq
0}\left(\frac{1}{2}t^{2}-\frac{1}{4^{*}}t^{4^{*}}\right)\mathcal{S}^{\frac{N}{4}}=\frac{2}{N}\mathcal{S}^{\frac{N}{4}}.$

The proof is complete. ∎

###### Proof of Theorem 1.3.

Similarly to the discussions in Section 3, we can obtain a bounded Palais-
Smale sequence (in $H^{2}(\mathbb{R}^{N})$)
$\\{u_{n}\\}\subset{\mathcal{Q}_{4^{*},q}(c)}$ for the constrained functional
$E_{4^{*},q}|_{S(c)}$ at the level $m_{4^{*},q}(c)$. By Lemma 4.1 and Lemma
2.5, $0<m_{4^{*},q}(c)<\frac{2}{N}\mathcal{S}^{\frac{N}{4}}$. After a suitable
translation in $\mathbb{R}^{N}$ and up to the extraction of a subsequence, we
claim that $\\{u_{n}\\}$ is non-vanishing. Indeed, if not, then applying Lemma
2.2, we infer that

$\lim\limits_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}|u_{n}|^{q}dx=0.$

Since $u_{n}$ is bounded in $H^{2}(\mathbb{R}^{N})$, passing to a subsequence
if necessary, we may assume that

$\lim_{n\rightarrow\infty}\|\Delta
u_{n}\|_{2}^{2}=\lim_{n\rightarrow\infty}\|u_{n}\|_{4^{*}}^{4^{*}}=m\in[0,\infty).$

By Lemma 2.5, $m>0$. Then, by the Sobolev inequality,
$\mathcal{S}m^{\frac{2}{4^{*}}}\leq m$, implying that
$m\geq\mathcal{S}^{\frac{4^{*}}{4^{*}-2}}=\mathcal{S}^{\frac{N}{4}}$.
Therefore, we derive a self-contradictory inequality

$\frac{2}{N}\mathcal{S}^{\frac{N}{4}}>m_{4^{*},q}(c)=\lim_{n\rightarrow\infty}E_{4^{*},q}(u_{n})=\frac{2}{N}m\geq\frac{2}{N}\mathcal{S}^{\frac{N}{4}}.$

Thus, the claim holds.

Up to a subsequence, there exists $\\{y_{n}^{1}\\}\subset\mathbb{R}^{N}$ and
$z^{1}$ such that $u_{n}(\cdot+y_{n}^{1})\rightharpoonup z^{1}\neq 0$ in
$H^{2}(\mathbb{R}^{N})$, $u_{n}(\cdot+y_{n}^{1})\rightarrow z^{1}$ in
$L_{loc}^{r}(\mathbb{R}^{N})$ for all $r\in(2,4^{*})$ and
$u_{n}(\cdot+y_{n}^{1})\rightarrow z^{1}$ almost everywhere in
$\mathbb{R}^{N}$, where $z^{1}\in B(c)=\\{u\in
H^{2}(\mathbb{R}^{N}),0<\|u\|_{2}^{2}\leq c\\}$. Let
$v_{n}^{1}:=u_{n}-z^{1}(\cdot-y_{n}^{1})$ for every $n\in\mathbb{N}^{+}$.
Mimicking the proof of Lemma 3.6, we can prove that $\|z^{1}\|_{2}^{2}=c$,
$Q_{4^{*},q}(v_{n}^{1})\rightarrow 0$, $E_{4^{*},q}(v_{n}^{1})\rightarrow 0$
and $\|v_{n}^{1}\|_{2}^{2}\rightarrow 0$. By Lemma 2.2, we have

$\lim\limits_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}|v_{n}^{1}|^{q}dx=0.$

Then,

$\|\Delta
v_{n}^{1}\|_{2}^{2}-\|v_{n}^{1}\|_{4^{*}}^{4^{*}}=Q_{4^{*},q}(v_{n}^{1})+\mu\gamma_{q}\|v_{n}^{1}\|_{q}^{q}\rightarrow
0,$ $\frac{1}{2}\|\Delta
v_{n}^{1}\|_{2}^{2}-\frac{1}{4^{*}}\|v_{n}^{1}\|_{4^{*}}^{4^{*}}=E_{4^{*},q}(v_{n}^{1})+\frac{\mu}{q}\gamma_{q}\|v_{n}^{1}\|_{q}^{q}\rightarrow
0,$

imply that $\|\Delta v_{n}^{1}\|_{2}^{2}\rightarrow
0,\|v_{n}^{1}\|_{4^{*}}^{4^{*}}\rightarrow 0$. Thus,
$u_{n}(\cdot+y_{n}^{1})\rightarrow z^{1}$ strongly in $H^{2}(\mathbb{R}^{N})$,
and we complete the proof. ∎

##  5 Non-existence result when $q=2+\frac{8}{N}$, $\mu
c^{\frac{4}{N}}\geq\frac{N+4}{NC^{q}_{N,q}}$

In this section, we write
$E_{p,q}^{\mu},Q_{p,q}^{\mu},\mathcal{Q}_{p,q}^{\mu}(c),m_{p,q}^{\mu}(c)$ in
place of $E_{p,q},Q_{p,q},\mathcal{Q}_{p,q}(c),m_{p,q}(c)$.

######  Lemma 5.1.

Let $N\geq 5$, $c>0$, $\mu>0$ and $2+\frac{8}{N}=q<p\leq 4^{*}$. We further
assume that $\mu c^{\frac{4}{N}}\geq\frac{N+4}{NC^{q}_{N,q}}$.

$(i)\ $ For any $u\in S(c)$ such that $\frac{\|\Delta
u\|_{2}^{2}}{\|u\|_{q}^{q}}>\frac{N}{N+4}\mu$, there exists a unique
$t_{u}\in(0,+\infty)$ for which $u_{t_{u}}\in\mathcal{Q}_{p,q}^{\mu}(c)$.
Moreover, $t_{u}$ is the unique critical point of $E_{p,q}^{\mu}(u_{t})$ such
that $E_{p,q}^{\mu}(u_{t_{u}})=\max_{t\in(0,+\infty)}E_{p,q}^{\mu}(u_{t}).$

$(ii)\ $ For any $u\in S(c)$ such that $\frac{\|\Delta
u\|_{2}^{2}}{\|u\|_{q}^{q}}\leq\frac{N}{N+4}\mu$,
$u_{t}\notin\mathcal{Q}_{p,q}^{\mu}(c),\forall t\in(0,\infty)$. Moreover,
$\mathcal{Q}_{p,q}^{\mu}(c)=\\{u_{t_{u}}:u\in S(c),\frac{\|\Delta
u\|_{2}^{2}}{\|u\|_{q}^{q}}>\frac{N}{N+4}\mu\\}.$

$(iii)\ \mathcal{Q}_{p,q}^{\mu}(c)\not=\emptyset$ if $5\leq N\leq 8$.

$(iv)\ $ If $\mathcal{Q}_{p,q}^{\mu}(c)\not=\emptyset$, then
$m_{p,q}^{\mu}(c)=0$.

###### Proof.

First we let

$Q_{p,q}^{\mu}(u_{t})=t^{2}\left(\|\Delta
u\|_{2}^{2}-\frac{N}{N+4}\mu\|u\|_{q}^{q}-\gamma_{p}t^{\frac{N(p-2)}{4}-2}\|u\|_{p}^{p}\right).$

When $\frac{\|\Delta u\|_{2}^{2}}{\|u\|_{q}^{q}}>\frac{N}{N+4}\mu$, we
complete the proof of $(i)$ similar to the one of Lemma 2.3. When
$\frac{\|\Delta u\|_{2}^{2}}{\|u\|_{q}^{q}}\leq\frac{N}{N+4}\mu$, it is easy
to see that $Q_{p,q}^{\mu}(u_{t})<0$ for all $t>0$, and we complete the proof
of $(ii)$.

When $5\leq N\leq 8$, for all $\mu>0$, we can always choose $u\in S(c)$ such
that $\frac{\|\Delta u\|_{2}^{2}}{\|u\|_{q}^{q}}>\frac{N}{N+4}\mu$. Indeed, we
consider $v_{\epsilon}$ given in the proof of Lemma 4.1. The proof of Lemma
4.1 shows that $\|v_{\epsilon}\|_{2}^{2}=c$,
$\|v_{\epsilon}\|_{q}^{q}\rightarrow 0$ if $5\leq N\leq 8$ and $\|\Delta
v_{\epsilon}\|_{2}^{2}\rightarrow\mathcal{S}^{\frac{N}{4}}$ as
$\epsilon\rightarrow 0$. Thus $v_{\epsilon}$ satisfies our condition for
$\epsilon$ small enough. Then we know that
$\mathcal{Q}_{p,q}^{\mu}(c)\not=\emptyset$.

Finally, for $\mu>0$ such that $\mathcal{Q}_{p,q}^{\mu}(c)\not=\emptyset$, we
can choose $u\in S(c)$ such that $\frac{\|\Delta
u\|_{2}^{2}}{\|u\|_{q}^{q}}>\frac{N}{N+4}\mu$. Noticing that

$E_{p,q}^{\mu}(u)=(\frac{\gamma_{p}}{2}-\frac{1}{p})\|u\|_{p}^{p},\forall
u\in\mathcal{Q}_{p,q}^{\mu}(c),$ (5.1)

we obtain that $m_{p,q}^{\mu}(c)\geq 0$ immediately. We will show that
$m_{p,q}^{\mu}(c)\leq 0$ to complete the proof.

Claim 1: There exists $\\{w_{n}\\}\subset S(c)$ such that $w_{n}\rightarrow
w_{\infty}$ in $H^{2}$, $\frac{\|\Delta
w_{n}\|_{2}^{2}}{\|w_{n}\|_{q}^{q}}>\frac{N}{N+4}\mu$ and $\frac{\|\Delta
w_{\infty}\|_{2}^{2}}{\|w_{\infty}\|_{q}^{q}}=\frac{N}{N+4}\mu$.

By the proof of $(iii)$, we can take $\phi_{1}\in S(c)$ such that
$\frac{\|\Delta\phi_{1}\|_{2}^{2}}{\|\phi_{1}\|_{q}^{q}}>\frac{N}{N+4}\mu$.
Furthermore, we can take a minimizer $\phi_{2}$ of the Gagliardo-Nirenberg
inequality (2.1) such that $\|\phi_{2}\|_{2}^{2}=c$. Since $\mu
c^{\frac{4}{N}}\geq\frac{N+4}{NC^{q}_{N,q}}$, we have

$\|\Delta\phi_{2}\|_{2}^{2}=\frac{1}{C_{N,q}^{q}c^{\frac{4}{N}}}\|\phi_{2}\|_{q}^{q}\leq\frac{N}{N+4}\mu\|\phi_{2}\|_{q}^{q}.$

Obviously, $\phi_{1}\neq-\phi_{2}$. Hence, we can consider

$\rho_{\tau}=\sqrt{c}\frac{\tau\phi_{1}+(1-\tau)\phi_{2}}{\|\tau\phi_{1}+(1-\tau)\phi_{2}\|_{2}}\in
S_{c},\tau\in[0,1].$

By the continuity of $\rho_{\tau}$ in $H^{2}$, there exists $\tau_{0}\in[0,1)$
such that
$\|\Delta\rho_{\tau_{0}}\|_{2}^{2}=\frac{N}{N+4}\mu\|\rho_{\tau_{0}}\|_{q}^{q}$,
$\|\Delta\rho_{\tau}\|_{2}^{2}>\frac{N}{N+4}\mu\|\rho_{\tau}\|_{q}^{q}$ for
all $\tau>\tau_{0}$. Take $w_{\infty}=\rho_{\tau_{0}}$,
$w_{n}=\rho_{\tau_{n}}$ with $\tau_{n}>\tau_{0}$,
$\tau_{n}\rightarrow\tau_{0}$. Thus the Claim 1 is proved.

Claim 2: $t_{n}=t_{w_{n}}\rightarrow 0$ where $t_{w_{n}}$ is given by $(i)$.

By the proof of $(i)$, we have

$t_{n}^{\frac{N(p-2)}{4}-2}=\frac{\|\Delta
w_{n}\|_{2}^{2}-\frac{N}{N+4}\mu\|w_{n}\|_{q}^{q}}{\gamma_{p}\|w_{n}\|_{p}^{p}}\rightarrow\frac{\|\Delta
w_{\infty}\|_{2}^{2}-\frac{N}{N+4}\mu\|w_{\infty}\|_{q}^{q}}{\gamma_{p}\|w_{\infty}\|_{p}^{p}}=0.$

Since $\frac{N(p-2)}{4}-2>0$, we complete the proof of Claim 2.

Claims 1, 2 yield that $E_{p,q}^{\mu}((w_{n})_{t_{n}})\rightarrow 0$. Noticing
that $(w_{n})_{t_{n}}\in\mathcal{Q}_{p,q}^{\mu}(c)$, we get

$m_{p,q}^{\mu}(c)=\inf_{u\in\mathcal{Q}_{p,q}^{\mu}(c)}E_{p,q}^{\mu}(u)\leq\lim_{n\rightarrow\infty}E_{p,q}^{\mu}((w_{n})_{t_{n}})=0.$

The proof is complete. ∎

###### Proof of Theorem 1.4.

Suppose the contrary that for $\mu
c^{\frac{4}{N}}\geq\frac{N+4}{NC^{q}_{N,q}}$, $m_{p,q}^{\mu}(c)$ is attained
by some $u_{\mu,c}$. By Lemma 5.1 $(ii)$, either
$u_{\mu,c}\in\mathcal{Q}_{p,q}^{\mu}(c)$ or $\|\Delta
u_{\mu,c}\|_{2}^{2}=\frac{N}{N+4}\mu\|u_{\mu,c}\|_{q}^{q}$,
$\|u_{\mu,c}\|_{2}^{2}=c$. If $u_{\mu,c}\in\mathcal{Q}_{p,q}^{\mu}(c)$, then

$0=m_{p,q}^{\mu}(c)=E_{p,q}^{\mu}(u_{\mu,c})=(\frac{\gamma_{p}}{q\gamma_{q}}-\frac{1}{p})\|u_{\mu,c}\|_{p}^{p},$

in a contradiction with $u_{\mu,c}\in S_{c}$. If $\|\Delta
u_{\mu,c}\|_{2}^{2}=\frac{N}{N+4}\mu\|u_{\mu,c}\|_{q}^{q}$, then

$0=m_{p,q}^{\mu}(c)=E_{p,q}^{\mu}(u_{\mu,c})=-\frac{1}{p}\|u_{p,q}\|_{p}^{p},$

which is impossible since $\|u_{\mu,c}\|_{2}^{2}=c$. ∎

##  6 Instability

In this section, we consider the instability of standing waves associated with
the ground states. We will consider our problem in
$H^{2}_{rad}(\mathbb{R}^{N})$. In fact, we need to use the local virial
identity to prove the instability by blowup and it is not clear how to control
errors term without radial symmetry.

  * Remark:

For now, we haven’t removed the radial symmetry assumption. The strategy here
was introduced in [36], where Ogawa and Tsutsumi showed blowup for radial
solutions for mass supercritical NLS with radial $u_{0}\in
H^{1}(\mathbb{R}^{N})$ with infinite variance ($i.e.$ we may not have
$xu_{0}\not\in L^{2}(\mathbb{R}^{N})$). We requires a careful analysis of the
time evolution for the localized virial identity $M_{\varphi_{R}}[u(t)]$,
which is essential in the proof of blowup result.

Firstly, from [37] we recall the local well-posedness of the Cauchy problem
(1.2) holds for $2+\frac{8}{N}\leq q<p\leq 4^{*}$.

######  Proposition 6.1.

Let $2+\frac{8}{N}\leq q<p\leq 4^{*}$. For any $u_{0}\in
H^{2}(\mathbb{R}^{N})$, there exists $T>0$ and a unique solution $u(t)\in
C([0,T);H^{2}(\mathbb{R}^{N}))$ to (1.2), which satisfies the conservation
laws along the time, that is, for any $t\in[0,T)$

$\|u(t)\|_{2}=\|u_{0}\|_{2},\ E_{p,q}(u(t))=E_{p,q}(u_{0}).$

Moreover, for the Sobolev subcritical case, we have the following blowup
alternative: either $T=\infty$, or $\lim_{t\rightarrow T}\|\Delta
u(t)\|_{2}=\infty$. For the Sobolev critical case $p=4^{*}$, we have a blowup
alternative that involves a critical Strichartz norm in space time, $i.e.$ it
holds that $0<T<\infty$ and

$\int_{0}^{T}\int_{\mathbb{R}^{N}}|u(t,x)|^{\frac{2(N+4)}{N-4}}dxdt=+\infty.$
(6.1)

Next we recall the localized virial to (1.2) introduced in [12], which will
play an important role to deduce the occurrence of blowup. Let
$\varphi:\mathbb{R}^{N}\rightarrow\mathbb{R}$ be a radial function such that
$D^{i}(\varphi)\in L^{\infty}(\mathbb{R}^{N})$, $1\leq i\leq 6$,

$\varphi:=\begin{cases}\frac{r^{2}}{2},\qquad\qquad r\leq 1\\\
const\qquad\quad r\geq 10,\\\ \end{cases}$

and $\varphi^{\prime\prime}(r)\leq 1$ for all $r\geq 0$. We set
$\varphi_{R}(r)=R^{2}\varphi(\frac{r}{R})$ for $R>0$. Then we record some
properties of $\varphi_{R}$ which will be often used in the following.

######  Lemma 6.2.

For all $r\geq 0$, we have

$(i)\ 1-\varphi^{\prime\prime}_{R}(r)\geq 0,\quad
1-\frac{\varphi^{\prime}_{R}(r)}{r}\geq 0,$

$(ii)\ N-\Delta\varphi_{R}(r)\geq 0.$

###### Proof.

$(i)\ $ By the definition of $\varphi_{R}$, we have
$\varphi_{R}^{\prime}(r)=R\varphi^{\prime}(\frac{r}{R})$ and
$\varphi_{R}^{\prime\prime}(r)=\varphi^{\prime\prime}(\frac{r}{R})$. Since
$\varphi^{\prime\prime}(r)\leq 1$ for $r\geq 0$, we get
$\varphi_{R}^{\prime\prime}(r)\leq 1$. Then integrating differential
inequality $\varphi_{R}^{\prime\prime}\leq 1$ on $[0,r]$, we have
$\int_{0}^{r}\varphi_{R}^{\prime\prime}(s)ds\leq r$. It follows that
$\varphi_{R}^{\prime}(r)\leq r$, then we get our conclusion.

$(ii)\ $ By simple calculations, we can get
$\Delta\varphi_{R}(r)=\varphi_{R}^{\prime\prime}(r)+\varphi_{R}^{\prime}(r)\frac{N-1}{r}$.
Hence, according to $(i)$, it follows that

$\displaystyle N-\Delta\varphi_{R}(r)$
$\displaystyle=N-\varphi_{R}^{\prime\prime}(r)-\varphi_{R}^{\prime}(r)\frac{N-1}{r}$
$\displaystyle=1-\varphi_{R}^{\prime\prime}(r)+N-1-\varphi_{R}^{\prime}(r)\frac{N-1}{r}$
$\displaystyle=1-\varphi_{R}^{\prime\prime}(r)+(N-1)(1-\frac{\varphi_{R}^{\prime}(r)}{r})\geq
0$

∎

######  Lemma 6.3.

We have

$(i)$

$\nabla\varphi_{R}(r)=R\varphi^{\prime}_{R}(\frac{r}{R})\frac{x}{|x|}=\begin{cases}x,\qquad
r\leq R\\\ 0,\qquad r\geq 10R,\\\ \end{cases}$

$(ii)$

$\|\nabla^{j}\varphi_{R}\|_{L^{\infty}}\leq R^{2-j},\quad\mathrm{for}\ 0\leq
j\leq 6,$

$(iii)$

$\mathrm{supp}(\nabla^{j}\varphi_{R})\subset\begin{cases}\\{|x|\leq
10R\\},\qquad\qquad\mathrm{for}\ j=1,2\\\ \\{R\leq|x|\leq
10R\\},\qquad\mathrm{for}\ 3\leq j\leq 6.\\\ \end{cases}$

###### Proof.

These properties can be checked easily from the definition of $\varphi$. ∎

For $u\in H^{2}(\mathbb{R}^{N})$, we define the localized virial
$M_{\varphi_{R}}$ of $u$ by

$M_{\varphi_{R}}[u]:=\big{<}u,-i(\nabla\varphi_{R}\cdot\nabla+\nabla\cdot\nabla\varphi_{R})u\big{>}=2Im\int_{\mathbb{R}^{N}}\overline{u}\nabla\varphi_{R}\cdot\nabla
udx.$ (6.2)

By Cauchy-Schwartz inequality and Hölder inequality, we have
$|M_{\varphi_{R}}[u]|\lesssim R\|u\|^{\frac{3}{2}}_{2}\|\Delta
u\|^{\frac{1}{2}}_{2}$. In particular, the localized virial is well-defined in
$H^{2}(\mathbb{R}^{N})$.

The following lemma reveals a key point on the evolution of this quantity.

######  Lemma 6.4.

Let $2+\frac{8}{N}\leq q<p\leq 4^{*}$ and $R>0$. Suppose that $u\in
C([0,T),H^{2}_{rad}(\mathbb{R}^{N}))$ is a solution to (1.2). Then for any
$t\in[0,T)$, we have the following inequality:

$\displaystyle\frac{d}{dt}M_{\varphi_{R}}[u(t)]$ $\displaystyle\leq
2N(q-2)E_{p,q}(u)-(N(q-2)-8)\int_{\mathbb{R}^{N}}|\Delta
u(t)|^{2}-\frac{2N(p-q)}{p}\int_{\mathbb{R}^{N}}|u(t)|^{p}$
$\displaystyle+O(R^{-4}+R^{-2}\|\Delta
u(t)\|_{2}+R^{-\frac{(q-2)(N-1)}{2}}\|\Delta
u\|_{2}^{\frac{q-2}{4}}+R^{-\frac{(p-2)(N-1)}{2}}\|\Delta
u\|_{2}^{\frac{p-2}{4}})$
$\displaystyle=8Q_{p,q}(u(t))+O(R^{-4}+R^{-2}\|\Delta
u(t)\|_{2}+R^{-\frac{(q-2)(N-1)}{2}}\|\Delta
u\|_{2}^{\frac{q-2}{4}}+R^{-\frac{(p-2)(N-1)}{2}}\|\Delta
u\|_{2}^{\frac{p-2}{4}}).$

###### Proof.

First, we rewrite $M_{\varphi_{R}}[u]$ as

$M_{\varphi_{R}}[u]=\big{<}u(t),\Gamma_{\varphi_{R}}u(t)\big{>}\quad\mathrm{with}\
\Gamma_{\varphi_{R}}=-i(\nabla\varphi_{R}\cdot\nabla+\nabla\cdot\nabla\varphi_{R}).$

Recall that if u satisfies the equation $i\partial_{t}u=Hu$, then

$\frac{d}{dt}\big{<}u,Au\big{>}=\big{<}u,[H,iA]u\big{>},$

where $[H,A]=HA-AH$ is a commutator operator. Then by taking the time
derivative and using that $i\partial_{t}u$ is given in (1.2), we get that

$\frac{d}{dt}M_{\varphi_{R}}[u(t)]=A_{R}^{1}[u(t)]+B_{R}^{2}[u(t)]+B_{R}^{3}[u(t)]$
(6.3)

with

$A_{R}^{1}[u]=:\big{<}u(t),[\Delta^{2}u+i\Gamma_{\varphi_{R}}]u(t)\big{>},$
$B_{R}^{2}[u]=:\big{<}u(t),[-\mu|u|^{q-2}u+i\Gamma_{\varphi_{R}}]u(t)\big{>}\quad\mathrm{and}\quad
B_{R}^{3}[u]=:\big{<}u(t),[-|u|^{p-2}u+i\Gamma_{\varphi_{R}}]u(t)\big{>}.$

For convenience, we state some identities to be used in the next. By using the
fact that $\Delta
A+A\Delta=2\partial_{k}A\partial_{k}+[\partial_{k},[\partial_{k},A]]$, we
observe that

$[\Delta^{2},i\Gamma_{\varphi_{R}}]=\Delta[\Delta,i\Gamma_{\varphi_{R}}]+[\Delta,i\Gamma_{\varphi_{R}}]\Delta=2\partial_{k}[\Delta,i\Gamma_{\varphi_{R}}]\partial_{k}+[\partial_{k},[\partial_{k},[\Delta,i\Gamma_{\varphi_{R}}]]].$
(6.4)

Then by simple calculations, it follows that

$[\Delta,i\Gamma_{\varphi_{R}}]=[\Delta,\nabla\varphi_{R}\cdot\nabla+\nabla\cdot\nabla\varphi_{R}]=4\partial_{k}(\partial^{2}_{kl}\varphi_{R})\partial_{l}+\Delta^{2}\varphi_{R}.$
(6.5)

Combining (6.4) and (6.5), we obtain the identity

$[\Delta^{2},i\Gamma_{\varphi_{R}}]=8\partial^{2}_{kl}(\partial^{2}_{lm}\varphi_{R})\partial^{2}_{mk}+4\partial_{k}(\partial^{2}_{kl}\Delta\varphi_{R})\partial_{l}+2\partial_{k}(\Delta^{2}\varphi_{R})\partial_{k}+\Delta^{3}\varphi_{R}.$
(6.6)

We recall that the Hessian of sufficiently regular and radial function
$f:\mathbb{R}^{N}\rightarrow\mathbb{C}$ is given by

$\partial^{2}_{kl}f=(\delta_{kl}-\frac{x_{k}x_{l}}{r^{2}})\frac{\partial_{r}f}{r}+\frac{x_{k}x_{l}}{r^{2}}\partial^{2}_{r}f.$
(6.7)

Applying (6.7) to $\varphi_{R}(r)$ and $u(t,r)$, a calculation combined with
integration by parts yields that

$\displaystyle
8\big{<}u,8\partial^{2}_{kl}(\partial^{2}_{lm}\varphi_{R})\partial^{2}_{mk}u\big{>}$
$\displaystyle=8\int_{\mathbb{R}^{N}}(\partial^{2}_{kl}\overline{u})(\partial^{2}_{lm}\varphi_{R})(\partial^{2}_{mk}u)$
$\displaystyle=8\int_{\mathbb{R}^{N}}(\partial_{r}^{2}\varphi_{R}|\partial_{r}^{2}u|^{2}+\frac{d-1}{r^{2}}\frac{\partial_{r}\varphi_{R}}{r}|\partial_{r}u|^{2})$
$\displaystyle=8\int_{\mathbb{R}^{N}}|\Delta
u|^{2}-(1-\partial_{r}^{2}\varphi_{R})|\partial_{r}^{2}u|^{2}-(1-\frac{\partial_{r}\varphi_{R}}{r})\frac{d-1}{r^{2}}|\partial_{r}u|^{2}.$

Noticing that $\int_{\mathbb{R}^{N}}|\Delta
u|^{2}=\int_{\mathbb{R}^{N}}|\partial_{r}^{2}u|^{2}+\frac{d-1}{r^{2}}|\partial_{r}u|^{2}$
for radial $u\in H^{2}(\mathbb{R}^{N})$, according to the inequalities in
Lemma 6.2, we have

$8\big{<}u,8\partial^{2}_{kl}(\partial^{2}_{lm}\varphi_{R})\partial^{2}_{mk}u\big{>}\leq
8\int_{\mathbb{R}^{N}}|\Delta u|^{2}.$ (6.8)

For the rest terms in (6.6), direct arguments yield that

$\displaystyle\begin{array}[]{cc}&|\big{<}u,\partial_{k}(\partial^{2}_{kl}\Delta\varphi_{R})\partial_{l}u\big{>}|\lesssim\|\partial^{2}_{kl}\Delta\varphi_{R}\|_{L^{\infty}}\|\nabla
u\|^{2}_{2}\lesssim R^{-2}\|\nabla u\|^{2}_{2}\lesssim R^{-2}\|u\|_{2}\|\Delta
u\|_{2},\\\
&|\big{<}u,\partial_{k}(\Delta^{2}\varphi_{R})\partial_{k}u\big{>}|\lesssim\|\Delta^{2}\varphi_{R}\|_{L^{\infty}}\|\nabla
u\|^{2}_{2}\lesssim R^{-2}\|\nabla u\|^{2}_{2}\lesssim R^{-2}\|u\|_{2}\|\Delta
u\|_{2},\\\
&|\big{<}u,\Delta^{3}\varphi_{R}u\big{>}||\lesssim\|\Delta^{3}\varphi_{R}\|_{L^{\infty}}\|u\|^{2}_{2}\lesssim
R^{-4}\|u\|^{2}_{2}.\\\ \end{array}$ (6.12)

By (6.8) and (6.12), we conclude that

$A_{R}^{1}[u(t)]\leq 8\int_{\mathbb{R}^{N}}|\Delta
u(t)|^{2}+O(R^{-4}+R^{-2}\|\Delta u(t)\|_{2}).$ (6.13)

Next, we turn to the combined nonlinearity. By using the identity
$\nabla(|u|^{q})=\frac{q}{q-2}\nabla(|u|^{q-2})|u|^{2}$, we note that
integration by parts yields that

$\displaystyle B_{R}^{2}[u]$
$\displaystyle=:-\big{<}u,[\mu|u|^{q-2}u,\nabla\varphi_{R}\cdot\nabla+\nabla\cdot\nabla\varphi_{R}]u\big{>}=2\mu\int_{\mathbb{R}^{N}}|u|^{2}\nabla\varphi_{R}\nabla(|u|^{q-2})$
$\displaystyle=-\mu\frac{2(q-2)}{q}\int_{\mathbb{R}^{N}}\Delta\varphi_{R}|u|^{q}.$

By the definition of $\varphi_{R}(r)$, we get that
$\Delta\varphi_{R}(r)-N\equiv 0$ for $r\leq R$, it follows that

$\displaystyle B_{R}^{2}[u]$
$\displaystyle=\mu\frac{-2N(q-2)}{q}\int_{\mathbb{R}^{N}}|u|^{q}-\mu\frac{2(q-2)}{q}\int_{|x|\geq
R}(\Delta\varphi_{R}-N)|u|^{q}$
$\displaystyle=\mu\frac{-2N(q-2)}{q}\int_{\mathbb{R}^{N}}|u|^{q}+O(R^{-\frac{(q-2)(N-1)}{2}}\|\Delta
u\|_{2}^{\frac{q-2}{4}}),$

where the last term follows from the fact
$\|\Delta\varphi_{R}(r)-N\|_{L^{\infty}}\lesssim 1$ and applying the Strauss
inequality, which gives us

$\int_{|x|\geq R}|u|^{q}\lesssim\|u\|_{2}^{2}\|u\|_{L^{\infty}}^{q-2}\lesssim
R^{-\frac{(q-2)(N-1)}{2}}\|u\|_{2}^{\frac{q+2}{2}}\|\nabla
u\|_{2}^{\frac{q-2}{2}}\lesssim
R^{-\frac{(q-2)(N-1)}{2}}\|u\|_{2}^{\frac{3q+2}{4}}\|\Delta
u\|_{2}^{\frac{q-2}{4}}.$

Similarly, we can also obtain that

$\displaystyle B_{R}^{3}[u]$
$\displaystyle=\frac{-2N(p-2)}{p}\int_{\mathbb{R}^{N}}|u|^{p}-\frac{2(p-2)}{p}\int_{|x|\geq
R}(\Delta\varphi_{R}-N)|u|^{p}$
$\displaystyle=\frac{-2N(p-2)}{p}\int_{\mathbb{R}^{N}}|u|^{p}+O(R^{-\frac{(p-2)(N-1)}{2}}\|\Delta
u\|_{2}^{\frac{p-2}{4}}).$

Finally, from the above, we conclude that

$\displaystyle\frac{d}{dt}M_{\varphi_{R}}[u(t)]$ $\displaystyle\leq
2N(q-2)E_{p,q}(u)-(N(q-2)-8)\int_{\mathbb{R}^{N}}|\Delta
u(t)|^{2}-\frac{2N(p-q)}{p}\int_{\mathbb{R}^{N}}|u(t)|^{p}$
$\displaystyle+O(R^{-4}+R^{-2}\|\Delta
u(t)\|_{2}+R^{-\frac{(q-2)(N-1)}{2}}\|\Delta
u\|_{2}^{\frac{q-2}{4}}+R^{-\frac{(p-2)(N-1)}{2}}\|\Delta
u\|_{2}^{\frac{p-2}{4}})$
$\displaystyle=8Q_{p,q}(u(t))+O(R^{-4}+R^{-2}\|\Delta
u(t)\|_{2}+R^{-\frac{(q-2)(N-1)}{2}}\|\Delta
u\|_{2}^{\frac{q-2}{4}}+R^{-\frac{(p-2)(N-1)}{2}}\|\Delta
u\|_{2}^{\frac{p-2}{4}}).$

This completes the proof. ∎

###  6.1 Sobolev Subcritical case: $2+\frac{8}{N}\leq q<p<4^{*}$

###### Proof of Theorem 1.6.

By Theorem 1.2, let $u\in S(c)$ be a ground state solution to (1.3) at the
level $m_{p,q}(c)>0$. Since we work in the space
$H^{2}_{rad}(\mathbb{R}^{N})$, then $u$ is radially symmetric. By Lemma 2.3,
we can have $\|u_{s}-u\|_{H^{2}}\rightarrow 0$ and $E_{p,q}(u_{s})\rightarrow
m_{p,q}(c)$ as $s\rightarrow 1^{+}$. Then we set $\psi_{0}=u_{s}$. By
Proposition 6.1, we know that there exists a unique solution $\psi(t)\in
C([0,T);H_{rad}^{2}(\mathbb{R}^{N}))$ to (1.2) with the initial datum
$\psi_{0}=u_{s}$, where $T$ is the maximal time of existence. In view of
Definition 1.5, to prove that $u$ is strongly unstable, we need to show that
$\psi(t)$ blows up in a finite time. We divide the proof into five steps:

Step $1$: $\mathcal{M}_{p,q}(c)$ is nonempty, where $\mathcal{M}_{p,q}(c)$ is
defined as follows:

$\mathcal{M}_{p,q}(c)=\\{v\in S(c),E_{p,q}(v)<E_{p,q}(u),Q_{p,q}(v)<0\\}.$
(6.14)

By Lemma 2.3 and the conservation laws, it follows that
$\psi_{0}=u_{s}\in\mathcal{M}_{p,q}(c)$ for any $s>1$, which means
$\mathcal{M}_{p,q}(c)\not=\emptyset$.

Step $2$: We claim that $\mathcal{M}_{p,q}(c)$ is invariant under the flow of
(1.2).

Since $\psi_{0}\in\mathcal{M}_{p,q}(c)$, by Proposition 6.1, there exist $T>0$
and a unique solution $\psi(t)\in C([0,T);H_{rad}^{2}(\mathbb{R}^{N}))$ to
(1.2) with the initial datum $\psi_{0}$. First, by conservation law, we know
that $E_{p,q}(\psi(t))=E_{p,q}(\psi_{0})<E_{p,q}(u)$. Then we show that
$Q_{p,q}(\psi(t))<0$. If not, by the continuity, there exists a
$t_{0}\in(0,T)$ such that $Q_{p,q}(\psi(t_{0}))=0$. Hence

$m_{p,q}(c)\leq E_{p,q}(\psi(t_{0}))<E_{p,q}(u)=m_{p,q}(c),$

which is a contradiction.

Step $3$: We claim that there exists a constant $a>0$ such that
$Q_{p,q}(\psi(t))\leq-a$ for all $t\in[0,T)$.

For convenience, we shall write $\psi=\psi(t)$. Since $Q_{p,q}(\psi(t))<0$, by
Lemma 2.3, there exists a $s_{\psi}\in(0,1)$ such that
$Q_{p,q}(\psi_{s_{\psi}})=0$. Recalling that $s\rightarrow E_{p,q}(\psi_{s})$
is concave on $[s_{\psi},\infty)$, it follows that

$E_{p,q}(\psi_{s_{\psi}})-E_{p,q}(\psi)\leq(s_{\psi}-1)\frac{d}{ds}E_{p,q}(\psi_{s})|_{s=1}=(s_{\psi}-1)Q_{p,q}(\psi).$

Since $Q_{p,q}(\psi)<0$ and $m_{p,q}(c)\leq E_{p,q}(\psi_{s_{\psi}})$, by the
conservation of energy, we have

$Q_{p,q}(\psi)<(1-s_{\psi})Q_{p,q}(\psi)\leq
E_{p,q}(\psi)-E_{p,q}(\psi_{s_{\psi}})\leq
E_{p,q}(\psi_{0})-m_{p,q}(c):=-a<0.$

Step $4$: Let us show that there exist a $\delta>0$ such that

$\frac{d}{dt}M_{\varphi_{R}}[\psi(t)]\leq-\delta\|\Delta\psi(t)\|_{2}^{2}\quad\mathrm{for}\
t\in[0,T)$ (6.15)

and a $t_{0}>0$ such that

$M_{\varphi_{R}}[\psi(t)]<0,\quad\mathrm{for}\ t>t_{0}.$ (6.16)

First, we claim that there exists a constant $C$ such that

$\|\Delta\psi\|_{2}^{2}\geq C,$ (6.17)

for $t\in[0,T)$. If not, there existed $t_{k}\subset[0,T)$ such that
$\|\Delta\psi(t_{k})\|_{2}^{2}\rightarrow 0$. By the Gagliardo-Nirenberg
inequality and conservation law, we have

$\|\psi(t_{k})\|_{q}^{q}\leq
C^{q}_{N,q}\|\Delta\psi(t_{k})\|_{2}^{q\gamma_{q}}\|\psi(t_{k})\|_{2}^{q(1-\gamma_{q})}\rightarrow
0$

and

$\|\psi(t_{k})\|_{p}^{p}\leq
C^{p}_{N,p}\|\Delta\psi(t_{k})\|_{2}^{p\gamma_{q}}\|\psi(t_{k})\|_{2}^{p(1-\gamma_{p})}\rightarrow
0$

as $k\rightarrow\infty$. It follows that

$Q_{p,q}(\psi(t_{k}))=\|\Delta\psi(t_{k})\|_{2}^{2}-\mu\gamma_{q}\|\psi(t_{k})\|_{q}^{q}-\gamma_{p}\|\psi(t_{k})\|_{p}^{p}\rightarrow
0,$

which is a contradiction to the fact $Q_{p,q}(\psi(t))\leq-a$ for all
$t\in[0,T)$. Since the solution $\psi$ is radial, we recall that

$\displaystyle\frac{d}{dt}M_{\varphi_{R}}[\psi(t)]$ $\displaystyle\leq
8\|\Delta\psi(t)\|_{2}^{2}-8\mu\gamma_{q}\|\psi(t)\|_{q}^{q}-8\gamma_{p}\|\psi(t)\|_{p}^{p}$
$\displaystyle+O(R^{-4}+R^{-2}\|\Delta\psi(t)\|_{2}+R^{-\frac{(q-2)(N-1)}{2}}\|\Delta\psi(t)\|_{2}^{\frac{q-2}{4}}+R^{-\frac{(p-2)(N-1)}{2}}\|\Delta\psi(t)\|_{2}^{\frac{p-2}{4}}),$

for all $t\in[0,T)$ and $R>1$. Then we apply the Young inequality to obtain
for any $\xi>0$,

$R^{-2}\|\Delta\psi(t)\|_{2}\leq
C\xi\|\Delta\psi(t)\|^{2}_{2}+\xi^{-1}R^{-4},$
$R^{-\frac{(q-2)(N-1)}{2}}\|\Delta\psi(t)\|_{2}^{\frac{q-2}{4}}\leq
C\xi\|\Delta\psi(t)\|^{2}_{2}+\xi^{-\frac{q-2}{10-q}}R^{-\frac{4(q-2)(N-1)}{10-q}},$

and

$R^{-\frac{(p-2)(N-1)}{2}}\|\Delta\psi(t)\|_{2}^{\frac{p-2}{4}}\leq
C\xi\|\Delta\psi(t)\|^{2}_{2}+\xi^{-\frac{p-2}{10-p}}R^{-\frac{4(p-2)(N-1)}{10-p}}.$

Therefore, we obtain

$\displaystyle\frac{d}{dt}M_{\varphi_{R}}[\psi(t)]$ $\displaystyle\leq
8\|\Delta\psi(t)\|_{2}^{2}-8\mu\gamma_{q}\|\psi(t)\|_{q}^{q}-8\gamma_{p}\|\psi(t)\|_{p}^{p}$
$\displaystyle+C\xi\|\Delta\psi(t)\|^{2}_{2}+O(R^{-4}+\xi^{-\frac{q-2}{10-q}}R^{-\frac{4(q-2)(N-1)}{10-q}}+\xi^{-\frac{p-2}{10-p}}R^{-\frac{4(p-2)(N-1)}{10-p}}),$

for all $t\in[0,T)$, any $R>1$, any $\xi>0$ and some constant $C>0$. For
simplicity, we denote
$\mathcal{K}=R^{-4}+\xi^{-\frac{q-2}{10-q}}R^{-\frac{4(q-2)(N-1)}{10-q}}+\xi^{-\frac{p-2}{10-p}}R^{-\frac{4(p-2)(N-1)}{10-p}}$.

When $2+\frac{8}{N}<q<p<4^{*}$, we consider two cases. Before that, we set

$\rho=\frac{4N(q-2)|E_{p,q}(\psi_{0})|+2}{N(q-2)-8}.$ (6.18)

$Case\ 1:$ Assume that $t\in[0,T)$ such that

$\|\Delta\psi(t)\|^{2}_{2}\leq\rho.$ (6.19)

Then we have

$\frac{d}{dt}M_{\varphi_{R}}[\psi(t)]\leq-8a+C\xi\rho+O(\mathcal{K})\quad\mathrm{for}\
t\in[0,T).$

Taking $\xi>0$ small enough and $R>1$ large enough depending on $\xi$, it
follows that

$\frac{d}{dt}M_{\varphi_{R}}[\psi(t)]\leq-4a\leq\frac{-4a}{\rho}\|\Delta\psi(t)\|_{2}^{2}\quad\mathrm{for}\
t\in[0,T).$ (6.20)

where $a$ is determined in the last step.

$Case\ 2:$ Assume that $t\in[0,T)$ such that

$\|\Delta\psi(t)\|^{2}_{2}>\rho.$ (6.21)

In this case by the conservation law, it follows that

$\displaystyle Q_{p,q}(\psi(t))$
$\displaystyle=2N(q-2)E_{p,q}(\psi(t))-(N(q-2)-8)\|\Delta\psi(t)\|_{2}^{2}-\frac{2N(p-q)}{p}\|\psi(t)\|_{p}^{p}$
$\displaystyle\leq\rho\frac{N(q-2)-8}{2}-1-\frac{N(q-2)-8}{2}\rho-\frac{N(q-2)-8}{2}\|\Delta\psi(t)\|_{2}^{2}.$

Thus we get that

$\frac{d}{dt}M_{\varphi_{R}}[\psi(t)]\leq-1-\frac{N(q-2)-8}{2}\|\Delta\psi(t)\|_{2}^{2}+C\xi\|\Delta
u(t)\|^{2}_{2}+O(\mathcal{K}).$ (6.22)

Since $2+\frac{8}{N}<q<p<4^{*}$ and $N\geq 5$, we first choose $\xi>0$ small
enough such that

$\frac{N(q-2)-8}{2}-C\xi>\frac{N(q-2)-8}{4}.$

Then taking $R>1$ large enough depending on $\xi$, we can get that

$-1+O(\mathcal{K})\leq 0.$

Hence, we can obtain that

$\frac{d}{dt}M_{\varphi_{R}}[\psi(t)]\leq-\frac{N(q-2)-8}{4}\|\Delta\psi(t)\|_{2}^{2}.$
(6.23)

When $2+\frac{8}{N}=q<p<4^{*}$, we can still get

$\frac{d}{dt}M_{\varphi_{R}}[\psi(t)]\leq-\delta\|\Delta\psi(t)\|_{2}^{2}\quad\mathrm{for}\
t\in[0,T).$ (6.24)

In fact, we also need to distinguish into two cases. First we denote

$\rho_{0}=\frac{8\beta_{2}\sigma|E_{p,q}(\psi_{0})|+C_{\eta}\frac{8\mu\sigma(\beta_{2}-\beta_{1})}{q}\|\psi_{0}\|_{2}^{2}+2}{\beta_{2}\sigma-4},$
(6.25)

where $\beta_{1}=\frac{N(q-2)}{2}$, $\beta_{2}=\frac{N(p-2)}{2}$and $C_{\eta}$
will be chosen later. In particular, we can find a small constant $\epsilon$
such that $p>2+\frac{8+\epsilon}{N}$, it follows that
$\sigma=\frac{8+\epsilon}{(p-2)N}<1$. If
$\|\Delta\psi(t)\|^{2}_{2}\leq\rho_{0}$, we can get the same conclusion as
above.

Now we assume that $\|\Delta\psi(t)\|^{2}_{2}>\rho_{0}$. Then by the
conservation of energy, we can get

$\displaystyle Q_{p,q}(\psi(t))$
$\displaystyle=8\|\Delta\psi(t)\|^{2}_{2}-\frac{4\mu\beta_{1}}{q}\|\psi(t)\|_{q}^{q}-\frac{4\beta_{2}}{p}\|\psi(t)\|_{p}^{p}$
$\displaystyle=4\beta_{2}\sigma
E_{p,q}(\psi_{0})+(8-2\beta_{2}\sigma)\|\Delta\psi(t)\|^{2}_{2}+(\frac{4\mu\beta_{2}\sigma}{q}-\frac{4\mu\beta_{1}}{q})\|\psi(t)\|_{q}^{q}+(\frac{4\beta_{2}\sigma}{p}-\frac{4\beta_{2}}{p})\|\psi(t)\|_{p}^{p}$
$\displaystyle\leq 4\beta_{2}\sigma
E_{p,q}(\psi_{0})+(8-2\beta_{2}\sigma)\|\Delta\psi(t)\|^{2}_{2}+\frac{4\mu\sigma(\beta_{2}-\beta_{1})}{q}\|\psi(t)\|_{q}^{q}-\frac{4\beta_{2}(1-\sigma)}{p}\|\psi(t)\|_{p}^{p}.$

By the Young inequality, we can obtain that

$\displaystyle\frac{4\mu\sigma(\beta_{2}-\beta_{1})}{q}\|\psi(t)\|_{q}^{q}$
$\displaystyle=\frac{4\mu\sigma(\beta_{2}-\beta_{1})}{q}\int_{\mathbb{R}^{N}}|\psi(t)|^{q}dx=\frac{4\mu\sigma(\beta_{2}-\beta_{1})}{q}\int_{\mathbb{R}^{N}}|\psi|^{\frac{2(p-q)}{p-2}}|\psi|^{q-\frac{2(p-q)}{p-2}}dx$
$\displaystyle\leq\frac{4\mu\sigma(\beta_{2}-\beta_{1})}{q}\big{[}\frac{p-q}{p-2}\int_{\mathbb{R}^{N}}(\eta^{-\frac{q-\frac{2(p-q)}{p-2}}{p}}|\psi|^{\frac{2(p-q)}{p-2}})^{\frac{p-2}{p-q}}dx$
$\displaystyle+\frac{q-\frac{2(p-q)}{p-2}}{p}\int_{\mathbb{R}^{N}}(\eta^{\frac{q-\frac{2(p-q)}{p-2}}{p}}|\psi|^{q-\frac{2(p-q)}{p-2}})^{\frac{p}{q-\frac{2(p-q)}{p-2}}}dx\big{]}$
$\displaystyle\leq\frac{4\mu\sigma(\beta_{2}-\beta_{1})}{q}\big{[}C_{\eta}\int_{\mathbb{R}^{N}}|\psi(t)|^{2}dx+\eta|\psi(t)|^{p}dx\big{]}.$

Choosing $\eta$ small enough such that

$\frac{4\mu\sigma(\beta_{2}-\beta_{1})}{q}\eta<\frac{4\beta_{2}(1-\sigma)}{p},$

which, together with the definition of $\rho_{0}$, implies that

$\displaystyle Q_{p,q}(\psi(t))$ $\displaystyle\leq 4\beta_{2}\sigma
E_{p,q}(\psi_{0})+(8-2\beta_{2}\sigma)\|\Delta\psi(t)\|^{2}_{2}+\frac{4\mu\sigma(\beta_{2}-\beta_{1})}{q}\|\psi(t)\|_{q}^{q}-\frac{4\beta_{2}(1-\sigma)}{p}\|\psi(t)\|_{p}^{p}$
$\displaystyle\leq 4\beta_{2}\sigma
E_{p,q}(\psi_{0})-(2\beta_{2}\sigma-8)\|\Delta\psi(t)\|^{2}_{2}+\frac{4\mu\sigma(\beta_{2}-\beta_{1})}{q}C_{\eta}\|\psi\|^{2}_{2}$
$\displaystyle\leq\frac{\beta_{2}\sigma-4}{2}\rho_{0}-1-\frac{\beta_{2}\sigma-4}{2}\rho_{0}-\frac{3(\beta_{2}\sigma-4)}{2}\|\Delta\psi(t)\|^{2}_{2}.$

By a similar argument above, we can obtain that

$\frac{d}{dt}M_{\varphi_{R}}[\psi(t)]\leq-\frac{\beta_{2}\sigma-4}{2}\|\Delta\psi(t)\|_{2}^{2}.$
(6.26)

Noticing that

$|\frac{d}{dt}M_{\varphi_{R}}[\psi(t)]|\geq\min\\{\frac{4a}{\rho},\frac{N(q-2)-8}{4},\frac{4a}{\rho_{0}},\frac{\beta_{2}\sigma-4}{2}\\}\|\Delta\psi(t)\|_{2}^{2}$

and

$M_{\varphi_{R}}[\psi(t)]=M_{\varphi_{R}}[\psi(0)]+\int_{0}^{t}\frac{d}{ds}M_{\varphi_{R}}[\psi(s)]ds,$

we get that there exists a $t_{0}>0$ such that for any $t>t_{0}$, (6.16)
holds.

Step $5$: We now conclude that the solution $\psi(t)$ with the initial data
$\psi_{0}=u_{s}$ blows up in finite time. Suppose by contradiction that
$T=\infty$. Then integrating (6.15) on $[t_{0},t]$, we have that

$M_{\varphi_{R}}[\psi(t)]\leq-\int_{t_{0}}^{t}\delta\|\Delta\psi(t)\|_{2}^{2}dx.$
(6.27)

Now using the Cauchy-Schwartz’s inequality, we get from the definition of
$M_{\varphi_{R}}[\psi(t)]$ that

$|M_{\varphi_{R}}[\psi(t)]|^{4}\leq C\|\Delta\psi(t)\|_{2}^{2}.$ (6.28)

Thus combining (6.27) and (6.28), we have for some $C>0$,

$M_{\varphi_{R}}[\psi(t)]\leq-C\int_{t_{0}}^{t}|M_{\varphi_{R}}[\psi(t)]|^{4}dx$
(6.29)

Setting $k(t)=\int_{t_{0}}^{t}|M_{\varphi_{R}}[\psi(\tau)]|^{4}d\tau$, we get
from (6.29) that

$k^{\prime}(t)\geq Ck(t)^{4}.$ (6.30)

Then integrating differential inequality (6.30) on $[t_{0},t]$, we obtain that

$M_{\varphi_{R}}[\psi(t)]\leq-
Ck(t)\leq\frac{-Ck(t_{0})}{(1-3Ck^{3}(t_{0})(t-t_{0}))^{\frac{1}{3}}}.$

This shows $M_{\varphi_{R}}[\psi(t)]\rightarrow-\infty$ as $t$ tends to some
limit time $t^{*}$. Hence the solution $\psi(t,x)$ cannot exist globally.
Recalling the blowup alternative we conclude our proof. ∎

###  6.2 Sobolev critical case: $2+\frac{8}{N}\leq q<p=4^{*}$

###### Proof of Theorem 1.7.

By Theorem 1.3, let $u\in S(c)$ be a ground state solution to (1.3) at the
level $m_{4^{*},q}(c)>0$. Similar in Sobolev subcritical case, $u$ is radially
symmetric since we work in the space $H^{2}_{rad}(\mathbb{R}^{N})$. By Lemma
2.3, we can have $\|u_{s}-u\|_{H^{2}}\rightarrow 0$ and
$E_{4^{*},q}(u_{s})\rightarrow m_{4^{*},q}(c)$ as $s\rightarrow 1^{+}$. Then
we set $\psi_{0}=u_{s}$. By Proposition 6.1, we know that there exists a
unique solution $\psi(t)\in C([0,T);H_{rad}^{2}(\mathbb{R}^{N}))$ to (1.2)
with the initial datum $\psi_{0}=u_{s}$. According to Definition 1.5, we need
to show that $\psi(t)$ blows up in a finite time. As before, we also divide
the proof into five steps:

Step $1$: $\mathcal{M}_{4^{*},q}(c)$ is nonempty, where
$\mathcal{M}_{4^{*},q}(c)$ is defined as follows:

$\mathcal{M}_{4^{*},q}(c)=\\{v\in
S(c),E_{4^{*},q}(v)<E_{4^{*},q}(u),Q_{4^{*},q}(v)<0\\}.$ (6.31)

Step $2$: $\mathcal{M}_{4^{*},q}(c)$ is invariant under the flow of (1.2).

Step $3$: There exists a constant $a^{*}>0$ such that
$Q_{4^{*},q}(\psi(t))\leq-a^{*}$ for all $t\in[0,T)$.

For the first three steps, we omit the proof since it turns out to be the same
as that of the Sobolev subcritical case.

Step $4$: There exist a $\delta^{*}>0$ such that

$\frac{d}{dt}M_{\varphi_{R}}[\psi(t)]\leq-\delta^{*}\|\Delta\psi(t)\|_{2}^{2},\quad\mathrm{for}\
t\in[0,T)$ (6.32)

and a $t_{0}>0$ such that

$M_{\varphi_{R}}[\psi(t)]<0,\quad\mathrm{for}\ t>t_{0}.$ (6.33)

First, we claim that there exists a costant $C$ such that

$\|\Delta\psi\|_{2}^{2}\geq C,$ (6.34)

for $t\in[0,T)$. If not, there existed $t_{k}\subset[0,T)$ such that
$\|\Delta\psi(t_{k})\|_{2}^{2}\rightarrow 0$. By (7.2), the Gagliardo-
Nirenberg inequality and conservation law, we have

$\|\psi(t_{k})\|_{q}^{q}\leq
C^{q}_{N,q}\|\Delta\psi(t_{k})\|_{2}^{q\gamma_{q}}\|\psi(t_{k})\|_{2}^{q(1-\gamma_{q})}\rightarrow
0$

and

$\|\psi(t_{k})\|^{4^{*}}_{4^{*}}\leq\mathcal{S}^{\frac{4^{*}}{2}}\|\Delta\psi(t_{k})\|^{4^{*}}_{2}\rightarrow
0$

as $k\rightarrow\infty$. It follows that

$Q_{4^{*},q}(\psi(t_{k}))=\|\Delta\psi(t_{k})\|_{2}^{2}-\mu\gamma_{q}\|\psi(t_{k})\|_{q}^{q}-\|\psi(t_{k})\|_{4^{*}}^{4^{*}}\rightarrow
0,$

which is a contradiction to the fact $Q_{4^{*},q}(\psi(t))\leq-a^{*}$ for
$t\in[0,T)$. Then we recall that

$\displaystyle\frac{d}{dt}M_{\varphi_{R}}[\psi(t)]$ $\displaystyle\leq
8\|\Delta\psi(t)\|_{2}^{2}-8\mu\gamma_{q}\|\psi(t)\|_{q}^{q}-8\|\psi(t)\|_{4^{*}}^{4^{*}}$
$\displaystyle+O(R^{-4}+R^{-2}\|\Delta\psi(t)\|_{2}+R^{-\frac{(q-2)(N-1)}{2}}\|\Delta\psi(t)\|_{2}^{\frac{q-2}{4}}+R^{-\frac{(4^{*}-2)(N-1)}{2}}\|\Delta\psi(t)\|_{2}^{\frac{4^{*}-2}{4}}),$

for all $t\in[0,T)$ and $R>1$. Then we apply the Young inequality to obtain
for any $\xi>0$,

$R^{-2}\|\Delta\psi(t)\|_{2}\leq
C\xi\|\Delta\psi(t)\|^{2}_{2}+\xi^{-1}R^{-4},$
$R^{-\frac{(q-2)(N-1)}{2}}\|\Delta\psi(t)\|_{2}^{\frac{q-2}{4}}\leq
C\xi\|\Delta\psi(t)\|^{2}_{2}+\xi^{-\frac{q-2}{10-q}}R^{-\frac{4(q-2)(N-1)}{10-q}},$

and

$R^{-\frac{(4^{*}-2)(N-1)}{2}}\|\Delta\psi(t)\|_{2}^{\frac{4^{*}-2}{4}}\leq
C\xi\|\Delta\psi(t)\|^{2}_{2}+\xi^{-\frac{4^{*}-2}{10-4^{*}}}R^{-\frac{4(4^{*}-2)(N-1)}{10-4^{*}}}(N\geq
6),$
$R^{-\frac{(4^{*}-2)(N-1)}{2}}\|\Delta\psi(t)\|_{2}^{\frac{4^{*}-2}{4}}=R^{-16}\|\Delta\psi(t)\|_{2}^{2}(N=5).$

Therefore, we obtain

$\displaystyle\frac{d}{dt}M_{\varphi_{R}}[\psi(t)]$ $\displaystyle\leq
8\|\Delta\psi(t)\|_{2}^{2}-8\mu\gamma_{q}\|\psi(t)\|_{q}^{q}-8\|\psi(t)\|_{4^{*}}^{4^{*}}$
$\displaystyle+C\xi\|\Delta\psi(t)\|^{2}_{2}+O(R^{-4}+\xi^{-\frac{q-2}{10-q}}R^{-\frac{4(q-2)(N-1)}{10-q}}+\xi^{-\frac{4^{*}-2}{10-4^{*}}}R^{-\frac{4(4^{*}-2)(N-1)}{10-4^{*}}})(N\geq
6),$ $\displaystyle\frac{d}{dt}M_{\varphi_{R}}[\psi(t)]$ $\displaystyle\leq
8\|\Delta\psi(t)\|_{2}^{2}-8\mu\gamma_{q}\|\psi(t)\|_{q}^{q}-8\|\psi(t)\|_{4^{*}}^{4^{*}}$
$\displaystyle+(C\xi+R^{-16})\|\Delta\psi(t)\|^{2}_{2}+O(R^{-4}+\xi^{-\frac{q-2}{10-q}}R^{-\frac{4(q-2)(N-1)}{10-q}})(N=5),$

for all $t\in[0,T)$, any $R>1$, any $\xi>0$ and some constant $C>0$. For
simplicity, we denote
$\mathcal{K}=R^{-4}+\xi^{-\frac{q-2}{10-q}}R^{-\frac{4(q-2)(N-1)}{10-q}}+\xi^{-\frac{4^{*}-2}{10-4^{*}}}R^{-\frac{4(4^{*}-2)(N-1)}{10-4^{*}}}$
if $N\geq 6$ and
$\mathcal{K}=R^{-4}+\xi^{-\frac{q-2}{10-q}}R^{-\frac{4(q-2)(N-1)}{10-q}}$ if
$N=5$.

When $2+\frac{8}{N}<q<p=4^{*}$, we consider two cases. Before that, we set

$\rho^{*}=\frac{4N(q-2)|E_{4^{*},q}(\psi_{0})|+2}{N(q-2)-8}.$ (6.35)

$Case\ 1:$ Assume that $t\in[0,T)$ such that

$\|\Delta\psi(t)\|^{2}_{2}\leq\rho^{*}.$ (6.36)

Then we have

$\frac{d}{dt}M_{\varphi_{R}}[\psi(t)]\leq-8a^{*}+(C\xi+R^{-16})\rho+O(\mathcal{K})\quad\mathrm{for}\
t\in[0,T).$

Taking $\xi>0$ small enough and $R>1$ large enough depending on $\xi$, it
follows that

$\frac{d}{dt}M_{\varphi_{R}}[\psi(t)]\leq-4a^{*}\leq\frac{-4a^{*}}{\rho^{*}}\|\Delta\psi(t)\|_{2}^{2}\quad\mathrm{for}\
t\in[0,T),$ (6.37)

where $a^{*}$ is determined in Step 2.

$Case\ 2:$ Assume that $t\in[0,T)$ such that

$\|\Delta\psi(t)\|^{2}_{2}>\rho^{*}.$ (6.38)

In this case by the conservation law, it follows that

$\displaystyle Q_{4^{*},q}(\psi(t))$
$\displaystyle=2N(q-2)E_{4^{*},q}(\psi_{0})-(N(q-2)-8)\|\Delta\psi(t)\|_{2}^{2}-\frac{2N(4^{*}-q)}{4^{*}}\|\psi(t)\|_{4^{*}}^{4^{*}}$
$\displaystyle\leq\rho^{*}\frac{N(q-2)-8}{2}-1-\frac{N(q-2)-8}{2}\rho^{*}-\frac{N(q-2)-8}{2}\|\Delta\psi(t)\|_{2}^{2}.$

Thus we get that

$\frac{d}{dt}M_{\varphi_{R}}[\psi(t)]\leq-1-\frac{N(q-2)-8}{2}\|\Delta\psi(t)\|_{2}^{2}+(C\xi+R^{-16})\|\Delta\psi(t)\|^{2}_{2}+O(\mathcal{K}).$
(6.39)

Since $2+\frac{8}{N}<q<p=4^{*}$ and $N\geq 5$, we first choose $\xi>0$ small
enough and $R>1$ large enough depending on $\xi$ such that

$C\xi+R^{-16}-\frac{N(q-2)-8}{2}<-\frac{N(q-2)-8}{4}.$

Also, we can get that

$-1+O(\mathcal{K})\leq 0.$

Hence, we can obtain that

$\frac{d}{dt}M_{\varphi_{R}}[\psi(t)]\leq-\frac{N(q-2)-8}{4}\|\Delta\psi(t)\|_{2}^{2}.$
(6.40)

When $2+\frac{8}{N}=q<p=4^{*}$, we can still get

$\frac{d}{dt}M_{\varphi_{R}}[\psi(t)]\leq-\delta^{*}\|\Delta\psi(t)\|_{2}^{2}\quad\mathrm{for}\
t\in[0,T).$ (6.41)

In fact, we also need to distinguish into two cases. First we denote

$\rho_{0}^{*}=\frac{8\beta^{*}_{2}\sigma^{*}|E_{4^{*},q}(\psi_{0})|+C_{\eta}\frac{8\mu\sigma^{*}(\beta^{*}_{2}-\beta_{1})}{q}\|\psi_{0}\|_{2}^{2}+2}{\beta_{2}\sigma^{*}-4},$

where $\beta^{*}_{2}=\frac{N(4^{*}-2)}{2}$and $C_{\eta}$ will be chosen later.
In particular, we can find a small constant $\epsilon$ such that
$4^{*}>2+\frac{8+\epsilon}{N}$, it follows that
$\sigma^{*}=\frac{8+\epsilon}{(4^{*}-2)N}<1$. If
$\|\Delta\psi(t)\|^{2}_{2}\leq\rho_{0}^{*}$, we can get the same conclusion as
above.

Now we assume that $\|\Delta\psi(t)\|^{2}_{2}>\rho_{0}^{*}$. Then by the
conservation of energy, we can get

$\displaystyle Q_{4^{*},q}(\psi(t))$
$\displaystyle=8\|\Delta\psi(t)\|^{2}_{2}-\frac{4\mu\beta_{1}}{q}\|\psi(t)\|_{q}^{q}-\frac{4\beta_{2}^{*}}{4^{*}}\|\psi(t)\|_{4^{*}}^{4^{*}}$
$\displaystyle=4\beta_{2}^{*}\sigma^{*}E_{4^{*},q}(\psi_{0})+(8-2\beta_{2}^{*}\sigma^{*})\|\Delta\psi(t)\|^{2}_{2}+(\frac{4\mu\beta_{2}^{*}\sigma^{*}}{q}-\frac{4\mu\beta_{1}}{q})\|\psi(t)\|_{q}^{q}+(\frac{4\beta_{2}^{*}\sigma^{*}}{4^{*}}-\frac{4\beta_{2}^{*}}{4^{*}})\|\psi(t)\|_{4^{*}}^{4^{*}}$
$\displaystyle\leq
4\beta_{2}^{*}\sigma^{*}E_{4^{*},q}(\psi_{0})+(8-2\beta_{2}^{*}\sigma^{*})\|\Delta\psi(t)\|^{2}_{2}+\frac{4\mu\sigma^{*}(\beta_{2}^{*}-\beta_{1})}{q}\|\psi(t)\|_{q}^{q}-\frac{4\beta_{2}^{*}(1-\sigma^{*})}{4^{*}}\|\psi(t)\|_{4^{*}}^{4^{*}}.$

By the Young inequality, we can obtain that

$\displaystyle\frac{4\mu\sigma^{*}(\beta_{2}^{*}-\beta_{1})}{q}\|\psi(t)\|_{q}^{q}$
$\displaystyle=\frac{4\mu\sigma^{*}(\beta_{2}^{*}-\beta_{1})}{q}\int_{\mathbb{R}^{N}}|\psi(t)|^{q}dx=\frac{4\mu\sigma^{*}(\beta_{2}^{*}-\beta_{1})}{q}\int_{\mathbb{R}^{N}}|\psi|^{\frac{2(4^{*}-q)}{4^{*}-2}}|\psi|^{q-\frac{2(4^{*}-q)}{4^{*}-2}}dx$
$\displaystyle\leq\frac{4\mu\sigma^{*}(\beta_{2}^{*}-\beta_{1})}{q}\big{[}\frac{4^{*}-q}{4^{*}-2}\int_{\mathbb{R}^{N}}(\eta^{-\frac{q-\frac{2(4^{*}-q)}{4^{*}-2}}{4^{*}}}|\psi|^{\frac{2(4^{*}-q)}{4^{*}-2}})^{\frac{4^{*}-2}{4^{*}-q}}dx$
$\displaystyle+\frac{q-\frac{2(4^{*}-q)}{4^{*}-2}}{4^{*}}\int_{\mathbb{R}^{N}}(\eta^{\frac{q-\frac{2(4^{*}-q)}{4^{*}-2}}{4^{*}}}|\psi|^{q-\frac{2(4^{*}-q)}{4^{*}-2}})^{\frac{4^{*}}{q-\frac{2(4^{*}-q)}{4^{*}-2}}}dx\big{]}$
$\displaystyle\leq\frac{4\mu\sigma^{*}(\beta_{2}^{*}-\beta_{1})}{q}\big{[}C_{\eta}\int_{\mathbb{R}^{N}}|\psi(t)|^{2}dx+\eta|\psi(t)|^{4^{*}}dx\big{]}.$

Choosing $\eta$ small enough such that

$\frac{4\mu\sigma^{*}(\beta_{2}^{*}-\beta_{1})}{q}\eta<\frac{4\beta^{*}_{2}(1-\sigma^{*})}{4^{*}},$

which, together with the definition of $\rho_{0}^{*}$, implies that

$\displaystyle Q_{4^{*},q}(\psi(t))$ $\displaystyle\leq
4\beta_{2}^{*}\sigma^{*}E_{4^{*},q}(\psi_{0})+(8-2\beta_{2}^{*}\sigma^{*})\|\Delta\psi(t)\|^{2}_{2}+\frac{4\mu\sigma^{*}(\beta_{2}^{*}-\beta_{1})}{q}\|\psi(t)\|_{q}^{q}-\frac{4\beta_{2}^{*}(1-\sigma^{*})}{4^{*}}\|\psi(t)\|_{4^{*}}^{4^{*}}$
$\displaystyle\leq
4\beta_{2}^{*}\sigma^{*}E_{4^{*},q}(\psi_{0})-(2\beta_{2}^{*}\sigma^{*}-8)\|\Delta\psi(t)\|^{2}_{2}+\frac{4\mu\sigma^{*}(\beta_{2}^{*}-\beta_{1})}{q}C_{\eta}\|\psi\|^{2}_{2}$
$\displaystyle\leq\frac{\beta_{2}^{*}\sigma^{*}-4}{2}\rho_{0}-1-\frac{\beta_{2}^{*}\sigma^{*}-4}{2}\rho_{0}-\frac{3(\beta_{2}^{*}\sigma^{*}-4)}{2}\|\Delta\psi(t)\|^{2}_{2}.$

By a similar argument above, we can obtain that

$\frac{d}{dt}M_{\varphi_{R}}[\psi(t)]\leq-\frac{\beta_{2}^{*}\sigma^{*}-4}{2}\|\Delta\psi(t)\|_{2}^{2}.$
(6.42)

Noticing that

$|\frac{d}{dt}M_{\varphi_{R}}[\psi(t)]|\geq\min\\{\frac{4a^{*}}{\rho^{*}},\frac{N(q-2)-8}{4},\frac{4a^{*}}{\rho_{0}^{*}},\frac{\beta_{2}^{*}\sigma^{*}-4}{2}\\}\|\Delta\psi(t)\|_{2}^{2}$

and

$M_{\varphi_{R}}[\psi(t)]=M_{\varphi_{R}}[\psi(0)]+\int_{0}^{t}\frac{d}{ds}M_{\varphi_{R}}[\psi(s)]ds,$

we get that there exists a $t_{0}>0$ such that for any $t>t_{0}$, (6.16)
holds.

Step $5$: We now conclude that the solution $\psi(t)$ with the initial data
$\psi_{0}=u_{s}$ blows up in finite time. Suppose by contradiction that
$T=\infty$. Then integrating (6.15) on $[t_{0},t]$, we have that

$M_{\varphi_{R}}[\psi(t)]\leq-\int_{t_{0}}^{t}\delta^{*}\|\Delta\psi(t)\|_{2}^{2}dx.$
(6.43)

With this estimate at hand, in the same fashion as we did before the solution
$\psi(t,x)$ cannot exist globally. Recalling the blowup alternative we
conclude our proof. ∎

##  7 Appendix

We consider the extremal function. Let

$U_{\epsilon}=R(\frac{\epsilon}{\epsilon^{2}+|x|^{2}})^{\frac{N-4}{2}},$ (7.1)

where $R=[(N-4)N(N^{2}-4)]^{\frac{N-4}{8}}$. We know that the best constant
$\mathcal{S}$ in the Sobolev embedding $H^{2}(\mathbb{R}^{N})\hookrightarrow
L^{4^{*}}(\mathbb{R}^{N})$, $i.e.$

$\mathcal{S}\|u\|^{2}_{4^{*}}\leq\|\Delta u\|^{2}_{2}$ (7.2)

was attained by $U_{\epsilon}$. We also select a suitable constant in order
that $U_{\epsilon}$ is the solution of the following critical equation

$\Delta^{2}u=|u|^{4^{*}-2}u.$

Then we have

$\begin{cases}\|\Delta
U_{\epsilon}\|_{2}^{2}=\|U_{\epsilon}\|_{4^{*}}^{4^{*}}\\\
\mathcal{S}\|U_{\epsilon}\|_{4^{*}}^{2}=\|\Delta U_{\epsilon}\|_{2}^{2}\\\
\end{cases}$ (7.3)

it follows that $\|U_{\epsilon}\|_{4^{*}}^{4^{*}}=\mathcal{S}^{\frac{N}{4}}$.
Then let $\varphi(x)\in C_{0}^{\infty}(\mathbb{R}^{N})$ is a cut-off function
satisfying

$(a)\ 0\leq\varphi(x)\leq 1$ and $x\in\mathbb{R}^{N}$.

$(b)\ \varphi(x)\equiv 1$ in $B_{1}$, where $B_{s}$ denotes the ball in
$\mathbb{R}^{N}$ of center at origin and radius $r$.

$(c)\ \varphi(x)\equiv 0$ in $\mathbb{R}^{N}\backslash B_{2}$.

######  Lemma 7.1.

Let $N\geq 5$, setting $u_{\epsilon}(x)=U_{\epsilon}(x)\varphi(x)$ and
denoting by $\omega$ the area of the unit sphere in $\mathbb{R}^{N}$. We have,
for $N\geq 5,$

$(i)$

$\|\Delta
u_{\epsilon}\|_{2}^{2}=\mathcal{S}^{\frac{N}{4}}+O(\epsilon^{N-4})\quad\mathrm{and}\quad\|u_{\epsilon}\|_{4^{*}}^{4^{*}}=\mathcal{S}^{\frac{N}{4}}+O(\epsilon^{N})$

$(ii)$

For some constant $K>0$ and $N\geq 5$,

$\|u_{\epsilon}\|_{q}^{q}=\begin{cases}K\epsilon^{N-\frac{(N-4)q}{2}}+o(\epsilon^{N-\frac{(N-4)q}{2}}),\qquad\qquad\qquad\qquad
q>\frac{N}{N-4},\\\
K\epsilon^{\frac{N}{2}}|\log\epsilon|+O(\epsilon^{\frac{N}{2}}),\qquad\qquad\qquad\qquad\quad\qquad
q=\frac{N}{N-4},\\\
K\epsilon^{\frac{(N-4)q}{2}}+o(\epsilon^{\frac{(N-4)q}{2}}),\qquad\qquad\qquad\qquad\qquad
1\leq q<\frac{N}{N-4}.\end{cases}$

$(iii)$

For some constant $K^{\prime}>0$

$\|u_{\epsilon}\|_{2}^{2}=\begin{cases}K^{\prime}\epsilon^{4}+o(\epsilon^{4}),\qquad\qquad\qquad\qquad\qquad\qquad\qquad
N>8,\\\
K^{\prime}\epsilon^{4}|\log\epsilon|+O(\epsilon^{4}),\qquad\qquad\qquad\
\qquad\quad\qquad N=8,\\\
K^{\prime}\epsilon^{N-4}+o(\epsilon^{(N-4)}),\qquad\qquad\qquad\qquad\qquad
5\leq N<8.\end{cases}$

  * Remark:

Let $0\leq r\leq R$, then

$\int_{r\leq|x|\leq
R}f(x)dx=\int_{r}^{R}\int_{\sum}f(rx^{\prime})r^{n-1}dx^{\prime}dr,$

where $\sum=\\{x^{\prime}\in\mathbb{R}^{N},|x^{\prime}|=1\\}$ and $f(x)$ is a
continuous function.

###### Proof.

$(i)$

$u_{\epsilon}=\varphi U_{\epsilon}=\begin{cases}U_{\epsilon},\qquad\qquad x\in
B_{1}\\\ 0\qquad\qquad\quad x\in\mathbb{R}^{N}\backslash B_{2},\\\
\end{cases}$

By some calculations, we have

$\Delta u_{\epsilon}=\Delta
U_{\epsilon}\varphi+U_{\epsilon}\Delta\varphi+2\nabla
U_{\epsilon}\cdot\nabla\varphi.$ (7.4)

And more accurately, we have

$\nabla
U_{\epsilon}=R\epsilon^{\frac{N-4}{2}}(4-N)\frac{x}{(\epsilon^{2}+|x|^{2})^{\frac{N-2}{2}}}$

and

$\Delta
U_{\epsilon}=R\epsilon^{\frac{N-4}{2}}(4-N)\big{[}\frac{2}{(\epsilon^{2}+|x|^{2})^{\frac{N-2}{2}}}+(2-N)\frac{|x|^{2}}{(\epsilon^{2}+|x|^{2})^{\frac{N}{2}}}\big{]}.$

Passing to radial coordinates and making a change of variable, we get

$\displaystyle\int_{\mathbb{R}^{N}}|\Delta U_{\epsilon}|^{2}dx$
$\displaystyle=\epsilon^{N-4}R^{2}(4-N)^{2}\int_{\mathbb{R}^{N}}\frac{4}{(\epsilon^{2}+|x|^{2})^{N-2}}+(2-N)^{2}\frac{|x|^{4}}{(\epsilon^{2}+|x|^{2})^{N}}+4(2-N)\frac{|x|^{2}}{(\epsilon^{2}+|x|^{2})^{N-1}}dx$
$\displaystyle=\epsilon^{N-4}R^{2}(4-N)^{2}\omega\int_{0}^{\infty}\frac{4r^{N-1}}{(\epsilon^{2}+r^{2})^{N-2}}+(2-N)^{2}\frac{r^{N+3}}{(\epsilon^{2}+r^{2})^{N}}+4(2-N)\frac{r^{N+1}}{(\epsilon^{2}+r^{2})^{N-1}}dr$
$\displaystyle=R^{2}(4-N)^{2}\omega\int_{0}^{\infty}\frac{4r^{N-1}}{(1+r^{2})^{N-2}}+(2-N)^{2}\frac{r^{N+3}}{(1+r^{2})^{N}}+4(2-N)\frac{r^{N+1}}{(1+r^{2})^{N-1}}dr.$

While for $\Delta u_{\epsilon}$, recalling (7.4), we have

$\|\Delta u_{\epsilon}\|^{2}_{2}=\int_{\mathbb{R}^{N}}\big{|}\Delta
U_{\epsilon}\varphi+U_{\epsilon}\Delta\varphi+2\nabla
U_{\epsilon}\cdot\nabla\varphi\big{|}^{2}dx.$

And it can be decomposed into four parts $I_{1}(\epsilon)$, $I_{2}(\epsilon)$,
$I_{3}(\epsilon)$ and $I_{4}(\epsilon)$. First, we deal with

$I_{1}(\epsilon)=\int_{\mathbb{R}^{N}}|U_{\epsilon}\Delta\varphi||2\nabla
U_{\epsilon}\cdot\nabla\varphi|+|\Delta U_{\epsilon}\varphi||2\nabla
U_{\epsilon}\cdot\nabla\varphi|+|\Delta
U_{\epsilon}\varphi||U_{\epsilon}\Delta\varphi|dx.$

Since $\varphi(x)\in C_{0}^{\infty}(\mathbb{R}^{N})$ is cut-off function
satisfying $(a),\ (b)$ and $(c)$, there exists constants
$K^{\prime},K^{\prime\prime}>0$ such that $|\nabla\varphi|\leq K^{\prime}$ and
$|\Delta\varphi|\leq K^{\prime\prime}$ in $B_{2}\backslash B_{1}$. It follows
that the integral in $I_{1}(\epsilon)$ is converging and
$I_{1}(\epsilon)=O(\epsilon^{N-4})$. For the second part, we have

$I_{2}(\epsilon)=\int_{\mathbb{R}^{N}}|\Delta\varphi|^{2}|U_{\epsilon}|^{2}dx=\epsilon^{N-4}R^{2}\int_{B_{2}\backslash
B_{1}}\frac{|\Delta\varphi|^{2}}{(\epsilon^{2}+|x|^{2})^{N-4}}dx.$

By the definition of $\varphi(x)$, similarly, we can easily get that the
integral in $I_{2}(\epsilon)$ is converging and
$I_{2}(\epsilon)=O(\epsilon^{N-4})$. While for the third part,

$I_{3}(\epsilon)=4\int_{\mathbb{R}^{N}}|\nabla
U_{\epsilon}|^{2}|\nabla\varphi\big{|}^{2}dx=4\epsilon^{N-4}R^{2}(4-N)^{2}\int_{B_{2}\backslash
B_{1}}\frac{|x|^{2}|\nabla\varphi|^{2}}{(\epsilon^{2}+|x|^{2})^{N-2}}dx.$

Similarly, according to the definition of $\varphi(x)$, it follows that the
integral in $I_{3}(\epsilon)$ is converging and
$I_{3}(\epsilon)=O(\epsilon^{N-4})$. Finally, we concentrate on

$\displaystyle I_{4}(\epsilon)$
$\displaystyle=\epsilon^{N-4}R^{2}(4-N)^{2}\int_{\mathbb{R}^{N}}\frac{4|\varphi|^{2}}{(\epsilon^{2}+|x|^{2})^{N-2}}+(2-N)^{2}\frac{|x|^{4}|\varphi|^{2}}{(\epsilon^{2}+|x|^{2})^{N}}+4(2-N)\frac{|x|^{2}|\varphi|^{2}}{(\epsilon^{2}+|x|^{2})^{N-1}}dx$
$\displaystyle=K_{1}(\epsilon)+K_{2}(\epsilon)+K_{3}(\epsilon).$

Passing to radial coordinates, we get

$K_{1}(\epsilon)=\epsilon^{N-4}R^{2}(4-N)^{2}\omega\int_{0}^{2}\frac{4|\varphi(r)|^{2}r^{N-1}}{(\epsilon^{2}+|r|^{2})^{N-2}}dr$

that can be decomposed as

$\displaystyle K_{1}(\epsilon)$
$\displaystyle=\epsilon^{N-4}R^{2}(4-N)^{2}\omega\int_{0}^{2}\big{[}\frac{4r^{N-1}(|\varphi(r)|^{2}-1)}{(\epsilon^{2}+|r|^{2})^{N-2}}+\frac{4r^{N-1}}{(\epsilon^{2}+|r|^{2})^{N-2}}\big{]}dr$
$\displaystyle=L_{1}(\epsilon)+L_{2}(\epsilon).$

Since $\varphi(r)\equiv 1$ on $[0,1]$, the integral in $L_{1}(\epsilon)$ is
converging and $L_{1}(\epsilon)=O(\epsilon^{N-4})$. Then by making a change of
variable, we rewrite $L_{2}(\epsilon)$ as

$L_{2}(\epsilon)=R^{2}(4-N)^{2}\omega\int_{0}^{\frac{2}{\epsilon}}\frac{4r^{N-1}}{(1+r^{2})^{N-2}}dr.$

Hence we know that

$K_{1}(\epsilon)=R^{2}(4-N)^{2}\omega\int_{0}^{\frac{2}{\epsilon}}\frac{4r^{N-1}}{(1+r^{2})^{N-2}}dr+O(\epsilon^{N-4}).$

Similarly, we can also obtain

$K_{2}(\epsilon)=R^{2}(4-N)^{2}(2-N)^{2}\omega\int_{0}^{\frac{2}{\epsilon}}\frac{r^{N+3}}{(1+r^{2})^{N}}dr+O(\epsilon^{N-4})$

and

$K_{3}(\epsilon)=4R^{2}(4-N)^{2}(2-N)\omega\int_{0}^{\frac{2}{\epsilon}}\frac{r^{N+1}}{(1+r^{2})^{N}}dr+O(\epsilon^{N-4})$

Therefore, as $\epsilon\rightarrow 0$, we get that

$\displaystyle\|\Delta u_{\epsilon}\|_{2}^{2}$
$\displaystyle=R^{2}(4-N)^{2}\omega\int_{0}^{\frac{2}{\epsilon}}\big{[}\frac{4r^{N-1}}{(1+r^{2})^{N-2}}+\frac{(2-N)^{2}r^{N+3}}{(1+r^{2})^{N}}+\frac{4(2-N)r^{N+1}}{(1+r^{2})^{N}}\big{]}dr+O(\epsilon^{N-4})$
$\displaystyle=\|\Delta U_{\epsilon}\|_{2}^{2}+O(\epsilon^{N-4}).$

Then we deal with the second part of $(i)$. By simple calculations, we have

$\displaystyle\|u_{\epsilon}\|_{4^{*}}^{4^{*}}$
$\displaystyle=R^{4^{*}}\int_{\mathbb{R}^{N}}\varphi^{4^{*}}(x)(\frac{\epsilon}{\epsilon^{2}+|x|^{2}})^{N}dx$
$\displaystyle=R^{4^{*}}\int_{\mathbb{R}^{N}}\varphi^{4^{*}}(x)\frac{\epsilon^{N}}{(\epsilon^{2}+|x|^{2})^{N}}dx.$

Passing to radial coordinates, we get

$\|u_{\epsilon}\|_{4^{*}}^{4^{*}}=R^{4^{*}}\omega\epsilon^{N}\int_{0}^{2}\varphi^{4^{*}}(r)\frac{r^{N-1}}{(\epsilon^{2}+r^{2})^{N}}dr$

that can be decomposed as

$\displaystyle\|u_{\epsilon}\|_{4^{*}}^{4^{*}}$
$\displaystyle=R^{4^{*}}\omega\epsilon^{N}\int_{0}^{2}\frac{(\varphi^{4^{*}}(r)-1)r^{N-1}}{(\epsilon^{2}+r^{2})^{N}}dr+R^{4^{*}}\omega\epsilon^{N}\int_{0}^{2}\frac{r^{N-1}}{(\epsilon^{2}+r^{2})^{N}}dr$
$\displaystyle=I_{1}(\epsilon)+I_{2}(\epsilon).$

Since $\varphi(r)\equiv 1$ on $[0,1]$, the integral in $I_{1}(\epsilon)$ is
converging and thus $I_{1}(\epsilon)=O(\epsilon^{N})$. Now by making a change
of variable, we rewrite $I_{2}(\epsilon)$ as

$I_{2}(\epsilon)=R^{4^{*}}\omega\int_{0}^{\frac{2}{\epsilon}}\frac{r^{N-1}}{(1+r^{2})^{N}}dr.$

Then we notice that

$\|U_{\epsilon}\|_{4^{*}}^{4^{*}}=R^{4^{*}}\int_{\mathbb{R}^{N}}\frac{\epsilon^{N}}{(\epsilon^{2}+|x|^{2})^{N}}dx.$

Passing to radial coordinates, we obtain

$\|U_{\epsilon}\|_{4^{*}}^{4^{*}}=R^{4^{*}}\omega\int_{0}^{\infty}\frac{r^{N-1}}{(1+r^{2})^{N}}dr.$

As $\epsilon\rightarrow 0$, we find that
$I_{2}(\epsilon)\rightarrow\|U_{\epsilon}\|_{4^{*}}^{4^{*}}=\mathcal{S}^{\frac{N}{4}}$.
Hence we know that
$\|u_{\epsilon}\|_{4^{*}}^{4^{*}}=\mathcal{S}^{\frac{N}{4}}+O(\epsilon^{N})$.

$(ii)$ Here we concentrate on point $(ii)$. We have

$\|u_{\epsilon}\|_{q}^{q}=R^{q}\int_{\mathbb{R}^{N}}\varphi^{q}(x)(\frac{\epsilon}{\epsilon^{2}+|x|^{2}})^{\frac{(N-4)q}{2}}dx.$

passing radial coordinates, we get

$\|u_{\epsilon}\|_{q}^{q}=R^{q}\omega\epsilon^{\frac{(N-4)q}{2}}\int_{0}^{2}\frac{\varphi^{q}(r)r^{N-1}}{(\epsilon^{2}+r^{2})^{\frac{(N-4)q}{2}}}dr$

that can be decomposed as

$\displaystyle\|u_{\epsilon}\|_{q}^{q}$
$\displaystyle=R^{q}\omega\epsilon^{\frac{(N-4)q}{2}}\int_{0}^{2}\frac{(\varphi^{q}(r)-1)r^{N-1}}{(\epsilon^{2}+r^{2})^{\frac{(N-4)q}{2}}}dr+R^{q}\omega\epsilon^{\frac{(N-4)q}{2}}\int_{0}^{2}\frac{r^{N-1}}{(\epsilon^{2}+r^{2})^{\frac{(N-4)q}{2}}}dr$
$\displaystyle=I_{1}(\epsilon)+I_{2}(\epsilon).$

Since $\varphi(r)\equiv 1$ on $[0,1]$, the integral in $I_{1}(\epsilon)$ is
converging and thus $I_{1}(\epsilon)=O(\epsilon^{\frac{(N-4)q}{2}})$. Now by
making a change of variable, we rewrite $I_{2}(\epsilon)$ as

$I_{2}(\epsilon)=R^{q}\omega\epsilon^{N-\frac{(N-4)q}{2}}\int_{0}^{\frac{2}{\epsilon}}\frac{r^{N-1}}{(1+r^{2})^{\frac{(N-4)q}{2}}}dr.$

The integral is converging, as $\epsilon\rightarrow 0$, to a finite value if
and only if $q>\frac{N}{N-4}$. Hence we have that, for some constant $K>0$,

$I_{2}(\epsilon)=K\epsilon^{N-\frac{(N-4)q}{2}}+o(\epsilon^{N-\frac{(N-4)q}{2}}).$

And recording that $I_{1}(\epsilon)=O(\epsilon^{\frac{(N-4)q}{2}})$, we have

$\|u_{\epsilon}\|_{q}^{q}=K\epsilon^{N-\frac{(N-4)q}{2}}+o(\epsilon^{N-\frac{(N-4)q}{2}}).$

Now assuming that $q=\frac{N}{N-4}$, and proceeding as before, we get that

$\displaystyle\|u_{\epsilon}\|_{q}^{q}$
$\displaystyle=R^{q}\omega\epsilon^{\frac{N}{2}}\int_{0}^{2}\frac{(\varphi^{q}(r)-1)r^{N-1}}{(\epsilon^{2}+r^{2})^{\frac{N}{2}}}dr+R^{q}\omega\epsilon^{\frac{N}{2}}\int_{0}^{2}\frac{r^{N-1}}{(\epsilon^{2}+r^{2})^{\frac{N}{2}}}dr$
$\displaystyle=I_{1}(\epsilon)+I_{2}(\epsilon).$

with $I_{1}(\epsilon)=O(\epsilon^{\frac{N}{2}})$. Also,

$I_{2}(\epsilon)=R^{q}\omega\epsilon^{\frac{N}{2}}\int_{0}^{\frac{2}{\epsilon}}\frac{r^{N-1}}{(1+r^{2})^{\frac{N}{2}}}dr=R^{q}\omega\epsilon^{\frac{N}{2}}(|\log\epsilon|+O(1)).$

Summarizing, for $q=\frac{N}{N-4}$ we obtain that

$\|u_{\epsilon}\|_{q}^{q}=R^{q}\omega\epsilon^{\frac{N}{2}}|\log\epsilon|+O(\epsilon^{\frac{N}{2}}).$

It remains to study the case $2\leq q<\frac{N}{N-4}$. We can observe that for
all $r>0$,

$\lim_{\epsilon\rightarrow
0}\frac{\varphi^{q}(r)r^{N-1}}{(\epsilon^{2}+r^{2})^{\frac{N-4}{2}q}}=\frac{\varphi^{q}(r)}{r^{(N-4)q-(N-1)}}$

and also that, for all $\epsilon>0$, for some constant $C$

$\big{|}\frac{\varphi^{q}(r)r^{N-1}}{(\epsilon^{2}+r^{2})^{\frac{N-4}{2}q}}\big{|}\leq\frac{C}{r^{(N-4)q-(N-1)}}\in
L^{2}([0,2]).$

Then from lebesgue’s theorem, we can get that

$\displaystyle\|u_{\epsilon}\|_{q}^{q}$
$\displaystyle=R^{q}\omega\epsilon^{\frac{(N-4)q}{2}}\int_{0}^{2}\frac{\varphi^{q}(r)r^{N-1}}{(\epsilon^{2}+r^{2})^{\frac{(N-4)q}{2}}}dr$
$\displaystyle=\epsilon^{\frac{(N-4)q}{2}}R^{q}\omega\int_{0}^{2}\frac{\varphi^{q}(r)}{r^{(N-4)q-(N-1)}}dr+o(\epsilon^{\frac{(N-4)q}{2}}).$

This completes the proof. ∎

## Acknowledgements

X. J. Chang was supported by NSFC(11971095).

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