# Normalized bound state solutions of fractional Schrödinger equations with
general potential111Supported by National Natural Science Foundation of
China(No. 12271443) and Natural Science Foundation of Chongqing,
China(cstc2020jcyjjqX0029)

Xin Bao, Ying Lv, Zeng-Qi Ou 222Corresponding author. E-mail address:
bxinss@163.com(X. Bao), ly0904@swu.edu.cn(Y. Lv), ouzengqi@swu.edu.cn(Z.-Q.
Ou).
_School of Mathematics and Statistics, Southwest University, Chongqing
400715,_
_People’s Republic of China_

Abstract: In this paper, we study a class of fractional Schrödinger equation

$\left\\{\begin{aligned} &(-\Delta)^{s}u=\lambda u+a(x)|u|^{p-2}u,\\\
&\int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2},\ u\in
H^{s}(\mathbb{R}^{N}),\end{aligned}\right.$ (0.1)

where $N>2s$, $s\in(0,1)$ and $p\in(2,2+4s/N),c>0$. $a(x)\in
C(\mathbb{R}^{N},\mathbb{R})$ is a positive potential function. By using Fixed
Point Theorem of Brouwer, barycenter function and variational method, we
obtain the existence of normalized bound solutions for problem (0.1).

Keywords: Fractional Schrödinger equation; Normalized solutions; Variational
method

##  1 Introduction and main results

In this paper, we discuss the existence of positive solutions for a class of
fractional Schrödinger equation

$\left\\{\begin{aligned} &(-\Delta)^{s}u=\lambda u+a(x)|u|^{p-2}u,\\\
&\int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2},\ u\in
H^{s}(\mathbb{R}^{N}),\end{aligned}\right.$ (1.1)

where $N>2s$, $s\in(0,1)$ and $p\in(2,2+4s/N),c>0$. $a(x)\in
C(\mathbb{R}^{N},\mathbb{R})$ satisfies the following assumption:

$(A_{1})$ $0<\inf\limits_{x\in\mathbb{R}^{N}}a(x)\leq
a(x)\leq\lim\limits_{|x|\rightarrow\infty}a(x)=a_{\infty},a(x)\not\equiv
a_{\infty}$ in $\mathbb{R}^{N}$.

The fractional Laplacian $(-\Delta)^{s}$ is the nonlocal operator and defined
by

$(-\Delta)^{s}u(x)=C_{N,s}\
P.V.\int_{\mathbb{R}^{N}}\frac{u(x)-u(y)}{|x-y|^{N+2s}}dy,$

where $u:\mathbb{R}^{N}\rightarrow\mathbb{R}$ is sufficiently smooth, and
$P.V.$ is an abbreviation for the Cauchy principal value and $C_{N,s}$ stands
for a positive constant depending on $N$ and $s$(see [5]).

For $s\in(0,1)$ and $N>2s$, the fractional Sobolev space of order $s$ on
$\mathbb{R}^{N}$ is defined by

$H^{s}(\mathbb{R}^{N})=\left\\{u\in
L^{2}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}dxdy<\infty\right\\},$

endowed with the norm

$\|u\|=\left(\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}dx+\int_{\mathbb{R}^{N}}u^{2}dx\right)^{1/2},$

where

$\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx=\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}dxdy.$

From [5], the embedding $H^{s}(\mathbb{R}^{N})\hookrightarrow
L^{q}(\mathbb{R}^{N})$ is continuous for any $q\in[2,2^{*}_{s}]$, where
$2^{*}_{s}=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent.
Moreover, there is a positive constant $C=C(q,N,s)$ such that

$\|u\|_{q}\leq C\|u\|$ (1.2)

for any $u\in H^{s}(\mathbb{R}^{N})$, where $\|\cdot\|_{q}$ denotes the norm
of the Lebesgue space $L^{q}(\mathbb{R}^{N})$.

Now, we recall the Gagliardo-Nirenberg-Sobolev inequality(see [10]): for any
$p\in(2,2^{*}_{s})$, there exists a sharp constant $C_{N,p,s}>0$ such that

$\|u\|^{p}_{p}\leq
C_{N,p,s}\left(\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}dx\right)^{\frac{N(p-2)}{4s}}\|u\|_{2}^{p-\frac{N(p-2)}{2s}}$
(1.3)

for any $u\in H^{s}(\mathbb{R}^{N})$, where the best constant $C_{N,p,s}$ can
be achieved.

In the past years, many researchers have extensively studied the existence of
nontrivial solutions for the Schödinger equation with a prescribed
$L^{2}$-norm:

$\left\\{\begin{aligned} &-\Delta u+\lambda u=g(u),\quad
x\in\mathbb{R}^{N},\\\
&\int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2},\end{aligned}\right.$ (1.4)

and such solution is known as a normalized solution and $\lambda$ appears as a
Lagrange multiplier (see [2], [3], [4], [6], [7], [11], [12], [18], [20], [25]
and the references therein). In the classical paper [11], when $g(u)$ is
nonhomogeneous $L^{2}$-supercritical nonlinearity, by developing a mountain-
pass argument on $S_{c}=\\{u\in H^{1}(\mathbb{R}^{N}):\|u\|_{2}=c\\}$,
Jeanjean obtained the existence of a ground state normalized solution for
problem (1.4). When $N=2$ and $f$ satisfies an exponential critical growth in
problem (1.4), Chang, Liu and Yan in [4] proved the existence of normalized
ground state solution for any $c>0$ by using a constrained minimization method
and the Trudinger-Moser inequality in $\mathbb{R}^{2}$. In a mass subcritical
case but under general assumptions on the nonlinearity $g$, Jeanjean and Lu
in[12] obtained multiple nonradial solutions for problem (1.4) when $N=4$ or
$N\geq 6$, and all these solutions are sign-changing.

For the following Schödinger equation with potential

$\left\\{\begin{aligned} &-\Delta u+V(x)u+\lambda u=g(u),\quad
x\in\mathbb{R}^{N},\\\
&\int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2},\end{aligned}\right.$ (1.5)

if $V\leq 0$, the existence of ground state normalized solution for problem
(1.5) is obtained in [7], where $g$ satisfies some Ambrosetti-Rabinowitz type
conditions, and in [18] for the case $g(u)=|p|^{p-2}u\ (p\in(2,2^{*}))$. But
for the case $V\geq 0$, the problem becomes more delicate and difficult. When
$V\geq 0$ satisfies $V(x)\to 0$ as $|x|\to\infty$ and $g(u)=|p|^{p-2}u\
(p\in(2+\frac{4}{N},2^{*}))$, by constructing a suitable linking geometry,
Bartsch et al. [3] succeeded to obtain the existence of bound state solution
with high Morse index for problem (1.5).

In[6], Chen and Tang proved the existence of normalized ground state solutions
for the following Schrödinger equation

$\left\\{\begin{aligned} &-\Delta u=\lambda u+a(x)g(u),\quad
x\in\mathbb{R}^{N},\\\ &\|u\|_{2}=c,\end{aligned}\right.$

where $a(x)\in C(\mathbb{R}^{N},\mathbb{R})$ satisfies
$0<a_{\infty}=\lim\limits_{|x|\rightarrow\infty}a(x)\leq a(x)$ and $f\in
C(\mathbb{R},\mathbb{R})$ satisfies some Ambrosetti-Rabinowitz type condition.

Furthermore, by barycentric function and minimax method, Zhang and Zhang [25]
concerned with the existence and multiplicity of normalized bounded solutions
for the following Schödinger equations in exterior domain:

$\left\\{\begin{aligned} &-\Delta u=\lambda u+|u|^{p-2}u\ &\mbox{in}\
\Omega,\\\ &u=0\ &\mbox{on}\ \partial\Omega,\\\
&\int_{\Omega}|u|^{2}dx=c^{2},\end{aligned}\right.$

where $p\in(2,2+4/N)$ and $\Omega\subset\mathbb{R}^{N}$ is an exterior domain.

Recently, the existence of normalized solutions for Schrödinger equations with
fractional Laplacian operator have attracted much attention, see
[13],[15],[19],[21],[24] and the references therein. In [8], Du and Tian et
al. were devoted to study the existence and nonexistence of normalized
solutions for the following equation

$\left\\{\begin{aligned} &(-\Delta)^{s}u+V(x)u=\lambda u+\mu f(u)\ \ \text{in
}\mathbb{R}^{N},\\\ &\int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2},u\in
H^{s}(\mathbb{R}^{N}),\end{aligned}\right.$ (1.6)

where $\mu>0$, $V$ is a trapping potential function and $f$ is a subcritical
nonlinearity. If $f(u)=|u|^{p-2}u$ with $p\in(2+4s/N,2_{s}^{*})$ and $V(x)\in
L^{\infty}(\mathbb{R}^{N})$ or $V(x)\in L^{\frac{N}{2}}(\mathbb{R}^{N})$
satisfies appropriate assumptions, Peng and Xia in [19] proved existence of
normalized bounded solutions for problem (1.6) with $\mu=1$.

In [13], the authors considered the existence of normalized ground state
solution for the fractional Schrödinger equations with combined
nonlinearities:

$\left\\{\begin{aligned} &(-\Delta)^{s}u=\lambda u+\mu|u|^{p-2}u+|u|^{q-2}u\ \
\text{in }\mathbb{R}^{N},\\\ &\int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2},u\in
H^{s}(\mathbb{R}^{N}).\end{aligned}\right.$

Li and Zou [15], Zhen and Zhang [23] investigated existence of the normalized
ground state solution for the critical fractional Schrödinger equation,
respectively.

Motivated by above papers, in this paper, we will investigate the existence of
normalized bound solutions for problem (1.1) under $p\in(2,2+4s/N)$ and
suitable assumption $(A_{1})$.

Normalized solutions of problem (1.1) are obtained by looking for critical
point the following $C^{1}$ functional

$I(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}dx-\frac{1}{p}\int_{\mathbb{R}^{N}}a(x)|u|^{p}dx$
(1.7)

constrained on the $L^{2}$-spheres in $H^{s}(\mathbb{R}^{N})$:

$S_{c}:=\\{u\in H^{s}(\mathbb{R}^{N}):\|u\|_{2}=c\\},$

where $c>0$ is any fixed. Define

$m_{c}:=\inf_{u\in S_{c}}I(u),$

from $(A_{1})$ and (1.7), we will prove $m_{c}>-\infty$ for any
$p\in(2,2+4s/N)$ and $m_{c}=-\infty$ for any $p\in(2+4s/N,2_{s}^{*})$.
$\bar{p}=2+4s/N$ is called the $L^{2}$-critical exponent.

From [9], let $p\in(2,2_{s}^{\ast})$, the following fractional Schrödinger
equation:

$(-\Delta)^{s}u+u=u^{p-1},\ x\in\mathbb{R}^{N},\quad u>0,$ (1.8)

has an unique solution $w\in H^{s}(\mathbb{R}^{N})$, which is radial, radially
decreasing and positive. For $\lambda<0$, let
$w_{\lambda}(x):=(-\lambda)^{\frac{1}{p-2}}w((-\lambda)^{\frac{1}{2s}}x)$,
$w_{\lambda}$ is the unique positive solution (up to a translation) of

$(-\Delta)^{s}u=\lambda u+u^{p-1}\text{ in }\mathbb{R}^{N},\quad u>0.$ (1.9)

To restore compactness of $(PS)$-sequence for the functional $I$, we need to
consider the uniqueness of solutions for the limited problem corresponding to
problem (1.1):

$\left\\{\begin{aligned} &(-\Delta)^{s}u=\lambda u+|u|^{p-2}u,\\\
&\int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2},\ u\in
H^{s}(\mathbb{R}^{N}).\end{aligned}\right.$ (1.10)

Define the functional $I_{\infty}:H^{s}(\mathbb{R}^{N})\to\mathbb{R}$ as
follows:

$I_{\infty}(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}dx-\frac{1}{p}\int_{\mathbb{R}^{N}}|u|^{p}dx$
(1.11)

and

$m^{\infty}_{c}:=\inf_{u\in S_{c}}I_{\infty}(u).$ (1.12)

Now we state our main results of this paper.

Theorem 1.1. Assume that $(A_{1})$ holds, we have $m_{c}=m^{\infty}_{c}$, and
$m_{c}$ is not attained for the functional $I$ in $S_{c}$.

Theorem 1.2. Assume that $(A_{1})$ and

$(A_{2})$ $\sup\limits_{x\in\mathbb{R}^{N}}|a_{\infty}-a(x)|$ is sufficiently
small.

Then, problem (1.1) has a positive solution $u\in S_{c}$ for some $\lambda<0$.

Remark. In[17], Lehrer, Maia and Squassina exploited Pohožaev manifold and a
variational technique to prove existence of bound state solutions for the
asymptotically linear fractional Schrodinger equations. Recently, the
existence of normalized solutions of mass subcritical fractional Schrödinger
equations were obtained by Yu, Tang and Zhang (see[21]) in exterior domains.
In this paper, we consider the normalized bound state solutions of fractional
Schrödinger equations in the whole space. Hence our situation is different
from [17] and [21], and the main idea of our argument comes from [25], and to
obtain the non-negative Palais-Smale sequence of the functional $I$ in $S_{c}$
, we will apply the argument similar with the one of [1].

##  2 the Proof of Theorem 1.1

In this section, we first consider the minimization problem (1.12), and then
prove Theorem 1.1. First of all, by the change of variable, we can assume
$a_{\infty}=1$ for convenience.

Lemma 2.1. For any $c>0$, we have $m^{\infty}_{c}\in(-\infty,0)$. Moreover, if
$\\{u_{n}\\}\subset S_{c}$ is a non-negative minimizing sequence for the
functional $I_{\infty}$ in $S_{c}$, then up to a subsequence, there exists
$\\{y_{n}\\}\subset\mathbb{R}^{N}$ such that

$u_{n}(\cdot-y_{n})\rightarrow w_{\lambda}\ \text{ strongly in
}H^{s}(\mathbb{R}^{N})$

as $n\rightarrow\infty$, where $\lambda<0$ is determined by

$c^{2}=\|w_{\lambda}\|_{2}^{2}=(-\lambda)^{s_{p}}\|w\|_{2}^{2},\quad
s_{p}:=\frac{2}{p-2}-\frac{N}{2s}.$ (2.1)

In particular, $m^{\infty}_{c}$ is attained by the function $w_{\lambda}$ and
can be expressed as

$m^{\infty}_{c}=\left(\frac{c^{2}}{\|w\|_{2}^{2}}\right)^{\frac{s_{p}+1}{s_{p}}}I_{\infty}(w).$
(2.2)

where $w$ is an unique solution for problem (1.8).

Proof. By (1.3), for any $u\in S_{c}$, we have

$\displaystyle I_{\infty}(u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}dx-\frac{1}{p}\|u\|_{p}^{p}$
(2.3) $\displaystyle\geq$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}dx-\frac{C_{N,p,s}}{p}\left(\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}dx\right)^{\frac{N(p-2)}{4s}}\|u\|_{2}^{p-\frac{N(p-2)}{2s}}.$

Since $2<p<2+\frac{4s}{N}$, then $0<\frac{N(p-2)}{4s}<1$, and it follows from
(2.3) that $I_{\infty}$ is coercive in $S_{c}$. Therefore,
$m^{\infty}_{c}>-\infty$. Moreover, for any $u\in S_{c}$, $t>0$, one has
$t^{\frac{N}{2}}u(tx)\in S_{c}$ and

$I_{\infty}(t^{\frac{N}{2}}u(tx))=\frac{1}{2}t^{2s}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}dx-\frac{1}{p}t^{N(\frac{p}{2}-1)}\|u\|_{p}^{p}\rightarrow
0^{-}\ \mbox{as}\ t\to 0^{+}.$

One can get that $m^{\infty}_{c}<0$, that is, $m^{\infty}_{c}\in(-\infty,0)$.

Let $c>0$, $\theta>1$ and $\\{v_{n}\\}\subset S_{c}$ be a minimizing sequence
for $m^{\infty}_{c}$. From $\theta>1$, $p>2$, we have

$m^{\infty}_{\theta c}\leq I_{\infty}(\theta
v_{n})=\frac{\theta^{2}}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}v_{n}|^{2}dx-\frac{\theta^{p}}{p}\|v_{n}\|_{p}^{p}<\theta^{2}I_{\infty}(v_{n}),$

hence, it follows that $m^{\infty}_{\theta c}\leq\theta^{2}m^{\infty}_{c}$,
and the equality holds if and only if
$\lim_{n\rightarrow\infty}\|v_{n}\|_{p}^{p}=0$. We claim that
$m^{\infty}_{\theta c}<\theta^{2}m^{\infty}_{c}$. Suppose, by contradiction,
that $m^{\infty}_{\theta c}=\theta^{2}m^{\infty}_{c}$, then
$\lim_{n\rightarrow\infty}\|v_{n}\|_{p}^{p}=0$. Hence, we have

$m^{\infty}_{c}=\lim_{n\rightarrow\infty}I_{\infty}(v_{n})\geq\liminf_{n\rightarrow\infty}\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}v_{n}|^{2}dx\geq
0,$

which contradicts with $m^{\infty}_{c}<0$.

Let $c_{1}$, $c_{2}>0$ be such that $c_{1}^{2}+c_{2}^{2}=c^{2}$ and let
$\theta_{j}=\frac{c}{c_{j}}>1$ for $j=1,\ 2$ we have
$m^{\infty}_{c}=m^{\infty}_{\theta_{j}c_{j}}<\theta_{j}^{2}m^{\infty}_{c_{j}}$.
From $c_{1}^{2}+c_{2}^{2}=c^{2}$, we conclude

$m^{\infty}_{c}=\left(\frac{1}{\theta_{1}^{2}}+\frac{1}{\theta_{2}^{2}}\right)m^{\infty}_{c}<m^{\infty}_{c_{1}}+m^{\infty}_{c_{2}}.$

By Theorem $1.2$ of [14], up to a subsequence, there exist $u\in S_{c}$ and
$\\{y_{n}^{\prime}\\}\subset\mathbb{R}^{N}$ such that $u_{n}(\cdot-
y_{n}^{\prime})\rightarrow u$ strongly in $H^{s}(\mathbb{R}^{N})$ as
$n\rightarrow\infty$. In particular, $I_{\infty}(u)=m^{\infty}_{c}$. Since
$u_{n}\geq 0$ in $\mathbb{R}^{N}$ for any $n\geq 1$, one has $u\geq 0$ in
$\mathbb{R}^{N}$. By the Lagrange multiplier rule, there exists
$\lambda\in\mathbb{R}$ such that $u$ is a non-negative nontrivial weak
solution of

$(-\Delta)^{s}u-|u|^{p-2}u=\lambda u\ \text{ in }\ H^{s}(\mathbb{R}^{N}),$

which implies that

$\lambda
c^{2}=\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}dx-\|u\|_{p}^{p}=pm^{\infty}_{c}+\frac{2-p}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}dx<0,$

hence, we have $\lambda<0$. By the strong maximum principle, $u>0$ in
$\mathbb{R}^{N}$. The uniqueness of solutions for problem (1.9) guarantees
that there exists $y\in\mathbb{R}^{N}$ such that $u(\cdot-y)=w_{\lambda}$.
Denote $y_{n}=y_{n}^{\prime}+y$ for any $n\geq 1$, we have $u_{n}(\cdot-
y_{n})\rightarrow w_{\lambda}$ strongly in $H^{s}(\mathbb{R}^{N})$ as
$n\rightarrow\infty$. Particularly, $w_{\lambda}\in S_{c}$ and
$I_{\infty}(w_{\lambda})=m^{\infty}_{c}$. By the definition of $w_{\lambda}$,
(2.1) and (2.2) hold. $\Box$

Define the map $\Psi:\mathbb{R}^{N}\rightarrow S_{c}$ by

$\Psi(y)=w_{\lambda}(\cdot-y)\ \text{for all}\ y\in\mathbb{R}^{N},$

where $w_{\lambda}(x)$ is given in Lemma 2.1.

Proof of Theorem 1.1. Since $a(x)\leq 1$, one has $m_{c}\geq m^{\infty}_{c}$.
On the other hand, from

$\int_{\mathbb{R}^{N}}a(x)|w_{\lambda}(x-y)|^{p}dx\to\int_{\mathbb{R}^{N}}|w_{\lambda}(x)|^{p}dx\
\ \text{as}\ \ |y|\to\infty,$

it follows that

$m_{c}\leq I(\Psi(y))\rightarrow I_{\infty}(w_{\lambda})=m^{\infty}_{c}\ \
\text{as}\ \ |y|\to\infty.$

Hence, we have $m_{c}=m^{\infty}_{c}\in(-\infty,0)$.

Suppose, by contradiction, that there is $u\in S_{c}$ such that $I(u)=m_{c}$.
Note that $|u|\in S_{c}$ and

$\displaystyle\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}|u||^{2}dx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{||u(x)|-|u(y)||^{2}}{|x-y|^{N+2s}}dxdy$
(2.4) $\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}dxdy=\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx,$

from $a(x)\leq 1$, we have

$m_{c}=I(u)\geq I_{\infty}(u)\geq I_{\infty}(|u|)\geq m_{c}^{\infty}.$

Hence from Lemma 2.1, we have $u=w_{\lambda}$ up to a translation, where
$\lambda<0$ is determined by (2.1). From
$I(w_{\lambda})=I_{\infty}(w_{\lambda})=m_{c}^{\infty}$, we have

$\int_{\mathbb{R}^{N}}(1-a(x))|w_{\lambda}|^{p}dx=0,$

which is a contradiction with $w_{\lambda}>0$, $a(x)\leq 1$ and
$a(x)\not\equiv 1$ in $\mathbb{R}^{N}$. Hence $m_{c}$ is not attained for the
functional $I$ in $S_{c}$. $\Box$

##  3 the Proof of Theorem 1.2

In the subsection, we will introduce the Splitting Lemma, which can be found
in [17] and is crucial to ensure the convergence of the some non-negative
Palais-Smale sequence of the functional $I$ in $S_{c}$. And then, we will
prove some useful lemmas and Theorem 1.2.

First of all, we define the energy functionals associated with problem (1.1)
and problem (1.9) respectively:

$I_{\lambda}(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}dx-\frac{1}{p}\int_{\mathbb{R}^{N}}a(x)|u|^{p}dx-\frac{\lambda}{2}\int_{\mathbb{R}^{N}}|u|^{2}dx,$
$I_{\lambda,\infty}(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}dx-\frac{1}{p}\int_{\mathbb{R}^{N}}|u|^{p}dx-\frac{\lambda}{2}\int_{\mathbb{R}^{N}}|u|^{2}dx$

for any $u\in H^{s}(\mathbb{R}^{N})$.

Lemma 3.1. For any $\lambda<0$, let $\\{u_{n}\\}\subset H^{s}(\mathbb{R}^{N})$
be a non-negative Palais-Smale sequence of the functional $I_{\lambda}$. Then
up to a subsequence, there exist an integer $k\geq 0$, a non-negative function
$u_{0}\in H^{s}(\mathbb{R}^{N})$, $k$ sequences
$\\{x_{n}^{i}\\}\subset\mathbb{R}^{N}$ for $1\leq i\leq k$ such that
$|x_{n}^{i}|\rightarrow\infty$ as $n\rightarrow\infty$ and

$u_{n}=u_{0}+\sum_{i=1}^{k}w_{\lambda}(\cdot+x_{n}^{i})+o(1)\ \text{ strongly
in }H^{s}(\mathbb{R}^{N}).$

Furthermore, $u_{0}$ is a weak solution of

$(-\Delta)^{s}u=\lambda u+a(x)|u|^{p-2}u\ \text{ in }\mathbb{R}^{N},$

and $w_{\lambda}(x)$ is a weak solution of

$(-\Delta)^{s}u=\lambda u+|u|^{p-2}u\ \text{ in }\mathbb{R}^{N}.$

Moreover, we have

$\|u_{n}\|_{2}^{2}=\|u_{0}\|_{2}^{2}+k\|w_{\lambda}\|_{2}^{2}+o(1),$ (3.1)
$I_{\lambda}(u_{n})=I_{\lambda}(u_{0})+kI_{\lambda,\infty}(w_{\lambda})+o(1).$
(3.2)

Lemma 3.2. Let $N>2s$, $2<p<2+\frac{4s}{N}$, $c>0$, there exists a positive
constant $\eta=\eta(c)\in(2^{-1/s_{p}},1)$ such that if $\\{u_{n}\\}\subset
S_{c}$ is a non-negative Palais-Smale sequence of $I|_{S_{c}}$ at level $d$
with $m^{\infty}_{c}<d<\eta m^{\infty}_{c}$, then up to a subsequence, there
exists $u_{0}\in S_{c}$ such that $u_{n}\rightarrow u_{0}$ strongly in
$H^{s}(\mathbb{R}^{N})$ as $n\rightarrow\infty$. Moreover, $u_{0}$ is a
positive solution of problem (1.1) for some $\lambda<0$.

Proof. Since $2<p<2+\frac{4s}{N}$, we have $s_{p}>0$ and $0<2^{-1/s_{p}}<1$.
Due to $m^{\infty}_{c}<0$, one has

$m^{\infty}_{c}<2^{-1/s_{p}}m^{\infty}_{c}<0.$

Set $\\{u_{n}\\}\subset S_{c}$ be a non-negative Palais-Smale sequence of
$I|_{S_{c}}$ at level $d\in(m^{\infty}_{c},2^{-1/s_{p}}m^{\infty}_{c})$.
Because of $I(u_{n})\rightarrow d<0$ as $n\rightarrow\infty$ and (2.3),
$\\{u_{n}\\}$ is bounded in $H^{s}(\mathbb{R}^{N})$. Note that
$(I|_{S_{c}})^{\prime}(u_{n})=o(1)$, according to the Lagrange multiplier
rule, there exists $\\{\lambda_{n}\\}\subset\mathbb{R}$ such that

$\int_{\mathbb{R}^{N}}(-\Delta)^{\frac{s}{2}}u_{n}(-\Delta)^{\frac{s}{2}}\psi
dx-\lambda_{n}\int_{\mathbb{R}^{N}}u_{n}\psi
dx-\int_{\mathbb{R}^{N}}a(x)u_{n}^{p-1}\psi dx=o(1)\|\psi\|$ (3.3)

for any $\psi\in H^{s}(\mathbb{R}^{N})$. Since $\\{u_{n}\\}\subset S_{c}$ is
bounded in $H^{s}(\mathbb{R}^{N})$, using (3.3), we have

$-\lambda_{n}c^{2}=\int_{\mathbb{R}^{N}}a(x)|u_{n}|^{p}dx-\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u_{n}|^{2}dx+o(1),$

which implies that $\\{\lambda_{n}\\}$ is bounded in $\mathbb{R}$. Thus, up to
a subsequence, there exists $\lambda\in\mathbb{R}$ such that
$\lambda_{n}\rightarrow\lambda$ as $n\rightarrow\infty$. Moreover, since $p>2$
and $I(u_{n})\rightarrow d<0$ as $n\rightarrow\infty$, we obtain

$\displaystyle-\lambda_{n}c^{2}$
$\displaystyle=\int_{\mathbb{R}^{N}}a(x)|u_{n}|^{p}dx-\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u_{n}|^{2}dx+o(1)$
$\displaystyle=-pd+\frac{p-2}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u_{n}|^{2}dx+o(1)$
$\displaystyle\geq-pd+o(1).$

Letting $n\rightarrow\infty$, we can get

$-\lambda\geq\frac{-pd}{c^{2}}>0,$ (3.4)

That is, $\lambda<0$. From (3.3) and the boundedness of $\\{u_{n}\\}$ in
$H^{s}(\mathbb{R}^{N})$, we deduce that $\\{u_{n}\\}$ is a non-negative
Palais-Smale sequence of the functional $I_{\lambda}$. By Lemma $3.1$, up to a
subsequence, there exists an integer $k\geq 0$, a non-negative function
$u_{0}\in H^{s}(\mathbb{R}^{N})$ and $k$ sequences
$\\{x_{n}^{i}\\}\subset\mathbb{R}^{n}$ for $1\leq i\leq k$ such that
$|x_{n}^{i}|\rightarrow\infty$ as $n\rightarrow\infty$ and

$u_{n}=u_{0}+\sum_{i=1}^{k}w_{\lambda}(\cdot+x_{n}^{i})+o(1)\ \text{ strongly
in }H^{s}(\mathbb{R}^{N}).$ (3.5)

Moreover, $u_{0}$ is a weak solution of

$(-\Delta)^{s}u=\lambda u+a(x)u^{p-2}u\ \text{ in }\mathbb{R}^{N}$ (3.6)

and

$c^{2}=\|u_{0}\|_{2}^{2}+k\|w_{\lambda}\|_{2}^{2},$ (3.7)
$I_{\lambda}(u_{n})=I_{\lambda}(u_{0})+kI_{\lambda,\infty}(w_{\lambda})+o(1).$
(3.8)

Because

$I_{\lambda}(v)=I(v)-\frac{\lambda}{2}\|v\|_{2}^{2},\ \ \
I_{\lambda,\infty}(v)=I_{\infty}(v)-\frac{\lambda}{2}\|v\|_{2}^{2}$

for any $v\in H^{s}(\mathbb{R}^{N})$, by (3.7), (3.8) and Lemma $2.1$, we can
see

$\displaystyle d$ $\displaystyle=$ $\displaystyle
I(u_{0})+km_{\|w_{\lambda}\|_{2}}^{\infty}\geq
m_{\|u_{0}\|_{2}}^{\infty}+km_{\|w_{\lambda}\|_{2}}^{\infty}$ $\displaystyle=$
$\displaystyle
C_{0}\left((c^{2}-k\|w_{\lambda}\|_{2}^{2})^{\frac{s_{p}+1}{s_{p}}}+k(\|w_{\lambda}\|_{2}^{2})^{\frac{s_{p}+1}{s_{p}}}\right),$
(3.9)

where

$C_{0}:=I_{\infty}(w)(\|w\|_{2}^{2})^{-\frac{s_{p}+1}{s_{p}}}<0.$

We claim that there exists $\eta=\eta(c)\in(2^{-1/s_{p}},1)$ such that if
$m^{\infty}_{c}<d<\eta m^{\infty}_{c}$, then $k=0$. Once this claim is proved,
using (3.5), we can see that $u_{n}\rightarrow u_{0}$ strongly in
$H^{s}(\mathbb{R}^{N})$. It implies that $u_{0}\in S_{c}$ and $u_{0}$ is a
nontrivial non-negative solution of problem (1.1) with $\lambda<0$. The strong
maximum principle then yields that $u_{0}>0$ in $\mathbb{R}^{N}$. In the
following, we prove the above claim by three steps.

Step 1. If $m^{\infty}_{c}<d<2^{-1/s_{p}}m^{\infty}_{c}$, then
$\|w_{\lambda}\|_{2}^{2}\geq lc^{2}$, where

$l:=\frac{1}{2}\left[\frac{-pI_{\infty}(w)}{\|w\|_{2}^{2}}\right]^{s_{p}}\in(0,\frac{1}{2}).$

From Lemma 2.1 and (3.4), we obtain

$\displaystyle\|w_{\lambda}\|_{2}^{2}$
$\displaystyle=(-\lambda)^{s_{p}}\|w\|_{2}^{2}\geq(\frac{-pd}{c^{2}})^{s_{p}}\|w\|_{2}^{2}$
$\displaystyle\geq\left(\frac{-p2^{-1/s_{p}}m^{\infty}_{c}}{c^{2}}\right)^{s_{p}}\|w\|_{2}^{2}$
$\displaystyle=lc^{2}.$

Observe that $w$ is a solution of (1.8), it is clear that

$\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}w|^{2}dx+\|w\|_{2}^{2}=\|w\|_{p}^{p}.$

Hence

$I_{-1,\infty}(w)=\left(\frac{1}{2}-\frac{1}{p}\right)\left(\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}w|^{2}dx+\|w\|_{2}^{2}\right)>\left(\frac{1}{2}-\frac{1}{p}\right)\|w\|_{2}^{2}.$

In view of $I_{\infty}(w)=I_{-1,\infty}(w)-\frac{1}{2}\|w\|_{2}^{2}<0$, we
conclude that

$0<\frac{-pI_{\infty}(w)}{\|w\|_{2}^{2}}<1.$

Since $s_{p}>0$, we get $l\in\left(0,\frac{1}{2}\right)$.

Step 2. Define

$l_{1}:=l^{(s_{p}+1)/s_{p}}+(1-l)^{(s_{p}+1)/s_{p}}\in(2^{-1/s_{p}},1).$

If $m^{\infty}_{c}<d<l_{1}m^{\infty}_{c}$, then $k\leq 1$. Actually, for
$k\geq 1$, we set

$f_{k}(t):=(c^{2}-kt)^{\frac{s_{p}+1}{s_{p}}}+kt^{\frac{s_{p}+1}{s_{p}}}\
\mbox{for all}\ 0\leq t\leq\frac{c^{2}}{k}.$

Then $f_{k}\in C^{1}\left(\left[0,\frac{c^{2}}{k}\right]\right)$ and

$f_{k}^{\prime}(t)=\frac{s_{p}+1}{s_{p}}k[t^{1/s_{p}}-(c^{2}-kt)^{1/s_{p}}],\quad\forall
t\in\left[0,\frac{c^{2}}{k}\right].$

Hence, $f_{k}(t)$ has a critical point $t_{k}:=\frac{c^{2}}{k+1}$, and is
strictly decreasing in $[0,t_{k}]$ and strictly increasing in
$\left[t_{k},\frac{c^{2}}{k}\right]$. Moreover, we have

$\min_{t\in[0,c^{2}/k]}f_{k}(t)=f_{k}(t_{k})=(k+1)^{-\frac{1}{s_{p}}}(c^{2})^{\frac{s_{p}+1}{s_{p}}}>0.$

Step $1$ and (3.7) leads to $k\leq l^{-1}$. Define

$h(t):=(1-lt)^{(s_{p}+1)/s_{p}}+tl^{(s_{p}+1)/s_{p}},\quad\forall
t\in[1,l^{-1}],$

it follows that $h\in C^{1}([1,l^{-1}])$ and

$h^{\prime}(t)=l^{(s_{p}+1)/s_{p}}-l\frac{s_{p}+1}{s_{p}}(1-lt)^{\frac{1}{s_{p}}},\quad\forall
t\in[1,l^{-1}].$

Hence, $h(t)$ has a critical point
$t_{0}:=\frac{1}{l}-\left(\frac{s_{p}}{s_{p}+1}\right)^{s_{\mathrm{p}}}\in(1,l^{-1})$,
and is strictly decreasing in $[1,t_{0}]$ and strictly increasing in
$[t_{0},l^{-1}]$.

If $k\geq 2$, by (3), we deduce that

$\displaystyle d$ $\displaystyle\geq
C_{0}\max\\{f_{k}(lc^{2}),f_{k}(c^{2}/k)\\}=\max\\{h(k),k^{-1/s_{p}}\\}m^{\infty}_{c}$
$\displaystyle\geq\max\\{h(1),h(l^{-1}),2^{-1/s_{p}}\\}m^{\infty}_{c}$
$\displaystyle=l_{1}m^{\infty}_{c},$

which is a contradiction to $d<l_{1}m^{\infty}_{c}$. Therefore, we have $k\leq
1$.

Step 3. There exists $\eta=\eta(c)\in[l_{1},1)$ such that if
$m^{\infty}_{c}<d<\eta m^{\infty}_{c}$, then $k=0$.

From Step $2$, we have $k\leq 1$. If $u_{0}\equiv 0$, then by (3.7) and (3),
we get $d=0$ or $d=m^{\infty}_{c}$, which contradicts to $0>d>m^{\infty}_{c}$.
Hence, $u_{0}\not\equiv 0$. By (3.6), we get

$\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u_{0}|^{2}dx+(-\lambda)\|u_{0}\|_{2}^{2}=\int_{\mathbb{R}^{N}}a(x)|u_{0}|^{p}dx.$
(3.10)

On the one hand, from $a(x)\leq 1$, (1.2) and (3.10), there exists $C_{1}>0$
independent of $u_{0}$ and $\lambda$ such that

$\min\\{1,-\lambda\\}\|u_{0}\|^{2}\leq\|u_{0}\|_{p}^{p}\leq
C_{1}\|u_{0}\|^{p}.$

Because $u_{0}\not\equiv 0$, $\lambda<0$ and $p>2$, we conclude that

$\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u_{0}|^{2}dx+\|u_{0}\|_{2}^{2}=\|u_{0}\|^{2}\geq\left(\frac{\min\\{1,-\lambda\\}}{C_{1}}\right)^{\frac{2}{p-2}}>0.$
(3.11)

On the other hand, by (1.3), (3.10) and $\lambda<0$, we can see

$\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u_{0}|^{2}dx\leq\|u_{0}\|_{p}^{p}\leq
C_{N,p,s}\left(\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u_{0}|^{2}dx\right)^{\frac{N(p-2)}{4s}}\|u_{0}\|_{2}^{p-\frac{N(p-2)}{2s}}.$

From $2<p<2+\frac{4s}{N}$, let $\gamma_{p}:=\frac{N(p-2)}{4s}$, we get

$\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u_{0}|^{2}dx\leq
C_{N,p,s}^{\frac{1}{1-\gamma_{p}}}\|u_{0}\|_{2}^{\frac{p-2\gamma_{p}}{1-\gamma_{p}}}.$
(3.12)

Owing to (3.4), (3.11), (3.12) and $d<2^{-1/s_{p}}m^{\infty}_{c}$ that

$\begin{array}[]{l}C_{N,p,s}^{\frac{1}{1-\gamma_{p}}}(\|u_{0}\|_{2}^{2})^{\frac{p-2\gamma_{p}}{2-2\gamma_{p}}}+\|u_{0}\|_{2}^{2}\geq\left(\frac{1}{C_{1}}\min\left\\{1,-p2^{-1/s_{p}}\frac{m^{\infty}_{c}}{c^{2}}\right\\}\right)^{\frac{2}{p-2}}>0,\end{array}$

which means that there exists $C(c)>0$ such that

$\|u_{0}\|_{2}^{2}\geq C(c).$

Set

$\eta(c):=\left\\{\begin{array}[]{ll}l_{1},&\text{ if }c^{2}\leq C(c),\\\
\max\left\\{l_{1},\frac{f_{1}(c^{2}-C(c))}{(c^{2})^{\frac{s_{p}+1}{s_{p}}}}\right\\},&\text{
if }c^{2}>C(c).\end{array}\right.$

If $c^{2}\leq C(c)$, then $\|u_{0}\|_{2}^{2}\geq C(c)\geq c^{2}$. By (3.7) we
get $\|u_{0}\|_{2}^{2}=c^{2}$ and hence $k=0$. If $c^{2}>C(c)$, then
$0<c^{2}-C(c)<c^{2}$. Hence, we have

$0<f_{1}(c^{2}-C(c))<\max\\{f_{1}(0),f_{1}(c^{2})\\}=(c^{2})^{\frac{s_{p}+1}{s_{p}}},$

which implies that
$0<\frac{f_{1}(c^{2}-C(c))}{(c^{2})^{\frac{s_{p}+1}{s_{p}}}}<1$. In
particular, $\eta(c)\in[l_{1},1)$. Assume that $k=1$, it follows from Step $1$
and (3.7) that

$lc^{2}\leq\|w_{\lambda}\|_{2}^{2}\leq c^{2}-C(c).$

By (3), one has

$d\geq
C_{0}\max\\{f_{1}(lc^{2}),f_{1}(c^{2}-C(c))\\}\geq\eta(c)m^{\infty}_{c},$

which contradicts to $d<\eta(c)m^{\infty}_{c}$. Hence $k=0$. $\Box$

Let the function $w_{\lambda}(x)$ and $\eta=\eta(c)\in(0,1)$ be given in Lemma
2.1 and Lemma 3.1, respectively, now we replace $(A_{2})$ by a accurate
condition, and we have the following lemma.

Lemma 3.3. Assume that

$\sup_{x\in\mathbb{R}^{N}}|1-a(x)|<\frac{(\eta-1)m^{\infty}_{c}}{\|w_{\lambda}\|_{p}^{p}},$

we have $I(\Psi(y))<\eta m^{\infty}_{c}$.

Proof. Since $I_{\infty}$ is translation invariant, we have
$I_{\infty}(\Psi(y))=I_{\infty}(w_{\lambda})=m^{\infty}_{c}$, and

$\displaystyle I(\Psi(y))$
$\displaystyle=I_{\infty}(\Psi(y))+I(\Psi(y))-I_{\infty}(\Psi(y))$
$\displaystyle\leq
m^{\infty}_{c}+\int_{\mathbb{R}^{N}}(1-a(x))|\Psi(y)|^{p}dx$
$\displaystyle\leq
m^{\infty}_{c}+\sup_{\mathbb{R}^{N}}|1-a(x)|\|\Psi(y)\|_{p}^{p}$
$\displaystyle<m^{\infty}_{c}+\frac{(\eta-1)m^{\infty}_{c}}{\|w_{\lambda}\|_{p}^{p}}\|w_{\lambda}\|_{p}^{p}$
$\displaystyle=\eta m^{\infty}_{c}.\ \ \ \ \ \ \ \ \Box$

Definition 3.4.([16]) Define the barycenter function of a given function $u\in
H^{s}(\mathbb{R}^{N})\setminus\\{0\\}$ as follows: let

$\mu(u)(x)=\frac{1}{|B_{1}|}\int_{B_{1}(x)}|u(y)|dy,$

we have $\mu(u)\in L^{\infty}(\mathbb{R}^{N})$ and $\mu$ is a continuous
function. Subsequently, let

$\hat{u}(x)=\left[\mu(u)(x)-\frac{1}{2}\max\mu(u)\right]^{+},$

it follows that $\hat{u}\in C_{0}(\mathbb{R}^{N})$. Now define the barycenter
of $u$ by

$\beta(u)=\frac{1}{\|\hat{u}\|_{1}}\int_{\mathbb{R}^{N}}x\hat{u}(x)dx\in\mathbb{R}^{N}.$

Since $\hat{u}$ has compact support, by the definition, $\beta(u)$ is well
defined. The function $\beta$ satisfies the following properties:

$(a)$ $\beta$ is a continuous function in
$H^{s}(\mathbb{R}^{N})\backslash\\{0\\}$.

$(b)$ If $u$ is radial, then $\beta(u)=0$.

$(c)$ For any $y\in\mathbb{R}^{N}$ and $u\in
H^{s}(\mathbb{R}^{N})\setminus\\{0\\}$, we have $\beta(u(x-y))=\beta(u(x))+y$.

Let

$P=\\{u\in H^{s}(\mathbb{R}^{N}):u\geq 0\\},$

now we define

$b_{c}:=\inf\\{I(u):u\in S_{c}\cap P,\ \beta(u)=0\\}.$

It is clear that $b_{c}\geqslant m_{c}^{\infty}$. Moreover, we have the
following lemma.

Lemma 3.5. $b_{c}>m_{c}^{\infty}$.

Proof. By the definition of $b_{c}$, it is clear that $b_{c}\geqslant
m_{c}^{\infty}$. Suppose, by contradiction, that $b_{c}=m_{c}^{\infty}$. There
exists a sequence $\\{u_{n}\\}\subset S_{c}\cap P$ such that $\beta(u_{n})=0$
for any $n\geq 1$ and $I(u_{n})\rightarrow m_{c}^{\infty}$ as
$n\rightarrow\infty$. From $a(x)\leq 1$, we have $I_{\infty}(u_{n})\rightarrow
m_{c}^{\infty}$ as $n\rightarrow\infty$. By Lemma 2.1, up to a subsequence,
there exists a sequence $\\{x_{n}\\}\subset\mathbb{R}^{N}$ such that

$u_{n}(\cdot-x_{n})\to w_{\lambda}\ \mbox{ strongly in
}H^{s}(\mathbb{R}^{N}),$ (3.13)

where $\lambda<0$ is determined by (2.1).

Since $w_{\lambda}$ is radial, by the properties of $\beta$, we have

$x_{n}=\beta(u_{n})+x_{n}=\beta(u_{n}(\cdot-x_{n}))\to\beta(w_{\lambda})=0\
\mbox{as}\ n\to\infty.$

Combining with (3.13), it follows that

$u_{n}\to w_{\lambda}\ \mbox{ strongly in }H^{s}(\mathbb{R}^{N}).$

Since $I(u_{n})\rightarrow m_{c}=m_{c}^{\infty}$ as $n\rightarrow\infty$, we
have

$I(w_{\lambda})=m_{c},$

which is a contradiction with that $m_{c}$ can not be attained. Therefore,
$b_{c}>m_{c}^{\infty}$. $\Box$

Proof of Theorem 1.2. Under assumption $(A_{1})$, for all $u\in
H^{s}(\mathbb{R}^{N})\backslash\\{0\\}$, we get $I_{\infty}(u)<I(u)$. In
particular, $I_{\infty}(\Psi(y))<I(\Psi(y))$ for any $y\in\mathbb{R}^{N}$. It
follows from Lemma 3.5 and $I(\Psi(y))\rightarrow m^{\infty}_{c}$ as
$|y|\rightarrow\infty$, there exists $\bar{\rho}>0$ such that

$m^{\infty}_{c}<\max\limits_{|y|=\bar{\rho}}I(\Psi(y))<\frac{b_{c}+m_{c}^{\infty}}{2}.$
(3.14)

Now let us define

$Q:=\Psi(\overline{B_{\bar{\rho}}(0)}),\ \ \ \ S:=\\{u\in
H^{s}(\mathbb{R}^{N}):u\in S_{c}\cap P,\ \beta(u)=0\\},$ $\mathcal{H}=\\{h\in
C(S_{c}\cap P,S_{c}\cap P):h(u)=u,\ \mbox{if}\
I(u)<\frac{b_{c}+m_{c}^{\infty}}{2}\\},$

we first claim $h(Q)\cap S\neq\emptyset$ for any $h\in\mathcal{H}$. Indeed,
given $h\in\mathcal{H}$, we define
$T:\overline{B_{\bar{\rho}}(0)}\rightarrow\mathbb{R}^{N}$ for $T(y)=\beta\circ
h\circ\Psi(y)$. Because $T$ is the composition of continuous functions, then
it is continuous. Note that for any $|y|=\bar{\rho}$, we have
$\Psi(y)\in\partial Q$. Hence from (3.14) , we obtain $h\circ\Psi(y)=\Psi(y)$
and then $T(y)=y$. By the Fixed Point Theorem of Brouwer, we obtain that there
exists $\tilde{y}\in B_{\bar{\rho}}(0)$ such that $T(\tilde{y})=0$, which
implies $h(\Psi(\tilde{y}))\in S$. Hence, $h(Q)\cap S\neq\emptyset$.

Now define

$d=\inf_{h\in\mathcal{H}}\max_{u\in Q}I(h(u)),$

we have $d\in(m^{\infty}_{c},\eta m^{\infty}_{c})$. From (3.14) and
$\frac{b_{c}+m_{c}^{\infty}}{2}<b_{c}$, we have

$b_{c}=\inf_{S}I>\max_{\partial Q}I>m^{\infty}_{c}.$ (3.15)

Since $h(Q)\cap S\neq\emptyset$ for any $h\in\mathcal{H}$, let
$h\in\mathcal{H}$ be fixed, there exists $\omega\in S$ such that $\omega$ also
belongs to $h(Q)$, which means that $\omega=h(v)$ for some
$v\in\Psi(\overline{B_{\bar{\rho}}(0)})$. Hence, we have

$I(\omega)\geqslant\inf_{u\in S}I(u)\ \ \text{ and }\ \ \max_{u\in
Q}I(h(u))\geqslant I(h(v)).$

This implies

$\max_{u\in Q}I(h(u))\geqslant I(h(v))=I(\omega)\geqslant\inf_{u\in
S}I(u)=b_{c},$

and then

$d=\inf_{h\in\mathcal{H}}\max_{u\in Q}I(h(u))\geqslant b_{c},$

it follows from (3.15) that $d>m^{\infty}_{c}$. In particular, let
$h=id|_{\mathbb{R}^{N}}$, according to Lemma 3.3, we have

$\inf_{h\in\mathcal{H}}\max_{u\in Q}I(h(u))\leq\max_{u\in Q}I(u)<\eta
m^{\infty}_{c},$

which implies that $d<\eta m^{\infty}_{c}$. Hence, we obtain
$d\in(m^{\infty}_{c},\eta m^{\infty}_{c})$.

At last, we claim for any $\mu\in(0,d-\frac{b_{c}+m_{c}^{\infty}}{2})$, there
is $u_{\mu}\in I^{-1}(d-\mu,d+\mu)\cap S_{c}\cap P$ such that

$\|I^{\prime}(u_{\mu})\|_{*}<\mu,$

where

$\|I^{\prime}(u)\|_{*}=\sup\\{|I^{\prime}(u)v|:v\in T_{u}S_{c}\mbox{ and
}\|v\|\leq 1\\},\ \ \forall\ u\in S_{c}.$

Arguing by contradiction, suppose that there is $\mu_{0}>0$ with
$\mu_{0}<d-\frac{b_{c}+m_{c}^{\infty}}{2}$ such that

$\|I^{\prime}(u)\|_{*}\geq\frac{\mu_{0}}{2},\ \ \forall u\in
I^{-1}(d-\mu_{0},d+\mu_{0})\cap S_{c}\cap P.$

Applying a deformation lemma (see Lemma 5.15 in [22]), there exists $\tau\in
C([0,1]\times S_{c}\cap P,S_{c}\cap P)$ such that

$\tau(t,u)=u,\ \mbox{ if }u\not\in I^{-1}(d-\mu_{0},d+\mu_{0})$

and

$\tau(1,I^{d+\frac{\mu_{0}}{2}})\subset I^{d-\frac{\mu_{0}}{2}},$ (3.16)

where $I^{\gamma}=\\{u\in S_{c}\cap P:I(u)\leq\gamma\\}$.

From the definition of $d$, there is $h\in\mathcal{H}$ such that

$\max_{u\in Q}I(h(u))\leq d+\frac{\mu_{0}}{2},$

that is, $h(Q)\subset I^{d+\frac{\mu_{0}}{2}}$. Hence, from (3.16), we have

$\tau(1,h(Q))\subset I^{d-\frac{\mu_{0}}{2}},$

which implies that

$\max_{u\in\tau(1,h(Q))}I(u)\leq d-\frac{\mu_{0}}{2}.$ (3.17)

On the other hand, we define

$\tilde{h}:=\tau(1,\cdot)\circ h\in C(S_{c}\cap P,S_{c}\cap P).$

Note that if $I(u)<\frac{b_{c}+m_{c}^{\infty}}{2}<d-\mu_{0}$, we have

$h(u)=u\ \ \ \mbox{and}\ \ \ \tau(1,u)=u,$

thus, $\tilde{h}\in\mathcal{H}$ and $\tau(1,h(Q))=\tilde{h}(Q)$. By the
definition of $d$, we have

$d\leq\max_{u\in\tau(1,h(Q))}I(u),$

which is a contradiction with (3.17).

Now there is $u_{n}\in I^{-1}(d-\mu_{n},d+\mu_{n})\cap S_{c}\cap P$ with
$\mu_{n}\to 0$ such that

$\|I^{\prime}(u_{n})\|_{*}<\mu_{n},\ \ \forall n\in\mathbb{N}$

that is, $\\{u_{n}\\}$ is a non-negative Palais-Smale sequence for
$I|_{S_{c}}$ at level $d$. It follows from Lemma $3.2$ that $\\{u_{n}\\}$ has
a convergent subsequence in $H^{s}(\mathbb{R}^{N})$. Hence problem (1.1) has a
positive solution $u\in S_{c}$ for some $\lambda<0$. $\Box$

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