# Orbital stability of standing waves of a class of fractional Schrödinger
equations with a general Hartree-type integrand

Y. Cho  Department of Mathematics, and Institute of Pure and Applied
Mathematics, Chonbuk National University, Jeonju 561-756, South Korea.  M.M.
Fall  African Institute for Mathematical Sciences of Senegal, AIMS-Senegal, KM
2, Route de Joal, B.P. 14 18. Mbour, Sénégal.  H. Hajaiej  Department of
Mathematics, College of Science, King Saud University P.O. Box 2455, Riyadh
11451, Saudi Arabia.  P.A. Markowich  Division of Math & Computer Sc & Eng,
King Abdullah University of Science and Technology Thuwal 23955-6900, Saudi
Arabia.  S. Trabelsi

###### Abstract

This article is concerned with the mathematical analysis of a class of a
nonlinear fractional Schrödinger equations with a general Hartree-type
integrand. We prove existence and uniqueness of global-in-time solutions to
the associated Cauchy problem. Under suitable assumptions, we also prove the
existence of standing waves using the method of concentration-compactness by
studying the associated constrained minimization problem. Finally we show the
orbital stability of standing waves which are the minimizers of the associate
variational problem.

###### keywords:

Fractional Schrödinger equation, Hartree type nonlinearity, standing waves,
orbital stability

##  1 Introduction

A partial differential equation is called fractional when it involves
derivatives or integrals of fractional order. Various physical phenomena and
applications require the use of fractional derivatives, for instance quantum
mechanics, pseudo-chaotic dynamics, dynamics in porous media, kinetic theories
of systems with chaotic dynamics. The latter application is based on the so
called fractional Schrödinger equation. This equation was derived using the
path integral over a kind of Lévy quantum mechanical path approach by Laskin
in Ref. [14, 15, 16]. The mathematical analysis of the fractional nonlinear
Schrödinger equation has been growing continually during the last few decades.
Many results have been obtained and we refer for instance to [5] and
references therein. This paper deals with the analysis of the following Cauchy
problem

$\displaystyle\mathscr{S}:\quad\left\\{\begin{array}[]{l}i\partial_{t}\phi+(-\Delta)^{s}\phi=\left(G(|\phi|)\star
V(|x|)\right)G^{\prime}(\phi),\\\ \\\ \phi(t=0,x)=\phi_{0}.\end{array}\right.$

In the system $\mathscr{S}$, $\phi(t,x)$ is a complex-valued function on
$\mathbb{R}\times{\mathbb{R}^{N}}$ and $\phi_{0}$ is a prescribed initial data
in $H^{s}(\mathbb{R}^{N})$. The operator $(-\Delta)^{s}$ denotes the
fractional Laplacian of power $0<s<1$. It is defined as a pseudo-differential
operator
$\mathcal{F}[(-\Delta)^{s}\,\phi](\xi)=|\xi|^{2s}\,\mathcal{F}[\phi](\xi)$
with $\mathcal{F}$ being the Fourier transform. The symbol $\star$ denotes the
convolution operator in ${\mathbb{R}^{N}}$ with the potential
$V(|x|)=|x|^{\beta-N}$ where $\beta>0$ is such that $\beta>N-2s$. The function
$G$ is a differentiable function from
$\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$,
$G^{\prime}(\phi):=\frac{dG}{d\phi}:=F(|\phi|)\phi$, where
$F:\mathbb{R}\rightarrow\mathbb{R}$. The above Cauchy problem reduces to the
massless boson Schrödinger equation in three dimensions when
$G(\phi)=|\phi|^{2}$, $V(|x|)=|x|^{-1}$ and $s=\frac{1}{2}$. In this case,
standing waves of the system $\mathscr{S}$, i.e. solutions of the form
$\phi(t,x)=u(x)e^{-i\kappa t}$, satisfy the following semilinear partial
differential equation

$(-\Delta)^{1/2}u-(|x|^{-1}\ast\;u^{2})u+\kappa u=0.$ (1)

The associated variational problem

$\displaystyle\mathcal{I}_{\lambda}$
$\displaystyle=\inf\left\\{{|\\!|}|\xi|^{\frac{1}{2}}\mathcal{F}[{u}](\xi)\|^{2}_{L^{2}({\mathbb{R}^{N}})}-\int_{\mathbb{R}^{N}\times{\mathbb{R}^{N}}}\frac{|u(x)|^{2}|u(y)|^{2}}{|x-y|}dxdy,\right.$
$\displaystyle\hskip 150.0pt\left.u\in
H^{\frac{1}{2}}({\mathbb{R}^{N}}),\>\int_{{\mathbb{R}^{N}}}|u(x)|^{2}\,dx=\lambda\right\\},$
(2)

has played a fundamental role in the mathematical theory of gravitational
collapse of boson stars, [18]. In Ref. [12], the authors studied the
associated variational problem

$\displaystyle\mathcal{I}^{G}_{\lambda}$
$\displaystyle=\inf\left\\{{|\\!|}|\xi|^{s}\mathcal{F}[{u}](\xi)\|^{2}_{L^{2}({\mathbb{R}^{N}})}-\int_{\mathbb{R}^{N}\times{\mathbb{R}^{N}}}G(u(x))V(|x-y|)G(u(y))dxdy,\right.$
$\displaystyle\hskip 170.0pt\left.u\in
H^{s}({\mathbb{R}^{N}}),\>\int_{{\mathbb{R}^{N}}}|u(x)|^{2}\,dx=\lambda\right\\},$
(3)

for a general nonlinearity $G$, a kernel $V(|x|)=|x|^{\beta-N}$ and dimension
$N$, where here and the following

$H^{s}(\mathbb{R}^{N}):=\\{u\in
L^{2}({\mathbb{R}^{N}}^{N}),\>\,{|\\!|}|\xi|^{s}\mathcal{F}[{u}](\xi)\|^{2}_{L^{2}({\mathbb{R}^{N}})}<\infty\\}.$

In the critical case $2s={N-\beta}{}$, they were able to extend the results of
[18]. Moreover, in the subcritical $2s>{N-\beta}$, they have also proved the
existence and symmetry of all minimizers of (3) by using rearrangement
techniques. More precisely, they showed that under suitable assumptions on
$G$, one can always take a radial and radial by decreasing minimizing sequence
of problem (3). Another very important issue related to the nonlinear
fractional Schrödinger equation $\mathscr{S}$ is the orbital stability of
standing waves. For such an issue, it is essential to show that all the
minimizing sequences are relatively compact in $H^{s}(\mathbb{R}^{N})$. This
is the gist of the breakthrough paper [4]. The line of attach consists of:

  1. 1.

Prove the uniqueness of the solutions of $\mathscr{S}$.

  2. 2.

Prove the conservation of energy and mass of the solutions.

  3. 3.

Prove the relative compactness of all minimizing sequences of the problem (3).

Our first result concerns the well-posedness of the system $\mathscr{S}$.
Before stating it, we need to fix some conditions on $G$. We assume that $G$
is nonnegative and differentiable such that $G(0)=0$ and for all
$\psi\in\mathbb{R}_{+}$

$\mathcal{A}_{0}:\exists\,\mu\in\left[\left.2,1+\frac{2s+\beta}{N}\right)\right.\>\text{s.t.}\>\quad\left\\{\begin{array}[]{ll}&G(\psi)\leq\eta(|\psi|^{2}+|\psi|^{\mu}),\\\
&\\\ &|G^{\prime}(\psi)|\leq\eta(|\psi|+|\psi|^{\mu-1}).\end{array}\right.$

We have obtained the following

######  Theorem 1.1.

Let $N\geq 1,0<s<1,0<\beta<N,N-2s\leq\beta,\phi_{0}\in
H^{s}({\mathbb{R}^{N}})$ and $G$ such that $\mathcal{A}_{0}$ holds true. Then,
there exists a weak global-in-time solution $\phi(t,x)$ to the system
$\mathscr{S}$ such that

$\phi\in L^{\infty}(\mathbb{R}\,;\,H^{s}({\mathbb{R}^{N}}))\cap
W^{1,\infty}(\mathbb{R}\,;\,H^{-s}({\mathbb{R}^{N}})).$

Moreover, if $N=1$ and $\frac{1}{2}<s<1$ or if $N\geq 3$,
$\frac{N}{2(N-1)}<s<1$,
$N-s+\frac{1}{2}<\beta<\min(N,\frac{3N}{2}-s-\frac{N}{4s})$ and $\mu$ (in
$\mathcal{A}_{0}$) is such that

$\max\left(2,1+\frac{2\beta-N}{N-2s}\right)<\mu<2+\frac{N}{N-2s}\frac{2s-1-2N+2\beta}{2s-1+N},$

then the solution is unique.

The particular case $\mu=2$ and $2s=N-\beta$ was treated in Ref. [5] and for
lightness of the proofs, we shall sometimes omit it and focus on the case
$\mu\in\left(2,1+\frac{2s+\beta}{N}\right)$. The proof of the existence part
of Theorem 1.1 is based on a classical contraction argument and the
conservation laws associated to the dynamics of the system $\mathscr{S}$. The
uniqueness part for $N=1$ of Theorem 1.1 readily follows from the embedding
$H^{s}\hookrightarrow L^{\infty}$ for all $s>\frac{1}{2}$. The part for $N\geq
3$ is obtained using mixed norms to be defined later and weighted Strichartz
and convolution inequalities, which require $N\geq 3$. It would be very
interesting to find estimates to handle the uniqueness for $N=2$. Let us
mention that in Ref. [9] the authors showed the orbital stability of standing
waves in the case of power nonlinearities by assuming energy conservation and
time continuity without proving uniqueness, which is an inescapable and quite
hard step, especially in the fractional setting. As mentioned before, if
$\phi(t,x)=e^{i\kappa t}u(x)$ with $\kappa\in\mathbb{R}$ is a solution of the
system $\mathscr{S}$, then it is called a standing wave solution and $u(x)$
solves the following bifurcation problem

$\tilde{\mathscr{S}}:\quad(-\Delta)^{s}u-\kappa u=\left(G(|u|)\star
V(|x|)\right)G^{\prime}(u).$

In order to study the existence of a solution $(\kappa,u)$ to the stationary
equation $\tilde{\mathscr{S}}$, we use a variational method based on the
following minimization problem

$\mathcal{I}_{\lambda}=\inf\left\\{\mathcal{E}(u),\quad u\in
H^{s}(\mathbb{R}^{N}),\quad\int_{\mathbb{R}^{N}}|u(x)|^{2}\,dx=\lambda\right\\},$
(4)

where $\lambda$ is a positive prescribed number and

$\displaystyle\mathcal{E}(u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}{|\\!|}\nabla_{s}u{|\\!|}^{2}_{L^{2}({\mathbb{R}^{N}})}-\frac{1}{2}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}G(|u(x)|)\,V(|x-y|)\,G(|u(y)|)\,dxdy,$
$\displaystyle:=$
$\displaystyle\frac{1}{2}{|\\!|}\nabla_{s}u{|\\!|}^{2}_{L^{2}({\mathbb{R}^{N}})}-\frac{1}{2}\,\mathcal{D}(G(|u|),G(|u|)).$

The kinetic energy is precisely expressed by the formula for all function $u$
in the Schwarz class

$\|\nabla_{s}u\|^{2}_{L^{2}({\mathbb{R}^{N}})}=C_{N,s}\int_{\mathbb{R}^{N}\times{\mathbb{R}^{N}}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}dxdy,$
(5)

with $C_{N,s}$ being a positive normalization constant. In order to prove the
existence of critical points to the functional $\mathcal{E}$ and thereby
solutions to the problem $\tilde{\mathscr{S}}$, we will need some extra grows
condition on $G$: for all $\psi\in\mathbb{R}_{+}$

$\mathcal{A}_{1}:\quad\left\\{\begin{array}[]{l}\exists
0<\alpha<1+\frac{2s+\beta}{N}\>\>s.t.\>\>\forall\psi,\>0<\psi\ll 1,\quad
G(\psi)\geq\eta\,\psi^{\alpha},\\\ \\\
G(\theta\,\psi)\geq\theta^{1+\frac{2s+\beta}{2N}}\,G(\psi).\end{array}\right.$

Our next main result is contained in the following

######  Theorem 1.2.

Let $0<s<1,0<\beta<N,N-\beta\leq 2s$ and $G$ such that $\mathcal{A}_{0}$ and
$\mathcal{A}_{1}$ hold true. Then, for all $\lambda>0$, problem (4) has a
minimizer $u_{\lambda}\in H^{s}({\mathbb{R}^{N}})$ such that
$I_{\lambda}=\mathcal{E}(u_{\lambda})$.

In fact we will show that any minimizing sequence of problem 4 is –up to
suitable translations– relatively compact in $H^{s}({\mathbb{R}^{N}})$. The
proof of Theorem 1.2 is based on the concentration-compactness method of P-L.
Lions [17]. The last part of the paper deals with the stability of the
standing waves. For that purpose, we introduce the following problem

$\hat{\mathcal{I}}_{\lambda}=\inf\left\\{\mathcal{J}(z),\quad z\in
H^{s}(\mathbb{R}^{N}),\quad\int_{\mathbb{R}^{N}}|z|^{2}\,dx=\lambda\right\\},$

where $z=u+i\,v$ and

$\displaystyle\mathcal{J}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{2}{|\\!|}\nabla_{s}z{|\\!|}^{2}_{L^{2}({\mathbb{R}^{N}})}-\frac{1}{2}\,\mathcal{D}(G(|z(x)|),G(|z(x)|)),$
$\displaystyle=$
$\displaystyle\frac{1}{2}{|\\!|}\nabla_{s}u{|\\!|}^{2}_{L^{2}({\mathbb{R}^{N}})}+\frac{1}{2}{|\\!|}\nabla_{s}v{|\\!|}^{2}_{L^{2}({\mathbb{R}^{N}})}-\frac{1}{2}\,\mathcal{D}(G((u^{2}+v^{2})^{\frac{1}{2}}),G((u^{2}+v^{2})^{\frac{1}{2}})),$
$\displaystyle:=$ $\displaystyle\mathcal{J}(u,v).$

We have obviously $\mathcal{E}(u)=\mathcal{J}(u,0)$. Following Ref. [4], we
introduce the following set

$\hat{\mathcal{O}}_{\lambda}=\left\\{z\in
H^{s}(\mathbb{R}^{N}),\quad\int_{\mathbb{R}^{N}}|z|^{2}\,dx=\lambda\>\>:\>\>\mathcal{J}(z)=\hat{\mathcal{I}}_{\lambda}\right\\}.$

The set $\hat{\mathcal{O}}_{\lambda}$ is the so called orbit of the standing
waves of $\mathscr{S}$ with mass $\sqrt{\lambda}$. We define the stability of
$\hat{\mathcal{O}}_{\lambda}$ as follows

######  Definition 1.3.

Let $\phi_{0}\in H^{s}({\mathbb{R}^{N}})$ be an initial data and $\phi(t,x)\in
H^{s}({\mathbb{R}^{N}})$ the associated solution of problem $\mathscr{S}$. We
say that $\hat{\mathcal{O}}_{\lambda}$ is $H^{s}({\mathbb{R}^{N}})-$stable
with respect to the system $\mathscr{S}$ if

  * 1.

$\hat{\mathcal{O}}_{\lambda}\neq\varnothing$.

  * 2.

For all $\varepsilon>0$, there exists $\delta>0$ such that for any
$\phi_{0}\in H^{s}({\mathbb{R}^{N}})$ satisfying
$\inf_{z\in\hat{\mathcal{O}}_{\lambda}}|\phi_{0}-z|<\delta$, we have
$\inf_{z\in\hat{\mathcal{O}}_{\lambda}}|\phi(t,x)-z|<\epsilon$ for all
$t\in\mathbb{R}$.

The notion of stability depends then intimately on the well-posedness of the
Cauchy problem $\mathscr{S}$ and the existence of standing waves. Therefore,
having in hand Theorems 1.1 and 1.2, we prove the following

######  Theorem 1.4.

Let $N\geq 3$, $\frac{N}{2(N-1)}<s<1$,
$N-s+\frac{1}{2}<\beta<\min(N,\frac{3N}{2}-s-\frac{N}{4s})$ and let $G$
satisfying $\mathcal{A}_{0}$ and $\mathcal{A}_{1}$ with $\mu$ (in
$\mathcal{A}_{0}$) such that

$\max\left(2,1+\frac{2\beta-N}{N-2s}\right)<\mu<2+\frac{N}{N-2s}\,\frac{2s-1-2N+2\beta}{2s-1+N}.$

Let $\phi_{0}\in H^{s}({\mathbb{R}^{N}})$ and $\phi(t,x)\in
H^{s}({\mathbb{R}^{N}})$ the associated solution to the problem $\mathscr{S}$.
Then $\hat{\mathcal{O}}_{\lambda}$ is $H^{s}({\mathbb{R}^{N}})-$stable with
respect to the system $\mathscr{S}$.

The paper is divided into three sections. The first one is dedicated to the
analysis of the dynamics of the system $\mathscr{S}$. More precisely, in this
section we prove Theorem 1.1. First of all, we prove a local-in-time existence
of solutions. Second we show that under extra assumptions, this solution is
actually unique. Eventually, we use the conservation laws to show the global-
in-time well-posedness. The second section is devoted to the proof of
existence of solution to the problem $\tilde{\mathscr{S}}$. For that purpose,
we use the classical concentration compactness method [17] to prove Theorem
1.2. The last section is dedicated to the proof of stability of standing
waves, namely Theorem 1.4. Here, we use ideas and techniques developed in
[13]. From this point onward, $\eta$ will denote variant universal constants
that may change from line to line of inequalities. When $\eta$ depends on some
parameter, we will write $\eta(\cdot)$ instead of $\eta$. In order to lighten
the notation and the calculation, we shall use $L^{p}$ and $H^{s}$ instead of
$L^{p}({\mathbb{R}^{N}})$ and $H^{s}({\mathbb{R}^{N}})$ respectively for real
or complex valued functions. Also, we shall use ${|\\!|}\cdot{|\\!|}_{p}$
instead of ${|\\!|}\cdot{|\\!|}_{L^{p}({\mathbb{R}^{N}})}$ for all
$p\in[1,\infty]$. The exponent $p^{\prime}$ will denotes the conjugate
exponent of $p$, that is $\frac{1}{p}+\frac{1}{p^{\prime}}=1$. For a more
detailed account about the Sobolev spaces $H^{s}$, we refer the reader to any
textbook of functional analysis (see [3] for instance).

##  2 Well-posedness of the system $\mathscr{S}$

In this section we consider the local and global well-posedness of the problem
$\mathscr{S}$ and prove Theorem 1.1. Let us denote the nonlinear term
$[V(|x|)\star G(\phi)]G^{\prime}(\phi)$ by $\mathcal{N}(\phi)$. Since the
well-posedness of the case $\mu=2,2s=N-\beta$ was treated in [5], in this
paper we consider the initial value problem $\mathscr{S}$ with
$\mu\in\left(2,1+\frac{2s+\beta}{N}\right)$. Let $g=G^{\prime}$, that is,
$\int_{0}^{|z|}g(\alpha)\,d\alpha=G(z)$, and assume that
$g(z)=\frac{z}{|z|}g(|z|),z\neq 0$, $G(z)\geq 0$. Then, with
$\mathcal{A}_{0}$, the function $g$ satisfies obviously

$\displaystyle|g(z)|+|g^{\prime}(z)z|\leq C(|z|+|z|^{\mu-1})\;\;\mbox{for
all}\;\;z\in\mathbb{C}.$ (6)

###  2.1 Weak solutions

We first show existence of weak solutions to $\mathscr{S}$ in $H^{s}$. For
this purpose we prove that $\mathcal{N}$ is Lipschitz map from
$L^{p^{\prime}}$ to $L^{r}$ for some
$p,r\in\left.\left[2,\frac{2N}{N-2s}\right)\right.$. Then the rest of the
proof is quite straightforward from the Lipschitz map and well-known
regularizing arguments and we refer the readers to the book [3].

######  Proposition 2.1.

Let $N\geq 2$, $0<s<1$, $0<\beta<N$ and $2s\geq N-\beta$. If $g$ satisfies (6)
with $\mu\in\left(2,1+\frac{2s+\beta}{N}\right)$. Then there exists a weak
solution $\phi$ such that

$\displaystyle\phi\in L^{\infty}(-T_{min},T_{max};H^{s})\cap
W^{1,\infty}(-T_{min},T_{max};H^{-s}),$
$\displaystyle{|\\!|}\phi(t){|\\!|}_{2}={|\\!|}\phi_{0}{|\\!|}_{2},\;\;\mathcal{J}(\phi(t))\leq\mathcal{J}(\phi_{0}).$

for all $t\in(-T_{min},T_{max})$, where $(-T_{min},T_{max})$ is the maximal
existence time interval of $\phi$ for given initial data $\phi_{0}$.

###### Proof.

Let us introduce the following cut-off for the function $g$,
$g_{1}(\alpha)=\chi_{\\{0\leq\alpha<1\\}}g(\alpha)$ and
$g_{2}(\alpha)=\chi_{\\{\alpha\geq 1\\}}g(\alpha)$ and
$G_{i}(z)=\int_{0}^{|z|}g_{i}(\alpha)\,d\alpha$ with obvious definition of the
Euler function $\chi$. Then, one can writes

$\mathcal{N}(\phi)=\sum_{i,j=1,2}\mathcal{N}_{ij}(\phi)\>\>\text{where}\>\>\mathcal{N}_{ij}(\phi)=\int_{{\mathbb{R}^{N}}}|x-y|^{-(N-\beta)}G_{i}(|\phi|)\,dy\,g_{j}(\phi).$

We claim that there exist
$p_{ij},r_{ij}\in\left[\left.2,\frac{2N}{N-2s}\right)\right.$111If $N=1$ and
$\frac{1}{2}\leq s<1$, then $\frac{2N}{N-2s}$ is interpreted as $\infty$. such
that

$\displaystyle{|\\!|}\mathcal{N}_{ij}(\phi)-\mathcal{N}_{ij}(\psi){|\\!|}_{p^{\prime}_{ij}}\leq\eta(K){|\\!|}\phi-\psi{|\\!|}_{r_{ij}},$
(7)

for some constant $\eta(K)$ with $\eta(K)\leq\eta\,K^{a_{i\\!j}}$,
$a_{i\\!j}>0$ for all $1\leq i,j\leq 2$, provided
${|\\!|}\phi{|\\!|}_{H^{s}}+{|\\!|}\psi{|\\!|}_{H^{s}}\leq K$. This implies
that $\mathcal{N}:H^{s}\to H^{-s}$ is a Lipschitz map on a bounded sets of
$H^{s}$. Indeed, let $\mu_{1}=2$ and $\mu_{2}=\mu$. Then we have

$\displaystyle|\mathcal{N}_{ij}(\phi)-\mathcal{N}_{ij}(\psi)|$
$\displaystyle\leq\eta\int_{{\mathbb{R}^{N}}}|x-y|^{-(N-\beta)}(|\phi|^{\mu_{i}-1}+|\psi|^{\mu_{i}-1})|\phi-\psi|\,dy|\phi|^{\mu_{j}-1}$
$\displaystyle+\eta\int_{{\mathbb{R}^{N}}}|x-y|^{-(N-\beta)}|\psi|^{\mu_{i}}\,dy(|\phi|^{\mu_{j}-2}+|\psi|^{\mu_{j}-2})|\phi-\psi|.$

By Hölder’s and Hardy-Littlewood-Sobolev inequalities with indices
$p_{ij},r_{ij}$ such that

$\displaystyle
1-\frac{1}{p_{ij}}=\frac{\mu_{i}}{r_{ij}}-\frac{\beta}{N}+\frac{\mu_{j}-1}{r_{ij}},\;\;\frac{\mu_{i}}{r_{ij}}>\frac{\beta}{N},$
(8)

we obtain

$\displaystyle{|\\!|}\mathcal{N}_{ij}(\phi)-\mathcal{N}_{ij}(\psi){|\\!|}_{p^{\prime}_{ij}}$
$\displaystyle\leq\eta\,\left[({|\\!|}\phi{|\\!|}_{r_{ij}}^{\mu_{i}-1}+{|\\!|}\psi{|\\!|}_{r_{ij}}^{\mu_{i}-1}){|\\!|}\phi{|\\!|}_{r_{ij}}^{\mu_{j}-1}\right.$
$\displaystyle\left.+{|\\!|}\psi{|\\!|}_{r_{ij}}^{\mu_{i}}({|\\!|}\phi{|\\!|}_{r_{ij}}^{\mu_{j}-2}+{|\\!|}\psi{|\\!|}_{r_{ij}}^{\mu_{j}-2})\right]{|\\!|}\phi-\psi{|\\!|}_{r_{ij}}.$

Thus if $p_{ij},r_{ij}\in\left.\left[2,\frac{2N}{N-2s}\right)\right.$, then
Sobolev inequality shows (7). Now we show that there exist
${p_{ij}},{r_{ij}}\in[2,\frac{2N}{N-2s})$ such that the combinations (8) hold
true. If ${p_{ij}},{r_{ij}}$ satisfy (8), then they are on the line

$\displaystyle\frac{1}{{r_{ij}}}=\frac{1}{\mu_{i}+\mu_{j}-1}(1+\frac{\beta}{N}-\frac{1}{{p_{ij}}}).$
(9)

Since $\frac{1}{\mu_{i}+\mu_{j}-1}(1+\frac{\beta}{N}-\frac{1}{2})<\frac{1}{2}$
and
$\frac{N-2s}{2N}<\frac{1}{\mu_{i}+\mu_{j}-1}(1+\frac{\beta}{N}-\frac{N-2s}{2N})$,
the line (9) of $(\frac{1}{{p_{ij}}},\frac{1}{{r_{ij}}})$ always passes
through the open square
$(\frac{N-2s}{2N},\frac{1}{2})\times(\frac{N-2s}{2N},\frac{1}{2})$. We have
only to find a pair $(\frac{1}{{p_{ij}}},\frac{1}{{r_{ij}}})$ of line (9) such
that $\frac{\mu_{i}}{{r_{ij}}}>\frac{\beta}{N}$. If
$\frac{\mu_{i}}{{r_{ij}}}>\frac{\beta}{N}$, then

$\frac{1}{{p_{ij}}}<1-\frac{\mu_{j}-1}{\mu_{i}}\frac{\beta}{N}.$

So, it suffices to show that

$\displaystyle\max\left(\frac{1}{p_{0}},\frac{N-2s}{2N}\right)<1-\frac{\mu_{j}-1}{\mu_{i}}\frac{\beta}{N},$
(10)

where $\frac{1}{p_{0}}$ is the point of line (9) when
$\frac{1}{{r_{ij}}}=\frac{1}{2}$, that is,
$\frac{1}{p_{0}}=1+\frac{\beta}{N}-\frac{\mu_{i}+\mu_{j}-1}{2}$. In fact, it
is an easy matter to show (10) from the condition
$\mu\in\left(2,1+\frac{\beta+2s}{N}\right)$ and we leave the proof to the
reader. The proof of Proposition 2.1 follows now by a straightforward
application of a contraction argument. ∎

###  2.2 Uniqueness

Since the case $N=1$ can be treated as in [3], we omit the details. When
$N\geq 3$, the uniqueness of weak solutions can be shown by a weighted
Strichartz and convolution estimates. For that purpose, we introduce the
following mixed norm for all $1\leq m,\widetilde{m}<\infty$

${|\\!|}h{|\\!|}_{L_{\rho}^{m}L_{\sigma}^{\widetilde{m}}}:=(\int_{0}^{\infty}(\int_{S^{N-1}}|h(\rho\sigma)|^{\widetilde{m}}\,d\sigma)^{\frac{m}{\widetilde{m}}}\,\rho^{n-1}d\rho)^{\frac{1}{m}}.$

The case $m=\infty$ or $\widetilde{m}=\infty$ can be defined is a usual way.
Then we have the following.

######  Proposition 2.2.

Let $N\geq 3$, $\frac{N}{2(N-1)}<s<1$,
$N-s+\frac{1}{2}<\beta<\min(N,\frac{3N}{2}-s-\frac{N}{4s})$, and $g$ such that
the condition (6) holds true with

$\max\left(2,1+\frac{2\beta-N}{N-2s}\right)<\mu<2+\frac{N}{N-2s}\,\frac{2s-1-2N+2\beta}{2s-1+N}.$

Then the $H^{s}$-weak solution to the problem $\mathscr{S}$ constructed in
proposition 2.1 is unique.

The dimension restriction $N\geq 3$ is necessary for $\frac{N}{2(N-1)}<s<1$
and $N-s+\frac{1}{2}<\beta<\frac{3N}{2}-s-\frac{N}{4s}$, which are needed for
the exponents appearing in (14).

###### Proof.

Let $U(t)=e^{it(-\Delta)^{s}}$, then the solution ${\phi}$ constructed in
Proposition 2.1 satisfies the integral equation

$\displaystyle{\phi}(t)=U(t)\varphi-i\int_{0}^{t}U(t-t^{\prime})\mathcal{N}({\phi}(t^{\prime}))\,dt^{\prime}\;\;\mbox{a.e.}\;t\in(-T_{min},T_{max}).$
(11)

Before going further, let us recall the following weighted Strichartz estimate
(see for instance Lemma 6.2 of [5] and Lemma 2 of [6]).

######  Lemma 2.3.

Let $N\geq 2$ and $2\leq q<4s$. Then, for all $\psi\in L^{2}$, we have

${|\\!|}|x|^{-\delta}U(t)\psi{|\\!|}_{L^{q}(-t_{1},t_{2};L_{\rho}^{q}L_{\sigma}^{\widetilde{q}})}\leq\eta\,{|\\!|}\psi{|\\!|}_{2},$

where $\delta=\frac{N+2s}{q}-\frac{N}{2}$,
$\frac{1}{\widetilde{q}}=\frac{1}{2}-\frac{1}{N-1}\left(\frac{2s}{q}-\frac{1}{2}\right)$
and $\eta$ is independent of $t_{1},t_{2}$.

In [5] it was shown that

${|\\!|}|x|^{-\delta}D_{\sigma}^{\frac{2s}{q}-\frac{1}{2}}U(t)\psi{|\\!|}_{L^{q}(-t_{1},t_{2};L_{\rho}^{q}L_{\sigma}^{2})}\leq\eta{|\\!|}\psi{|\\!|}_{2}.$

Lemma 2.3 can be derived by Sobolev embedding on the unit sphere. Here
$D_{\sigma}=\sqrt{1-\Delta_{\sigma}}$ where $\Delta_{\sigma}$ is the Laplace-
Beltrami operator on the unit sphere. Now, let us recall the following
weighted convolution inequality we shall use in the sequel

######  Lemma 2.4 (Lemma 4.3 of [7]).

Let $r\in[1,\infty]$ and $0\leq\delta\leq\gamma<N-1$. If
$\frac{1}{r}>\frac{\gamma}{N-1}$, then for all $f$ such that
$|x|^{-(\gamma-\delta)}f\in L^{1}$, we have

${|\\!|}|x|^{\delta}(|x|^{-\gamma}\ast
f){|\\!|}_{L_{\rho}^{\infty}L_{\sigma}^{r}}\leq\eta{|\\!|}|x|^{-(\gamma-\delta)}f{|\\!|}_{1}.$

Therefore, using Lemma 2.3 one can readily deduce that

$\displaystyle{|\\!|}|x|^{-\delta}\int_{0}^{t}U(t-t^{\prime})f(t^{\prime}){|\\!|}_{L^{q}(-t_{1},t_{2};L_{\rho}^{q}L_{\sigma}^{\widetilde{q}})}\leq\eta{|\\!|}f{|\\!|}_{L^{1}(-t_{1},t_{2};L^{2})}.$
(12)

Thus, if we set $f=\mathcal{N}({\phi})-\mathcal{N}({\psi})$ and
$\gamma=N-\beta$. Then from (11) we infer

$\displaystyle{|\\!|}{\phi}-{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};L^{2})}+{|\\!|}|x|^{-\delta}({\phi}-{\psi}){|\\!|}_{L^{q}(-t_{1},t_{2};L_{\rho}^{q}L_{\sigma}^{\widetilde{q}})}$
$\displaystyle\leq$
$\displaystyle\eta\sum_{i,j=1}^{2}\int_{-t_{1}}^{t_{2}}{|\\!|}\mathcal{N}_{ij}({\phi})-\mathcal{N}_{ij}({\psi}){|\\!|}_{2}\,dt^{\prime},$
$\displaystyle\leq$
$\displaystyle\eta\sum_{i,j=1}^{2}\int_{-t_{1}}^{t_{2}}{|\\!|}\int_{{\mathbb{R}^{N}}}|x-y|^{-\gamma}(|{\phi}|^{\mu_{i}-1}+|{\psi}|^{\mu_{i}-1})|{\phi}-{\psi}|\,dy|{\phi}|^{\mu_{j}-1}{|\\!|}_{2}\,dt^{\prime},$
$\displaystyle+$
$\displaystyle\eta\sum_{i,j=1}^{2}\int_{-t_{1}}^{t_{2}}{|\\!|}\int_{{\mathbb{R}^{N}}}|x-y|^{-\gamma}|{\psi}|^{\mu_{i}}\,dy(|{\phi}|^{\mu_{j}-2}+|{\psi}|^{\mu_{j}-2})|{\phi}-{\psi}|{|\\!|}_{2}\,dt^{\prime},$
$\displaystyle\equiv$
$\displaystyle\sum_{i,j=1}^{2}(\mathcal{T}^{1}_{ij}+\mathcal{T}^{2}_{ij}).$

We first estimate $\mathcal{T}^{1}_{ij}$ using Hölder’s and Hardy-Littlewood-
Sobolev inequalities. On the one side if $(i,j)=(1,2)$, since
$\mu\in\left(1+\frac{2\beta-N}{N-2s},1+\frac{\beta+2s}{N}\right)$, $0<\beta<N$
and $2s>\gamma=N-\beta$, we can find $r\in\left[2,\frac{2N}{N-2s}\right]$ such
that

$\frac{\beta}{N}=\frac{1}{r}+\frac{(\mu-1)(N-2s)}{2N},\;\;\frac{1}{r}+\frac{1}{2}>\frac{\beta}{N}.$

Thus, we can write

$\displaystyle\mathcal{T}^{1}_{12}$
$\displaystyle\leq\eta\int_{-t_{1}}^{t_{2}}({|\\!|}{\phi}{|\\!|}_{r}+{|\\!|}{\psi}{|\\!|}_{r}){|\\!|}{\phi}-{\psi}{|\\!|}_{2}|{\phi}|_{\frac{2N}{N-2s}}^{\mu-1}\,dt^{\prime},$
$\displaystyle\leq\eta(t_{1}+t_{2})({|\\!|}{\phi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};H^{s})}^{\mu}+{|\\!|}{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};H^{s})}^{\mu}){|\\!|}{\phi}-{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};L^{2})}.$

On the opposite side, if $(i,j)\neq(1,2)$, then we can choose
$r\in\left[2,\frac{2N}{N-2s}\right]$ such that

$\frac{\beta}{N}=\frac{\mu_{i}-1}{r}+\frac{(\mu_{j}-1)}{r},\;\;\frac{\mu_{i}-1}{r}+\frac{1}{2}>\frac{\beta}{N}.$

Such a combination is always possible thanks to our conditions on $\mu,\beta$
and $s$. Therefore, we get as above

$\displaystyle\mathcal{T}^{1}_{ij}$
$\displaystyle\leq\eta\int_{-t_{1}}^{t_{2}}({|\\!|}{\phi}{|\\!|}_{r}^{\mu_{i}-1}+{|\\!|}{\psi}{|\\!|}_{r}^{\mu_{i}-1}){|\\!|}{\phi}-{\psi}{|\\!|}_{2}{|\\!|}{\phi}{|\\!|}_{\frac{r}{\mu_{j}-1}}^{\mu_{j}-1}\,dt^{\prime},$
$\displaystyle\leq\eta(t_{1}+t_{2})({|\\!|}{\phi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};H^{s})}^{\mu_{i}+\mu_{j}-2}+{|\\!|}{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};H^{s})}^{\mu_{i}+\mu_{j}-2}){|\\!|}{\phi}-{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};L^{2})}.$

We are kept with the estimates of $\mathcal{T}^{2}_{ij}$. If $j=1$, then we
can use Hardy-Sobolev inequality such that for $0<q<N$ and $2\leq p<\infty$

$\displaystyle{|\\!|}|x|^{-\frac{q}{p}}f{|\\!|}_{p}\leq\eta{|\\!|}f{|\\!|}_{\dot{H}^{\frac{N}{2}-\frac{N-q}{p}}}.$
(13)

In fact, we have

$\displaystyle\mathcal{T}^{2}_{11}$
$\displaystyle\leq\int_{-t_{1}}^{t_{2}}{|\\!|}\int_{\mathbb{R}^{N}}|x-y|^{-\gamma}|{\psi}|^{2}\,dy{|\\!|}_{L_{x}^{\infty}}{|\\!|}{\phi}-{\psi}{|\\!|}_{2}\,dt^{\prime},$
$\displaystyle\leq\eta\int_{-t_{1}}^{t_{2}}{|\\!|}{\psi}{|\\!|}_{\dot{H}^{\frac{\gamma}{2}}}^{2}{|\\!|}{\phi}-{\psi}{|\\!|}_{2}\,dt^{\prime},$
$\displaystyle\leq\eta(t_{1}+t_{2}){|\\!|}{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};H^{s})}^{2}{|\\!|}{\phi}-{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};L^{2})}.$

Since $\frac{N}{2}-\frac{\beta}{\mu}\leq s$ we also have

$\displaystyle\mathcal{T}^{2}_{21}$
$\displaystyle\leq\int_{-t_{1}}^{t_{2}}{|\\!|}\int_{\mathbb{R}^{N}}|x-y|^{-\gamma}|{\psi}|^{\mu}\,dy{|\\!|}_{L_{x}^{\infty}}{|\\!|}{\phi}-{\psi}{|\\!|}_{2}\,dt^{\prime},$
$\displaystyle\leq
C\int_{-t_{1}}^{t_{2}}{|\\!|}{\psi}{|\\!|}_{\dot{H}^{\frac{N}{2}-\frac{\beta}{\mu}}}^{\mu}{|\\!|}{\phi}-{\psi}{|\\!|}_{2}\,dt^{\prime},$
$\displaystyle\leq
C(t_{1}+t_{2}){|\\!|}{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};H^{s})}^{2}{|\\!|}{\phi}-{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};L^{2})}.$

When $j=2$, we use the weighted convolution inequality (Lemma 2.4). The
hypothesis on $\beta,\mu$ guarantees the existence of exponents
$q,\widetilde{q}$ and $r$ satisfying the conditions of Lemmas 2.3, 2.4 and
also the following combination

$\displaystyle\frac{1}{2}=\frac{(\mu-2)(N-2s)}{2N}+\frac{1}{q}=\frac{1}{r}+\frac{(\mu-2)(N-2s)}{2N}+\frac{1}{\widetilde{q}}.$
(14)

Hence, using the Hardy-Sobolev inequality (13) we write

$\displaystyle\mathcal{T}^{2}_{i,2}$
$\displaystyle\leq\int_{-t_{1}}^{t_{2}}{|\\!|}|x|^{\delta}\int_{\mathbb{R}^{N}}|x-y|^{-\gamma}|{\psi}|^{\mu_{i}}\,dy{|\\!|}_{L_{\rho}^{\infty}L_{\sigma}^{r}}\left({|\\!|}{\phi}{|\\!|}_{\frac{2N}{N-2s}}^{\mu-2}+{|\\!|}{\psi}{|\\!|}_{\frac{2N}{N-2s}}^{\mu-2}\right)\times$
$\displaystyle\hskip
200.0pt\times{|\\!|}|x|^{-\delta}({\phi}-{\psi}){|\\!|}_{L_{\rho}^{q}L_{\sigma}^{\widetilde{q}}}\,dt^{\prime},$
$\displaystyle\leq\eta\int_{-t_{1}}^{t_{2}}{|\\!|}|x|^{(-\gamma-\delta)}|{\psi}|^{\mu_{i}}{|\\!|}_{1}\left({|\\!|}|{\phi}{|\\!|}_{H^{s}}^{\mu-2}+{|\\!|}{\psi}{|\\!|}_{H^{s}}^{\mu-2}\right){|\\!|}|x|^{-\delta}({\phi}-{\psi}){|\\!|}_{L_{\rho}^{q}L_{\sigma}^{\widetilde{q}}}\,dt^{\prime},$
$\displaystyle\leq\eta\int_{-t_{1}}^{t_{2}}{|\\!|}{\psi}{|\\!|}_{\dot{H}^{\frac{N}{2}-\frac{\beta+\delta}{\mu_{i}}}}^{\mu_{i}}\left({|\\!|}{\phi}{|\\!|}_{H^{s}}^{\mu-2}+{|\\!|}{\psi}{|\\!|}_{H^{s}}^{\mu-2}\right){|\\!|}|x|^{-\delta}({\phi}-{\psi}){|\\!|}_{L_{\rho}^{q}L_{\sigma}^{\widetilde{q}}}\,dt^{\prime},$
$\displaystyle\leq\eta(t_{1}+t_{2})^{1-\frac{1}{q}}\left({|\\!|}{\phi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};H^{s})}^{\mu_{i}+\mu-2}+{|\\!|}{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};H^{s})}^{\mu_{i}+\mu-2}\right)\times$
$\displaystyle\hskip
200.0pt\times{|\\!|}|x|^{-\delta}({\phi}-{\psi}){|\\!|}_{L^{q}(-t_{1},t_{2};L_{\rho}^{q}L_{\sigma}^{\widetilde{q}})}.$

Now, if $(-t_{1},t_{2})\subset[-T_{1},T_{2}]$ and
${|\\!|}{\phi}{|\\!|}_{L^{\infty}(-T_{1},T_{2};H^{s})}+{|\\!|}\psi{|\\!|}_{L^{\infty}(-T_{1},T_{2};H^{s})}\leq
K$, then by combining all the estimates above we infer

$\displaystyle{|\\!|}{\phi}-{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};L^{2})}+{|\\!|}|x|^{-\delta}({\phi}-\psi){|\\!|}_{L^{q}(-t_{1},t_{2};L_{\rho}^{q}L_{\sigma}^{\widetilde{q}})}\leq\eta(K^{2}+K^{2\mu-2})\times$
$\displaystyle\hskip
40.0pt\times(t_{1}+t_{2})^{1-\frac{1}{q}}\left({|\\!|}{\phi}-{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};L^{2})}+{|\\!|}|x|^{-\delta}({\phi}-\psi){|\\!|}_{L^{q}(-t_{1},t_{2};L_{\rho}^{q}L_{\sigma}^{\widetilde{q}})}\right).$

Thus, ${\phi}=\psi$ on $[-t_{1},t_{2}]$ for sufficiently small $t_{1},t_{2}$.
Let $I=(-a,b)$ be the maximal interval of $[-T_{1},T_{2}]$ with

${|\\!|}{\phi}-\psi{|\\!|}_{L^{\infty}(-c,d;L^{2})}+{|\\!|}|x|^{-\delta}({\phi}-\psi){|\\!|}_{L^{q}(-c,d;L_{\rho}^{q}L_{\sigma}^{\widetilde{q}})}=0,\>c<a,d<b.$

Assume that $a<T_{1}$ or $b<T_{2}$. Without loss of generality, we may also
assume that $a<T_{1}$ and $b<T_{2}$. Then for a small $\varepsilon>0$ we can
find $a<t_{1}<T_{1},b<t_{2}<T_{2}$ such that

$\displaystyle{|\\!|}{\phi}-\psi{|\\!|}_{L^{\infty}(-t_{1},t_{2};L^{2})}+{|\\!|}|x|^{-\delta}({\phi}-\psi){|\\!|}_{L^{q}(-t_{1},t_{2};L_{\rho}^{q}L_{\sigma}^{\widetilde{q}})}$
$\displaystyle\leq(K^{2}+K^{2\mu-2})(t_{1}+t_{2}-a-b)^{1-\frac{1}{q}}\times$
$\displaystyle\hskip
50.0pt\times\left({|\\!|}{\phi}-\psi{|\\!|}_{L^{\infty}(-t_{1},t_{2};L^{2})}+{|\\!|}|x|^{-\delta}({\phi}-\psi){|\\!|}_{L^{q}(-t_{1},t_{2};L_{\rho}^{q}L_{\sigma}^{\widetilde{q}})}\right),$
$\displaystyle\leq(1-\varepsilon)\left({|\\!|}{\phi}-\psi{|\\!|}_{L^{\infty}(-t_{1},t_{2};L^{2})}+{|\\!|}|x|^{-\delta}({\phi}-\psi){|\\!|}_{L^{q}(-t_{1},t_{2};L_{\rho}^{q}L_{\sigma}^{\widetilde{q}})}\right).$

This contradicts the maximality of $I$. Thus $I=[-T_{1},T_{2}]$. Since
$[-T_{1},T_{2}]$ is arbitrarily taken in $(-T_{min},T_{max})$, we finally get
the whole uniqueness and the Proposition 2.2 is now proved. ∎

###  2.3 Global well-posedness

Using the argument of [3], one can show that the uniqueness implies actually
well-posedness and conservation laws:

$\displaystyle\bullet\;{\phi}\in C(-T_{min},T_{max};H^{s})\cap
C^{1}(-T_{min},T_{max};H^{-s}),$
$\displaystyle\bullet\;{\phi}\;\;\mbox{depends continuously
on}\;\;\phi_{0}\;\;\mbox{in}\;\;H^{s},$
$\displaystyle\bullet\;{|\\!|}{\phi}(t){|\\!|}_{2}={|\\!|}\phi_{0}{|\\!|}_{2}\;\;\mbox{
and}\;\;\mathcal{J}({\phi}(t))=\mathcal{J}(\phi_{0})\;\;\forall\;t\in(-T_{min},T_{max}).$

The proofs of these points are standard, we omit them and refer to [3]. Now we
remark that the well-posedness is actually global by establishing a uniform
bound on the $H^{s}$ norm of $\phi(t)$ for all $t\in(-T_{min},T_{max})$. We
first consider the global existence of weak solutions. Suppose ${\phi}$ is a
weak solution on $(-T_{min},T_{max})$ as in Proposition 2.1. We show that
${|\\!|}{\phi}(t){|\\!|}_{H^{s}}$ is bounded for all $t\in(-T_{min},T_{max})$.
For this purpose let us introduce the following notation

$\mathcal{D}(G(|\phi|),G(|\phi|))=\sum_{i,j=1}^{2}\mathcal{D}_{i,j}(|\phi|),\quad\mathcal{D}_{i,j}(|\phi|):=\mathcal{D}(G_{i}(|\phi|),G_{j}(|\phi|)),$
(15)

where obviously we set $G_{i}:=\int_{0}^{|z|}g_{i}(\alpha)\,d\alpha$ and
recall that the $g_{i}$ are defined as
$g_{1}(\alpha)=\chi_{\\{0\leq\alpha<1\\}}g(\alpha)$ and
$g_{2}(\alpha)=\chi_{\\{\alpha\geq 1\\}}g(\alpha)$. Using Hardy-Littlewood-
Sobolev and the fractional Gagliardo-Nirenberg inequalities and the assumption
$\mathcal{A}_{0}$ we can write the following estimates

$\displaystyle\mathcal{D}_{1,1}\leq\eta\,{|\\!|}u{|\\!|}^{4}_{\frac{4N}{N+\beta}}\leq\eta{|\\!|}u{|\\!|}^{4-\frac{N-\beta}{s}}_{2}\,{|\\!|}u{|\\!|}^{\frac{N-\beta}{s}}_{\dot{H}^{s}},$
(16)
$\displaystyle\mathcal{D}_{2,2}\leq\eta\,{|\\!|}u{|\\!|}^{2\mu}_{\frac{2N\mu}{N+\beta}}\leq\eta{|\\!|}u{|\\!|}^{2\mu-\frac{N(\mu-1)-\beta}{s}}_{2}\,{|\\!|}u{|\\!|}^{\frac{N(\mu-1)-\beta}{s}}_{\dot{H}^{s}},$
(17)
$\displaystyle\mathcal{D}_{1,2},\,\mathcal{D}_{2,1}\leq\eta\,{|\\!|}u{|\\!|}^{\mu+2-\frac{N\mu-2\beta}{2s}}_{2}\,{|\\!|}u{|\\!|}^{\frac{N\mu-2\beta}{2s}}_{\dot{H}^{s}}.$
(18)

Since $N-\beta<2s$, then $0<\frac{N-\beta}{s}<2$ and $4-\frac{N-\beta}{s}>2$.
As well since $2\leq\mu<1+\frac{2s+\beta}{N}$, then
$0<\frac{N-\beta}{s}\leq\frac{N(\mu-1)-\beta}{s}<2$ and
$2<\mu-\frac{N(\mu-1)-\beta}{s}$. Eventually, we have
$0<\frac{N-\beta}{s}\leq\frac{N\mu-2\beta}{2s}$ and
$2\leq\mu<\mu+1+\frac{\beta-N}{2s}<\mu+2-\frac{N\mu-2\beta}{2s}$. The
estimates above can be summarized as follows with $\mu_{1}=2$ and
$\mu_{2}=\mu$.

$\displaystyle\mathcal{D}_{i,j}(|u|)$ $\displaystyle\leq$
$\displaystyle\eta\int_{{\mathbb{R}^{N}}\times{\mathbb{R}^{N}}}\frac{|u(x)|^{\mu_{i}}|u(y)|^{\mu_{j}}}{|x-y|^{N-\beta}}\,dxdy,$
(19) $\displaystyle\leq$
$\displaystyle\eta\,{|\\!|}u{|\\!|}^{\mu_{i}+\mu_{j}-\gamma_{i,j}}_{2}\,{|\\!|}u{|\\!|}^{\gamma_{i,j}}_{\dot{H}^{s}}$

where

$\gamma_{i,j}=\frac{N}{s}\left(1+\frac{\beta}{N}\right)-\left(\frac{N}{2s}-1\right)(\mu_{i}+\mu_{j}).$

Thus, we have clearly

$\displaystyle\frac{1}{2}{|\\!|}{\phi}{|\\!|}_{H^{s}}^{2}$
$\displaystyle=\frac{1}{2}{|\\!|}{\phi}{|\\!|}_{2}^{2}+\mathcal{J}({\phi})+\mathcal{D}(G(|\phi|),G(|\phi|)),$
$\displaystyle\leq\frac{1}{2}{|\\!|}\phi_{0}{|\\!|}_{2}^{2}+\mathcal{J}(\phi_{0})+\eta\sum_{i,j=1,2}{|\\!|}\phi_{0}{|\\!|}_{2}^{\frac{2\gamma_{ij}}{2-\mu_{i}-\mu_{j}+\gamma_{ij}}}+\frac{1}{4}{|\\!|}{\phi}{|\\!|}_{H^{s}}^{2}.$

Thus

${|\\!|}{\phi}{|\\!|}_{H^{s}}\leq\eta\left({|\\!|}\phi_{0}{|\\!|}_{H^{s}}\right),\quad\text{for
all}\>t\in(-T_{min},T_{max}).$

Therefore $T_{min}=T_{max}=\infty$. If $s,\beta,\mu$ satisfy the hypothesis of
Proposition 2.2, then we get the global well-posedness. Eventually, combining
this fact with the Propositions 2.1 and 2.2 prove Theorem 1.1.

##  3 Existence of standing waves

In this section we study the minimization problem $\tilde{\mathscr{S}}$. We
prove the existence of a solution to $\tilde{\mathscr{S}}$ using a variational
approach via the concentration-compactness method of P-L. Lions [17]. Indeed,
we aim to prove the existence of critical points to the energy functional

$\displaystyle\mathcal{E}(u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}{|\\!|}\nabla_{s}u{|\\!|}^{2}_{2}-\frac{1}{2}\,\mathcal{D}(G(|u|),G(|u|)).$

In other words, we look for a function $u_{\lambda}$ such that

$\mathcal{E}(u_{\lambda})=\mathcal{I}_{\lambda}=\inf\left\\{\mathcal{E}(u),\quad
u\in
H^{s}(\mathbb{R}^{N}),\quad\int_{\mathbb{R}^{N}}|u(x)|^{2}\,dx=\lambda\right\\}.$

As noticed in the introduction of this paper, this problem has been studied in
various situation depending on the value of $s$ and the conditions on $\beta$
and the integrand $G$ in Ref. [5, 12, 18]. In order to prove the existence of
critical points to the functional $\mathcal{E}$, we start with the following
claim

######  Proposition 3.1.

For all $\lambda>0$ and $G$ such that $\mathcal{A}_{0}$ and $\mathcal{A}_{1}$
hold true, we have

  * 1.

The functional $\mathcal{E}\in C^{1}(H^{s},\mathbb{R})$ and there exists a
constant $\eta>0$ such that

${|\\!|}\mathcal{E}^{\prime}(u){|\\!|}_{H^{-s}}\leq\eta\left({|\\!|}u{|\\!|}_{H^{s}}+{|\\!|}u{|\\!|}_{H^{s}}^{\frac{2s+\beta}{N}}\right).$

  * 2.

$-\infty<\mathcal{I}_{\lambda}<0$.

  * 3.

Each minimizing sequence for the problem $\mathcal{I}_{\lambda}$ is bounded in
$H^{s}$.

###### Proof.

Let us mention that only assumption $\mathcal{A}_{0}$ is needed to prove the
$C^{1}$ property of the energy functional $\mathcal{E}$. The proof of this
claim is standard and we refer the reader to Ref. [11] for details. Now, we
prove the second assertion. Let $u\in H^{s}(\mathbb{R}^{N})$ such that
${|\\!|}u{|\\!|}_{2}=\sqrt{\lambda}$ and assume $\mathcal{A}_{0}$. Then, on
the one hand, thanks to (16 –18), it is rather easy to show using Young’s
inequality that for all $\epsilon_{1},\epsilon_{2}$ and $\epsilon_{3}$, there
exist $C_{\epsilon_{1}},C_{\epsilon_{2}},C_{\epsilon_{3}}>0$ such that

$\displaystyle\mathcal{D}_{1,1}$ $\displaystyle\leq$
$\displaystyle\eta\left(\epsilon_{1}\,{|\\!|}u{|\\!|}^{2}_{\dot{H}^{s}}+C_{\epsilon_{1}}\lambda^{e_{1}}\right),\quad
e_{1}:=\frac{4s+\beta-N}{2s+\beta-N}.$ (20) $\displaystyle\mathcal{D}_{1,2}$
$\displaystyle\leq$
$\displaystyle\eta\left(\epsilon_{2}\,{|\\!|}u{|\\!|}^{2}_{\dot{H}^{s}}+C_{\epsilon_{2}}\lambda^{e_{2}}\right),\quad
e_{2}:=\frac{2s\mu+\beta-N(\mu-1)}{2s+\beta-N(\mu-1)}.$ (21)
$\displaystyle\mathcal{D}_{1,2},\,\mathcal{D}_{2,1}$ $\displaystyle\leq$
$\displaystyle\eta\left(\epsilon_{1}{|\\!|}u{|\\!|}^{2}_{\dot{H}^{s}}+C_{\epsilon_{3}}\lambda^{e_{3}}\right),\quad
e_{3}:=1+\frac{2s\mu}{4s-N\mu+2\beta}.$ (22)

Observe that $0<2s+\beta-N<4s+\beta-N$ so that $e_{1}>1$. Also,
$0<2s+\beta-N(\mu-1)<2s\mu+\beta-N(\mu-1)$ so that $e_{2}>1$. Eventually,
$4s-N\mu+2\beta>2s+\beta-N>0$ so that $\frac{2s\mu}{4s-N\mu+2\beta}>0$ and
$e_{3}>1$. Therefore, for sufficiently small $\epsilon_{1},\epsilon_{2}$ and
$\epsilon_{3}$, one has

$\displaystyle\mathcal{E}(u)$ $\displaystyle\geq$
$\displaystyle\left(\frac{1}{2}-\eta(\epsilon_{1}+\epsilon_{2}+\epsilon_{3})\right)\,{|\\!|}u{|\\!|}^{2}_{H^{s}}-\frac{1}{2}-\eta\left(C_{\epsilon_{1}}\lambda^{e_{1}}+C_{\epsilon_{2}}\lambda^{e_{2}}+C_{\epsilon_{3}}\lambda^{e_{3}}\right),$
$\displaystyle\geq$
$\displaystyle-\frac{1}{2}\lambda-\eta\left(C_{\epsilon_{1}}\lambda^{e_{1}}+C_{\epsilon_{2}}\lambda^{e_{2}}+C_{\epsilon_{3}}\lambda^{e_{3}}\right).$

Thus, we obtain $\mathcal{I}_{\lambda}>-\infty$. On the other hand, let us
introduce for all $\kappa\in\mathbb{R}$, the rescaled function
$u_{\kappa}=\kappa^{\frac{1}{2}}u(\kappa^{\frac{1}{N}}\cdot)$. Obviously, one
has $\int_{\mathbb{R}^{N}}|u_{\kappa}|^{2}=\lambda$ and using
$\mathcal{A}_{1}$

$\displaystyle\mathcal{E}(u_{\kappa})\leq\frac{1}{2}\kappa^{\frac{2s}{N}}\int_{\mathbb{R}^{N}}|(-\Delta)^{s}u(x)|^{2}dx-\frac{\kappa^{\alpha-\left(1+\frac{\beta}{N}\right)}}{2}\,\mathcal{D}(|u(x)|^{\alpha},|u(y)|^{\alpha}).$

We have $0<\alpha-\left(1+\frac{\beta}{N}\right)<\frac{2s}{N}$, therefore we
can take $\kappa$ small enough to get $\mathcal{E}(u_{\kappa})<0$. Thus,
$\mathcal{I}_{\lambda}\leq\mathcal{E}(u_{\kappa})<0$. We are kept with the
proof of the third assertion. Let $(u_{n})_{n\in\mathbb{N}}$ be a minimizing
sequence for the problem $\mathcal{I_{\lambda}}$. Therefore, thanks to
(20–22), we have for all $u\in H^{s}$

$\displaystyle\mathcal{D}(G(|u|),G(|u|))$ $\displaystyle\leq$
$\displaystyle\eta(\epsilon_{1}+\epsilon_{2}+\epsilon_{3})\,{|\\!|}u{|\\!|}^{2}_{\dot{H}^{s}}+\eta\left(C_{\epsilon_{1}}\lambda^{e_{1}}+C_{\epsilon_{2}}\lambda^{e_{2}}+C_{\epsilon_{3}}\lambda^{e_{3}}\right).$

Hence

$\displaystyle{|\\!|}u_{n}{|\\!|}^{2}_{H^{s}}$ $\displaystyle=$ $\displaystyle
2\,\mathcal{E}(u_{n})+{|\\!|}u_{n}{|\\!|}^{2}_{2}+\mathcal{D}(G(|u_{n}|),G(|u_{n}|)),$
$\displaystyle\leq$ $\displaystyle
2\,\mathcal{I}_{\lambda}+\lambda+\eta(\epsilon_{1}+\epsilon_{2}+\epsilon_{3})\,{|\\!|}u_{n}{|\\!|}^{2}_{H^{s}}+\eta\left(C_{\epsilon_{1}}\lambda^{e_{1}}+C_{\epsilon_{2}}\lambda^{e_{2}}+C_{\epsilon_{3}}\lambda^{e_{3}}\right).$

Eventually, we pick $\epsilon_{1},\epsilon_{2}$ and $\epsilon_{3}$ such that
$\eta(\epsilon_{1}+\epsilon_{2}+\epsilon_{3})<1$, we get immediately that the
minimizing sequence $(u_{n})_{n\in\mathbb{N}}$ is bounded in $H^{s}$. ∎

Before going further, let us introduce the so called Lévy concentration
function

$\mathcal{Q}_{n}(r)=\sup_{y\in{\mathbb{R}^{N}}}\int_{B(y,r)}\,|u_{n}(x)|^{2}dx.$

It is known that each $\mathcal{Q}_{n}$ is nondecreasing on $(0,+\infty)$.
Also, with the Helly’s selection Theorem, the sequence
$(\mathcal{Q}_{n})_{n\in\mathbb{N}}$ has a subsequence that we still denote
$(\mathcal{Q}_{n})_{n\in\mathbb{N}}$ by abuse of notation, such that there is
a nondecreasing function $\mathcal{Q}(r)$ satisfying

$\mathcal{Q}_{n}(r)\xrightarrow[n\to+\infty]{}\mathcal{Q}(r),\quad\text{for
all}\quad r>0.$

Since $0\leq\mathcal{Q}_{n}(r)\leq\lambda$, there exists $\beta\in\mathbb{R}$
such that $0\leq\beta\leq\lambda$ such that

$\mathcal{Q}(r)\xrightarrow[r\to+\infty]{}\gamma.$

Briefly speaking, a minimizing sequence $(u_{n})_{n\in\mathbb{N}}$ for the
problem $\mathcal{I_{\lambda}}$ can only be in one of the following
situations:

  * 1.

Vanishing, i.e. $\gamma=0$.

  * 2.

Dichotomy, i.e. $0<\gamma<\lambda$.

  * 3.

Compactness, i.e. $\gamma=\lambda$.

In the sequel we shall proceed by elimination and show that vanishing and
dichotomy do not occur. Therefore, compactness holds true and we are done. We
start with the following

######  Proposition 3.2.

Let $\lambda>0$ and $(u_{n})_{n\in\mathbb{N}}$ be a minimizing sequence of
problem $\mathcal{I}_{\lambda}$ with $G$ such that $\mathcal{A}_{0}$ and
$\mathcal{A}_{1}$ hold true. Then $\gamma>0$.

The proposition claims then that the situation of vanishing does not occurs.
In the proof of Proposition 3.2, we shall use, for all subset of
$A\subset{\mathbb{R}^{N}}$, the notation

$\mathcal{D}|_{A}(G(|u|),G(|u|)):=\int_{A\times
A}G(|u(x)|)\,V(|x-y|)\,G(|u(y)|)\,dxdy.$

###### Proof.

Let us first prove that $\mathcal{D}(G(|u_{n}|),G(|u_{n}|))$ is lower bounded.
In other words, we show that for $n\in\mathbb{N}$ large enough there exists
$\delta>0$ such that

$\delta<\mathcal{D}(G(|u_{n}|),G(|u_{n}|)).$ (23)

We argue by contradiction and assume that there exist no such $\delta$,
therefore $\liminf_{n\rightarrow+\infty}\mathcal{D}(G(|u_{n}|),G(|u_{n}|))\leq
0$, thus

$\displaystyle\mathcal{I}_{\lambda}=\lim_{n\rightarrow+\infty}\mathcal{E}(u_{n})$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow+\infty}\left(\frac{1}{2}{|\\!|}\nabla_{s}u_{n}{|\\!|}^{2}_{2}-\frac{1}{2}\,\mathcal{D}(G(|u_{n}|),G(|u_{n}|))\right)$
$\displaystyle\geq$
$\displaystyle-\frac{1}{2}\lim_{n\rightarrow+\infty}\,\mathcal{D}(G(|u_{n}|),G(|u_{n}|))\geq
0.$

The inequality above is in contradiction with the fact that
$\mathcal{I}_{\lambda}<0$. On the other hand, arguing by contradiction and
assuming that the minimizing sequence $(u_{n})_{n\in\mathbb{N}}$ vanishes,
i.e. assume that $\gamma=0$. Then there exists a subsequence
$(u_{n_{k}})_{k\in\mathbb{N}}$ of $(u_{n})_{n\in\mathbb{N}}$ and a radius
$\tilde{r}>0$ such that

$\sup_{y\in{\mathbb{R}^{N}}}\int_{B(y,\tilde{r})}\,|u_{n_{k}}(x)|^{2}dx\xrightarrow[k\to+\infty]{}0.$

Next, since the sequence $(u_{n_{k}})_{k\in\mathbb{N}}$ is bounded in $H^{s}$,
then one can find $r_{\epsilon}>0$ such that

$\mathcal{D}|_{|x-y|\geq
r_{\epsilon}}({G(|u_{n_{k}}|),G(|u_{n_{k}}|))}\leq\frac{\epsilon}{2}.$

Now, we cover ${\mathbb{R}^{N}}$ by balls of radius $r$ and centers $c_{i}$
for $i=1,2,\ldots$ such that each point of ${\mathbb{R}^{N}}$ is contained in
at most $N+1$ ball. Therefore, there exists $N_{\epsilon}$ ball and a
subsequence $(c_{i_{l}})_{l=1,\ldots,N_{\epsilon}}$ such that

$\displaystyle\mathcal{D}|_{|x-y|\geq
r_{\epsilon}}({G(|u_{n_{k}}|),G(|u_{n_{k}}|))}\leq\eta\sum_{p,q=1}^{2}{\mathcal{D}_{p,q}|_{|x-y|\geq
r_{\epsilon}}}(|u_{n_{k}}|),$ $\displaystyle\leq$
$\displaystyle\eta\sum_{p,q=1}^{2}\sum_{l=1}^{\infty}\sum_{i=1}^{N_{\epsilon}}\int_{B_{x}(c_{l},r)}\int_{B_{y}(c_{i_{l}},r)}\frac{|u_{n_{k}}(x)|^{\mu_{p}}|u_{n_{k}}(y)|^{\mu_{q}}}{|x-y|^{N-\beta}}dxdy,$
$\displaystyle\leq$
$\displaystyle\eta\sum_{p,q=1}^{2}\sum_{l=1}^{\infty}\sum_{i=1}^{N_{\epsilon}}{|\\!|}u_{n_{k}}{|\\!|}_{L^{2}(B_{x}(c_{l},r))}{|\\!|}\int_{B_{y}(c_{l_{i}},r)}\frac{|u_{n_{k}}(y)|^{\mu_{q}}}{|x-y|^{N-\beta}}dy\,|u_{n_{k}}|^{\mu_{p}-1}{|\\!|}_{L^{2}(B_{x}(c_{l},r))}\,,$
$\displaystyle\leq$ $\displaystyle
N_{\epsilon}\,\eta\left(\sum_{l=1}^{\infty}{|\\!|}u_{n_{k}}{|\\!|}_{L^{2}(B_{x}(c_{l},r))}\right){|\\!|}u_{n_{k}}{|\\!|}_{r}\,{|\\!|}u_{n_{k}}{|\\!|}^{\mu-1}_{\frac{2N}{N-2s}}\sup_{y\in\mathbb{R}^{N}}\,{|\\!|}u_{n_{k}}{|\\!|}_{L^{2}(B(y,r))}$
$\displaystyle+$ $\displaystyle
N_{\epsilon}\,\eta\sum_{(p,q)\neq(1,2),p,q=1}^{2}\left(\sum_{l=1}^{\infty}{|\\!|}u_{n_{k}}{|\\!|}_{L^{2}(B_{x}(c_{l},r))}\right)\,{|\\!|}u_{n_{k}}{|\\!|}^{\mu_{q}-1}_{{r_{pq}}}\,{|\\!|}u_{n_{k}}{|\\!|}^{\mu_{p}-1}_{{\frac{r_{pq}}{\mu_{p}-1}}}\times$
$\displaystyle\hskip
250.0pt\times\sup_{y\in\mathbb{R}^{N}}\,{|\\!|}u_{n_{k}}{|\\!|}_{L^{2}(B(y,r))},$

where $r$ and $r_{pq}$ are such that

$\displaystyle\frac{\beta}{N}=\frac{1}{r}+(\mu-1)\left(\frac{1}{2}-\frac{s}{N}\right),\quad\frac{1}{r}+\frac{1}{2}>\frac{\beta}{N},$
$\displaystyle\frac{\mu_{p}-1}{r_{pq}}+\frac{\mu_{q}-1}{r_{pq}}=\frac{\beta}{N},\quad\frac{\mu_{p}-1}{r_{pq}}+\frac{1}{2}>\frac{\beta}{N},\quad(p,q)\neq(1,2).$

Since $1+\frac{2\beta-N}{N-2s}<\mu<1+\frac{\beta+2s}{N},0<\beta<N$ and
$s>\frac{N-\beta}{2}$, it is rather clear that one can find (as in section 2)
$r,r_{p,q}\in\left[2,\frac{2N}{N-2s}\right]$. Consequently, we have obviously

$\displaystyle\mathcal{D}|_{|x-y|\geq
r_{\epsilon}}({G(|u_{n_{k}}|),G(|u_{n_{k}}|))}\leq(N+1)\,N_{\epsilon}\,\eta{|\\!|}u_{n_{k}}{|\\!|}_{{2}}\times$
$\displaystyle\hskip
50.0pt\times\left({|\\!|}u_{n_{k}}{|\\!|}^{\mu}_{H^{s}}+{|\\!|}u_{n_{k}}{|\\!|}^{2}_{H^{s}}+{|\\!|}u_{n_{k}}{|\\!|}^{2(\mu-1)}_{H^{s}}\right)\left(\sup_{y\in\mathbb{R}^{N}}\,\int_{B(y,r)}|u_{n_{k}}|^{2}\right)^{\frac{1}{2}}.$
$\displaystyle\hskip 300.0pt\xrightarrow[n\to+\infty]{}0.$

This shows that if the minimizing sequence $(u_{n})_{n\in\mathbb{N}}$
vanishes, then

$\mathcal{D}(G(|u_{n}|),G(|u_{n}|))\xrightarrow[n\to+\infty]{}0.$

This is in contradiction with the property (23), namely for $n\in\mathbb{N}$
large enough there exists $\gamma>0$ such that
$\mathcal{D}(G(|u_{n}|),G(|u_{n}|))>\gamma$. Thus, vanishing does not occurs.
∎

Now, we show the following

######  Proposition 3.3.

Let $0<\pi<\lambda$ and $G$ such that $\mathcal{A}_{0}$ and $\mathcal{A}_{1}$
hold true. Then the mapping $\lambda\mapsto\mathcal{I}_{\lambda}$ is
continuous and
$\mathcal{I}_{\lambda}<\mathcal{I}_{\pi}+\mathcal{I}_{\lambda-\pi}$.

###### Proof.

Let $\lambda>0$ and $(\lambda_{k})_{k\in\mathbb{N}}$ be a sequence of positive
numbers such that $\lambda_{k}\xrightarrow[k\to+\infty]{}\lambda$. Let
$\epsilon>0$ and $u\in H^{s}(\mathbb{R}^{N})$ such that
${|\\!|}u{|\\!|}_{2}=\sqrt{\lambda}$ and

$\mathcal{I}_{\lambda}\leq\mathcal{E}(u)\leq\mathcal{I}_{\lambda}+\frac{\epsilon}{2}.$

For all $k\in\mathbb{N}$, let $u_{k}=\sqrt{\frac{\lambda_{k}}{\lambda}}u$.
Obviously $u_{k}\in H^{s}(\mathbb{R}^{N})$ and
${|\\!|}u_{k}{|\\!|}^{2}_{2}=\lambda_{k}$ so that for all
$k\in\mathbb{N},\,\mathcal{I}_{\lambda_{k}}\leq\mathcal{E}(u_{k})$. Now, we
show that $\mathcal{E}(u_{k})\xrightarrow[k\to+\infty]{}\mathcal{E}(u)$.
First, for all $k\in\mathbb{N}$

${|\\!|}u_{k}-u{|\\!|}_{\dot{H}^{s}}\leq{|\\!|}u_{k}{|\\!|}_{\dot{H}^{s}}\,\left|1-\sqrt{\frac{\lambda_{k}}{\lambda}}\right|.$

Since any sequence of $\mathcal{I}_{\lambda}$ is bounded in
$H^{s}(\mathbb{R}^{N})$ and $\lambda_{k}\xrightarrow[k\to+\infty]{}\lambda$,
then we have obviously
$\frac{1}{2}{|\\!|}\nabla_{s}u_{k}{|\\!|}^{2}_{2}\xrightarrow[k\to+\infty]{}\frac{1}{2}{|\\!|}\nabla_{s}u{|\\!|}^{2}_{2}$.
Next, following the first assertion of Proposition 3.1, we have
$\mathcal{E}(u)\in C^{1}(H^{s}(\mathbb{R}^{N}),\mathbb{R})$. In particular,
one can easily see from the proof of this point that
$D(u):=\mathcal{D}(G(|u|),G(|u|))\in C^{1}(H^{s}(\mathbb{R}^{N}),\mathbb{R})$
and

$\left|\mathcal{D}^{\prime}(u)\right|\leq\eta\left({|\\!|}u{|\\!|}_{H^{s}}+{|\\!|}u{|\\!|}^{\frac{2s+\beta}{N}}_{H^{s}}\right).$
(24)

We refer to Ref. [11] for details. Therefore, we have

$\displaystyle\left|\mathcal{D}(u_{k})-\mathcal{D}(u)\right|$ $\displaystyle=$
$\displaystyle\left|\int_{0}^{t}\frac{d}{dt}\mathcal{D}(tu_{k}+(1-t)u)dt\right|,$
$\displaystyle\leq$
$\displaystyle\eta\sup_{u\in{H^{s}},{|\\!|}u{|\\!|}_{H^{s}}\leq\eta}{|\\!|}\mathcal{D}^{\prime}(u){|\\!|}_{H^{-s}}\,{|\\!|}u_{k}-u{|\\!|}_{H^{s}},$
$\displaystyle\leq$
$\displaystyle\eta\,{|\\!|}u_{k}{|\\!|}_{H^{s}}\,\left|1-\sqrt{\frac{\lambda_{k}}{\lambda}}\right|\,\xrightarrow[k\to+\infty]{}0.$

Thus, we have $\mathcal{E}(u_{k})\xrightarrow[k\to+\infty]{}\mathcal{E}(u)$.
Consequently, we have
$\mathcal{I}_{\lambda_{k}}\leq\mathcal{I}_{\lambda}+\epsilon$ for $k$ large
enough. Next, for all $k\in\mathbb{N}$, let us choose $\tilde{u}_{k}\in
H^{s}(\mathbb{R}^{N})$ such that
${|\\!|}\tilde{u}_{k}{|\\!|}_{2}=\sqrt{\lambda}_{k}$ and
$\mathcal{E}(\tilde{u}_{k})\leq\mathcal{I}_{\lambda_{k}}+\frac{1}{k}$.
Moreover, for all $k\in\mathbb{N}$, we set
$\bar{u}_{k}=\sqrt{\frac{\lambda}{\lambda_{k}}}\tilde{u}_{k}$. Obviously,
since $\bar{u}_{k}\in H^{s}(\mathbb{R}^{N})$ and
${|\\!|}\bar{u}_{k}{|\\!|}_{2}^{2}=\lambda$, we have
$\mathcal{I}_{\lambda}\leq\mathcal{E}(\bar{u}_{k})$. Exactly the same argument
as above shows that
$\mathcal{E}(\tilde{u}_{k})\xrightarrow[k\to+\infty]{}\mathcal{E}(\bar{u})$ so
that for $k$ large enough, we have
$\mathcal{I}_{\lambda}\leq\mathcal{I}_{\lambda_{k}}+\epsilon$. Whence,
$\lambda\mapsto\mathcal{I}_{\lambda}$ is continuous on
$\mathbb{R}_{+}^{\star}$. Eventually, using the energy estimates (16-18) or
(19), it is rather easy to show that $I_{\lambda}\xrightarrow[\lambda\to
0^{+}]{}0$. This shows that the mapping $\lambda\mapsto\mathcal{I}_{\lambda}$
is continuous.

Let us now prove the strict sub–additivity inequality. For that purpose, we
introduce $u_{\theta}=\theta^{\kappa}u(\theta^{\frac{\kappa}{N}})$ for all
$\kappa>\frac{N}{N+2s}$. Obviously $u_{\kappa}\in H^{s}(\mathbb{R}^{N})$ and
${|\\!|}u_{\theta}{|\\!|}_{{L^{2}(\mathbb{R}^{N})}}=\sqrt{\theta\lambda}$.
Moreover, using $\mathcal{A}_{1}$, we have

$\displaystyle\mathcal{E}(u_{\theta})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{{\mathbb{R}^{N}}}|(-\Delta)^{\frac{s}{2}}u_{\theta}|^{2}dx-\frac{1}{2}\mathcal{D}(G(|u_{\theta}|),G(|u_{\theta}|)),$
$\displaystyle\leq$
$\displaystyle\frac{\theta^{\kappa\left(1+\,\frac{2s}{N}\right)}}{2}\left(\int_{{\mathbb{R}^{N}}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx-\mathcal{D}(G(|u|),G(|u|))\right)={\theta^{\kappa\left(1+\,\frac{2s}{N}\right)}}\mathcal{E}(u).$

Thus, we deduce that
$\mathcal{I}_{\theta\lambda}\leq{\theta^{\kappa\left(1+\,\frac{2s}{N}\right)}}\mathcal{I}_{\lambda}$
for all $\theta>0$. Now we let $0<\pi<\lambda$, therefore since
$\kappa\left(1+\,\frac{2s}{N}\right)>1$ we have

$\displaystyle\mathcal{I}_{\lambda}\leq\lambda^{\kappa\left(1+\,\frac{2s}{N}\right)}\mathcal{I}_{1}$
$\displaystyle<$
$\displaystyle\pi^{\kappa\left(1+\,\frac{2s}{N}\right)}\mathcal{I}_{1}+(\lambda-\pi)^{\kappa\left(1+\,\frac{2s}{N}\right)}\mathcal{I}_{1},$
$\displaystyle\leq$
$\displaystyle\pi^{\kappa\left(1+\,\frac{2s}{N}\right)}\pi^{-\kappa\left(1+\,\frac{2s}{N}\right)}\mathcal{I}_{\pi}+(\lambda-\pi)^{\kappa\left(1+\,\frac{2s}{N}\right)}(\lambda-\pi)^{-\kappa\left(1+\,\frac{2s}{N}\right)}\mathcal{I}_{\lambda-\pi},$
$\displaystyle=$ $\displaystyle\mathcal{I}_{\pi}+\mathcal{I}_{\lambda-\pi}.$

In summary, for all $0<\pi<\lambda$, we have
$\mathcal{I}_{\lambda}<\mathcal{I}_{\pi}+\mathcal{I}_{\lambda-\pi}$. ∎

Now, we are able to claim the following

######  Proposition 3.4.

Let $\lambda>0$ and $(u_{n})_{n\in\mathbb{N}}$ be a minimizing sequence of
problem $\mathcal{I}_{\lambda}$ with $G$ such that $\mathcal{A}_{0}$ and
$\mathcal{A}_{1}$ hold true. Then dichotomy does not occur for
$(u_{n})_{n\in\mathbb{N}}$.

###### Proof.

Let us introduce $\xi$ and $\chi$ in $C^{\infty}$ such that $0\leq\xi,\chi\leq
1$ and

$\xi(x)=\left\\{\begin{array}[]{lcl}1&\text{if}&|x|\leq 1\\\ &&\\\
0&\text{if}&|x|\geq
2\end{array}\right.,\>\chi(x)=1-\xi(x),\>{|\\!|}\nabla\xi{|\\!|}_{\infty},{|\\!|}\nabla\chi{|\\!|}_{\infty}\leq
2.$

For all $r>0$, let $\xi_{r}(\cdot)=\xi(\frac{\cdot}{R})$ and
$\chi_{r}(\cdot)=\chi(\frac{\cdot}{R})$. we will show that dichotomy does not
occur by contradicting the fact that for all $0<\pi<\lambda$, we have
$\mathcal{I}_{\lambda}<\mathcal{I}_{\pi}+\mathcal{I}_{\lambda-\pi}$ proved in
Proposition 3.3. Indeed, let $(u_{n})_{n\in\mathbb{N}}$ be a minimizing
sequence of problem $\mathcal{I}_{\lambda}$ and assume that dichotomy holds.
Then, using the construction of [17], there exist

  * 1.

$0<\pi<\lambda$,

  * 2.

a sequence $(y_{n})_{n\in\mathbb{N}}$ of points in ${\mathbb{R}^{N}}$,

  * 3.

two increasing sequences of positive real number $(r_{1,n})_{n\in\mathbb{N}}$
and $(r_{2,n})_{n\in\mathbb{N}}$ such that

$r_{1,n}\xrightarrow[n\to+\infty]{}+\infty\quad\text{and}\quad\frac{r_{2,n}}{2}-r_{1,n}\xrightarrow[n\to+\infty]{}+\infty,$

such that the sequences $u_{1,n}=\xi_{r_{1,n}}(\cdot-y_{n})u_{n}$ and
$u_{2,n}=\chi_{r_{2,n}}(\cdot-y_{n})u_{n}$ satisfy

$\left\\{\begin{array}[]{lll}&u_{n}=u_{1,n}\>\text{on}\>B(y_{n},{r_{1,n}}),\\\
&\\\
&u_{n}=u_{2,n}\>\text{on}\>B^{c}(y_{n},{r_{2,n}})={\mathbb{R}^{N}}\setminus
B(y_{n},{r_{2,n}}),\\\ &\\\
&\int_{\mathbb{R}^{N}}|u_{1,n}|^{2}dx\xrightarrow[n\to+\infty]{}\pi,\>\int_{\mathbb{R}^{N}}|u_{1,n}|^{2}dx\xrightarrow[n\to+\infty]{}\lambda-\pi,\\\
&\\\
&{|\\!|}u_{n}-(u_{1,n}+u_{2,n}){|\\!|}_{p}\xrightarrow[n\to+\infty]{}0,\>\text{for
all}\>2\leq p<\frac{2N}{N-2s},\\\ &\\\
&{|\\!|}u_{n}{|\\!|}_{L^{p}(B(y_{n},r_{2,n})\setminus
B(y_{n},r_{1,n}))}\xrightarrow[n\to+\infty]{}0,\>\text{for all}\>2\leq
p<\frac{2N}{N-2s},\\\ &\\\
&\mathrm{dist}(\mathrm{Supp}(u_{1,n}),\mathrm{Supp}(u_{2,n}))\xrightarrow[n\to+\infty]{}+\infty.\end{array}\right.$

We have obviously

$\displaystyle\mathcal{E}(u_{n})$ $\displaystyle=$
$\displaystyle\mathcal{E}(u_{1,n})+\mathcal{E}(u_{2,n})+\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}\,u_{n}|^{2}-\frac{1}{2}\mathcal{D}(G(|u_{n}|),G(|u_{n}|))dx$
$\displaystyle-$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}\left(|(-\Delta)^{\frac{s}{2}}\,u_{1,n}|^{2}+|(-\Delta)^{\frac{s}{2}}\,u_{2,n}|^{2}\right)dx$
$\displaystyle+$
$\displaystyle\frac{1}{2}\left(\mathcal{D}(G(|u_{1,n}|),G(|u_{1,n}|))+\mathcal{D}(G(|u_{2,n}|),G(|u_{2,n}|))\right).$

Now we show the existence of $\epsilon>0$ such that for sufficiently large
radius $r_{1,n}$ and $r_{1,n}$ we have

$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}\left(|(-\Delta)^{\frac{s}{2}}\,u_{n}|^{2}-|(-\Delta)^{\frac{s}{2}}\,u_{1,n}|^{2}-|(-\Delta)^{\frac{s}{2}}\,u_{2,n}|^{2}\right)dx\geq-\eta\epsilon.$
(25)

Firs of all, it is rather easy to show that by construction of the sequences
$u_{i,n}$ for $i=1,2$, we have

$\displaystyle\int_{\mathbb{R}^{N}}\left(|(-\Delta)^{\frac{s}{2}}\,u_{n}|^{2}-|(-\Delta)^{\frac{s}{2}}\,u_{1,n}|^{2}-|(-\Delta)^{\frac{s}{2}}\,u_{2,n}|^{2}\right)dx$
$\displaystyle\hskip
70.0pt\geq-\,\int_{{\mathbb{R}^{N}}\times{\mathbb{R}^{N}}}\frac{|\xi_{r_{1,n}}(x-y_{n})-\xi_{r_{1,n}}(y-y_{n})|^{2}|u_{n}(x)|^{2}}{|x-y|^{N+2s}}dxdy$
$\displaystyle\hskip
85.0pt-\,\int_{{\mathbb{R}^{N}}\times{\mathbb{R}^{N}}}\frac{|\chi_{r_{2,n}}(x-y_{n})-\chi_{r_{2,n}}(y-y_{n})|^{2}|u_{n}(x)|^{2}}{|x-y|^{N+2s}}dxdy.$

Indeed, the estimate above is justified using the definition (5) combined with
the following basic fact for $u_{1,n}$

$\displaystyle|u_{1,n}(x)-u_{1,n}(y)|^{2}$
$\displaystyle=|\xi_{r_{1,n}}(x-y_{n})u_{n}(x)-\xi_{r_{1,n}}(y-y_{n})u_{n}(y)|^{2}$
$\displaystyle\leq\frac{1}{2}|\xi_{r_{1,n}}(x-y_{n})-\xi_{r_{1,n}}(y-y_{n})|^{2}\left(|u_{1,n}(x)|^{2}+|u_{1,n}(y)|^{2}\right)$
$\displaystyle+\frac{1}{2}\left(|\xi_{r_{1,n}}(x-y_{n})|^{2}+|\xi_{r_{1,n}}(y-y_{n})|^{2}\right)|u_{1,n}(x)-u_{1,n}(y)|^{2}.$

and equivalently for $u_{2,n}$

$\displaystyle|u_{2,n}(x)-u_{2,n}(y)|^{2}$
$\displaystyle=|\chi_{r_{2,n}}(x-y_{n})u_{n}(x)-\chi_{r_{2,n}}(y-y_{n})u_{n}(y)|^{2}$
$\displaystyle\leq\frac{1}{2}|\chi_{r_{2,n}}(x-y_{n})-\chi_{r_{2,n}}(y-y_{n})|^{2}\left(|u_{2,n}(x)|^{2}+|u_{2,n}(y)|^{2}\right)$
$\displaystyle+\frac{1}{2}\left(|\chi_{r_{2,n}}(x-y_{n})|^{2}+|\chi_{r_{2,n}}(y-y_{n})|^{2}\right)|u_{2,n}(x)-u_{2,n}(y)|^{2}.$

In order to show (25), it suffices to show that there exist $\epsilon>0$ such
that for large radius $r_{1,n}$ and $r_{2,n}$, we have

$\displaystyle\int_{{\mathbb{R}^{N}}\times{\mathbb{R}^{N}}}\frac{|\xi_{r_{1,n}}(x-y_{n})-\xi_{r_{1,n}}(y-y_{n})|^{2}|u_{n}(x)|^{2}}{|x-y|^{N+2s}}dxdy\leq\eta\epsilon,$
$\displaystyle\int_{{\mathbb{R}^{N}}\times{\mathbb{R}^{N}}}\frac{|\chi_{r_{2,n}}(x-y_{n})-\chi_{r_{2,n}}(y-y_{n})|^{2}|u_{n}(x)|^{2}}{|x-y|^{N+2s}}dxdy\leq\eta\epsilon.$

We prove the first assertion and the second one follows equivalently. Indeed,
we split the sum in two part as follows

$\displaystyle\int_{{\mathbb{R}^{N}}\times{\mathbb{R}^{N}}}\frac{|\xi_{r_{1,n}}(x-y_{n})-\xi_{r_{1,n}}(y-y_{n})|^{2}|u_{n}(x)|^{2}}{|x-y|^{N+2s}}dxdy$
$\displaystyle\hskip 30.0pt=\int_{|x-y|\leq
r_{1,n}}\frac{|\xi_{r_{1,n}}(x-y_{n})-\xi_{r_{1,n}}(y-y_{n})|^{2}|u_{n}(x)|^{2}}{|x-y|^{N+2s}}dxdy$
$\displaystyle\hskip
30.0pt+\int_{|x-y|>r_{1,n}}\frac{|\xi_{r_{1,n}}(x-y_{n})-\xi_{r_{1,n}}(y-y_{n})|^{2}|u_{n}(x)|^{2}}{|x-y|^{N+2s}}dxdy:=\mathcal{T}_{1}+\mathcal{T}_{2}$

Now, we write

$\displaystyle\mathcal{T}_{1}$ $\displaystyle\leq{r^{-2}_{1,n}}\int_{|x-y|\leq
r_{1,n}}\frac{|u_{n}(x)|^{2}}{|x-y|^{N+2s-2}}dxdy$
$\displaystyle\leq{r^{-2}_{1,n}}\,\int_{{\mathbb{R}^{N}}}|u_{n}(x)|^{2}dx\int_{|x|\leq
r_{1,n}}\frac{1}{|x|^{N+2s-2}}dx\leq\eta\,r^{-2s}_{1,n}\,\int_{{\mathbb{R}^{N}}}|u_{n}(x)|^{2}dx.$

Moreover,

$\displaystyle\mathcal{T}_{2}$ $\displaystyle\leq
r^{-s}_{1,n}\int_{|x-y|>r_{1,n}}\frac{|\xi_{r_{1,n}}(x-y_{n})-\xi_{r_{1,n}}(y-y_{n})|^{2}|u_{n}(x)|^{2}}{|x-y|^{N+s}}dxdy$
$\displaystyle\leq\eta\,r^{-s}_{1,n}\,\int_{{\mathbb{R}^{N}}}|u_{n}(x)|^{2}dx\int_{|x-y|>r_{1,n}}\frac{1}{|x-y|^{N+s}}dy\leq\eta\,r^{-s}_{1,n}\,\int_{{\mathbb{R}^{N}}}|u_{n}(x)|^{2}dx.$

Eventually summing up $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ and use the same
argument in order to handle the term
$\int_{{\mathbb{R}^{N}}\times{\mathbb{R}^{N}}}\frac{|\chi_{r_{2,n}}(x-y_{n})-\chi_{r_{2,n}}(y-y_{n})|^{2}|u_{n}(x)|^{2}}{|x-y|^{N+2s}}dxdy$,
one ends with

$\displaystyle\int_{\mathbb{R}^{N}}\left(|(-\Delta)^{\frac{s}{2}}\,u_{n}|^{2}-|(-\Delta)^{\frac{s}{2}}\,u_{1,n}|^{2}-|(-\Delta)^{\frac{s}{2}}\,u_{2,n}|^{2}\right)dx\,$
$\displaystyle\hskip
100.0pt\geq-\eta\,(r^{-2s}_{1,n}+\,r^{-s}_{1,n}+r^{-2s}_{2,n}+\,r^{-s}_{2,n})\,\int_{{\mathbb{R}^{N}}}|u_{n}(x)|^{2}dx.$

The estimate (25) follows for $r_{1,n}$ and $r_{2,n}$ large enough. Next,
observe that $|u_{n}-u_{1,n}-u_{2,n}|\leq
3\,\mathds{1}_{(B(y_{n},r_{2,n})\setminus B(y_{n},r_{1,n}))}$ where
$\mathds{1}_{(B(y_{n},r_{2,n})\setminus B(y_{n},r_{1,n}))}$ denotes the
characteristic function of $B(y_{n},r_{2,n})\setminus B(y_{n},r_{1,n})$. Now,
we have

$\displaystyle\left|\mathcal{D}(G(|u_{n}|),G(|u_{n}|))-\mathcal{D}(G(|v_{n}|),G(|v_{n}|))-\mathcal{D}(G(|w_{n}|),G(|w_{n}|))\right|$
$\displaystyle\leq$
$\displaystyle\int_{B(y_{n},2r)\setminus\bar{B}(y_{n},2r)}\left(\left|\frac{G(|u_{n}|)G(|u_{n}|)}{|x-y|^{N-\beta}}\right|+\left|\frac{G(|v_{n}|)G(|v_{n}|)}{|x-y|^{N-\beta}}\right|\right.$
$\displaystyle\hskip
200.0pt+\left.\left|\frac{G(|w_{n}|)G(|w_{n}|)}{|x-y|^{N-\beta}}\right|\right)dxdy,$
$\displaystyle\leq$
$\displaystyle\eta\left({|\\!|}u{|\\!|}^{4-\frac{N-\beta}{s}}_{L^{2}(B(y_{n},r_{2,n})\setminus
B(y_{n},r_{1,n}))}\,{|\\!|}u{|\\!|}^{\frac{N-\beta}{s}}_{H^{s}}+{|\\!|}u{|\\!|}^{2\mu-\frac{N(\mu-1)-\beta}{s}}_{L^{2}(B(y_{n},r_{2,n})\setminus
B(y_{n},r_{1,n}))}\,{|\\!|}u{|\\!|}^{\frac{N(\mu-1)-\beta}{s}}_{H^{s}}\right)$
$\displaystyle+$
$\displaystyle\eta\,{|\\!|}u{|\\!|}^{\mu+2-\frac{N\mu-2\beta}{2s}}_{L^{2}(B(y_{n},r_{2,n})\setminus
B(y_{n},r_{1,n}))}\,{|\\!|}u{|\\!|}^{\frac{N\mu-2\beta}{2s}}_{H^{s}}\xrightarrow[n\to+\infty]{}0.$

where we used the estimates (16–18). Thus, for $r_{2,n}$ and $r_{1,n}$ large
enough we have

$-\frac{1}{2}\left(\mathcal{D}(G(|u_{n}|),G(|u_{n}|))-\mathcal{D}(G(|v_{n}|),G(|v_{n}|))-\mathcal{D}(G(|w_{n}|),G(|w_{n}|))\right)\geq-\eta\epsilon.$
(26)

Summing up (25) and (26), we end up for large $r_{1,n}$ and $r_{2,n}$ with

$\mathcal{E}(u_{n})-\mathcal{E}(u_{1,n})-\mathcal{E}(u_{2,n})\geq-\eta\epsilon.$
(27)

Since we have
$\int_{\mathbb{R}^{N}}|u_{1,n}|^{2}dx\xrightarrow[n\to+\infty]{}\pi$ and
$\int_{\mathbb{R}^{N}}|u_{1,n}|^{2}dx\xrightarrow[n\to+\infty]{}\lambda-\pi$,
there exist two positive real sequences $(\mu_{1,n})_{n\in\mathbb{N}}$ and
$(\mu_{2,n})_{n\in\mathbb{N}}$ such that
$|\mu_{1,n}-1|,|\mu_{2,n}-1|<\epsilon$ and

$\int_{\mathbb{R}^{N}}|\mu_{1,n}u_{1,n}|^{2}dx=\pi,\quad\int_{\mathbb{R}^{N}}|\mu_{2,n}u_{2,n}|^{2}dx=\lambda-\pi,$

so that

$\displaystyle\mathcal{I}_{\pi}\leq\mathcal{E}(\mu_{1,n}u_{1,n})\leq\mathcal{E}(u_{1,n})+\frac{\eta\epsilon}{2},$
$\displaystyle\mathcal{I}_{\lambda-\pi}\leq\mathcal{E}(\mu_{2,n}u_{2,n})\leq\mathcal{E}(u_{2,n})+\frac{\eta\epsilon}{2}.$

Thus, with (27), we have and the continuity of the mapping
$\lambda\mapsto\mathcal{I}_{\lambda}$ for all $\lambda>0$, we have

$\mathcal{I}_{\pi}+\mathcal{I}_{\lambda-\pi}-3\eta\epsilon\leq\mathcal{E}(u_{1,n})+\mathcal{E}(u_{2,n})-\eta\epsilon\leq\mathcal{E}(u_{n})\xrightarrow[n\to+\infty]{}\mathcal{I}_{\lambda}.$

In summary, we proved that for all $0<\pi<\lambda$, we have
$\mathcal{I}_{\pi}+\mathcal{I}_{\lambda-\pi}\leq\mathcal{I}_{\lambda}$
contradicting the strict sub–additivity inequality proved above. Then, the
dichotomy does not occur. ∎

Now, we finish the proof of Theorem 1.2. Since vanishing and dichotomy do not
occur for any minimizing sequence $(u_{n})_{n\in\mathbb{N}}$ for the problem
$\mathcal{I}_{\lambda}$, then the compactness certainly occurs. Following the
concentration-compactness principle [17], we know that every minimizing
sequence $(u_{n})_{n\in\mathbb{N}}$ of $\mathcal{I}_{\lambda}$ satisfies (up
to extraction if necessary)

$\lim_{r\to+\infty}\lim_{n\to+\infty}\sup_{y\in{\mathbb{R}^{N}}}\int_{B(y,r)}|u_{n}(x)|^{2}dx=\lambda.$

That is, for all $\epsilon>0$, there exist $r_{\epsilon}>0$ and
$n_{\epsilon}\in\mathbb{N}^{\star}$ and $\\{y_{n}\\}\subset{\mathbb{R}^{N}}$
such that for all $r>r_{\epsilon}$ and $n\geq n_{\epsilon}$, we have

$\int_{B(y_{n},r)}|u_{n}(x)|^{2}dx=\lambda-\epsilon$

Now, let $w_{n}=u_{n}(x+y_{n})$, we have obviously that
${|\\!|}w_{n}{|\\!|}_{H^{s}}={|\\!|}u_{n}{|\\!|}_{H^{s}}$ is bounded in
$H^{s}(\mathbb{R}^{N})$, therefore $(w_{n})_{n\in\mathbb{N}}$ (up to
extraction if necessary) converges weakly to $w$ in $H^{s}(\mathbb{R}^{N})$.
In particular $(w_{n})_{n\in\mathbb{N}}$ converges weakly to $w$ in
${L^{2}(\mathbb{R}^{N})}$ and ${|\\!|}w_{n}{|\\!|}_{2}=\sqrt{\lambda}$. Now,
let $\tilde{r}_{\epsilon}>r_{\epsilon}$ such that
${|\\!|}w{|\\!|}_{L^{2}(B^{c}(0,\tilde{r}_{\epsilon}))}<\frac{\epsilon}{2}$.
Thus, there exists
$\tilde{n}_{\epsilon}\in\mathbb{N}^{\star},\>\tilde{n}_{\epsilon}>n_{\epsilon}$
such that for all $n\geq\tilde{n}_{\epsilon}$, we have
${|\\!|}w_{n}-w{|\\!|}_{L^{2}(B(0,\tilde{r}_{\epsilon}))}<\frac{\epsilon}{2}$.
Therefore, with the triangle inequality, we have

$\displaystyle{|\\!|}w{|\\!|}_{2}$ $\displaystyle\geq$
$\displaystyle{|\\!|}u_{n}{|\\!|}_{2}-{|\\!|}w_{n}-w{|\\!|}_{L^{2}(B(0,\tilde{r}_{\epsilon}))}-{|\\!|}w_{n}-w{|\\!|}_{L^{2}(B^{c}(0,\tilde{r}_{\epsilon}))},$
$\displaystyle\geq$
$\displaystyle{|\\!|}u_{n}{|\\!|}_{L^{2}(B(y_{n},\tilde{r}_{\epsilon}))}-{|\\!|}w_{n}-w{|\\!|}_{L^{2}(B(0,\tilde{r}_{\epsilon}))}-{|\\!|}w{|\\!|}_{L^{2}(B^{c}(0,\tilde{r}_{\epsilon}))}\geq\sqrt{\lambda-\epsilon}-\epsilon.$

Passing to the limit we get ${|\\!|}w{|\\!|}_{2}\geq\sqrt{\lambda}$. Since the
$L^{2}$ is lower semi continuous, we obtain that
${|\\!|}w{|\\!|}_{2}\leq\liminf_{n\to+\infty}{|\\!|}w_{n}{|\\!|}_{2}=\sqrt{\lambda}$.
Eventually, we get ${|\\!|}w{|\\!|}_{2}=\sqrt{\lambda}$, therefore the
sequence $(w_{n})_{n\in\mathbb{N}}$ converges strongly in
${L^{2}(\mathbb{R}^{N})}$ to $w$. Also, we have

$\displaystyle\left|\mathcal{D}(G(|w_{n}|),G(|w_{n}|))-D(G(|w|),G(|w|))\right|$
$\displaystyle\hskip
70.0pt\leq\left|\int_{0}^{t}\frac{d}{dt}\mathcal{D}(tG(|w_{n}|)+(1-t)G(|w|))\,dt\right|,$
$\displaystyle\hskip
70.0pt\leq\eta\sup_{u\in{H^{s}},{|\\!|}u{|\\!|}_{H^{s}}\leq\eta}{|\\!|}\mathcal{D}^{\prime}(u){|\\!|}_{H^{-s}}\,{|\\!|}w_{n}-w{|\\!|}_{H^{s}},$
$\displaystyle\hskip
70.0pt\leq\eta\,{|\\!|}w_{n}-w{|\\!|}_{2}+\eta{|\\!|}w_{n}-w{|\\!|}_{\frac{2s+\beta}{N}}\xrightarrow[n\to+\infty]{}0.$

In the last line we used 24-kind inequality and again we refer to [11] for a
proof. Using the lower semi-continuity of the $-s$ norm, we have
${|\\!|}w{|\\!|}_{H^{s}}\leq\liminf_{n\to+\infty}{|\\!|}w_{n}{|\\!|}_{H^{s}}$.
Summing up, we get clearly

$\mathcal{I}_{\lambda}\leq\mathcal{E}(w)\leq\liminf_{n\to+\infty}\mathcal{E}(w_{n})=\mathcal{I}_{\lambda}.$

This shows that $w$ is a minimizer of $\mathcal{I}_{\lambda}$ and
$w_{n}\xrightarrow[n\to+\infty]{}w$ in $H^{s}(\mathbb{R}^{N})$. Theorem 1.1 is
now proved.

##  4 Stability of standing waves

In this section, we prove the orbital stability of standing waves in the sense
of Definition 1.3. That is we prove Theorem 1.4.

We argue par contradiction. Assume that $\hat{\mathcal{O}}_{\lambda}$ is not
stable, then either $\hat{\mathcal{O}}_{\lambda}$ is empty or there exist
$w\in\hat{\mathcal{O}}_{\lambda}$ and a sequence $\phi^{n}_{0}\in H^{s}$ such
that ${|\\!|}\phi^{n}_{0}-w{|\\!|}_{H^{s}}\xrightarrow[n\to+\infty]{}0$ as
$n\rightarrow\infty$ but

$\displaystyle{\inf_{z\in\hat{\mathcal{O}}_{\lambda}}}{|\\!|}\phi^{n}(t_{n},.)-z{|\\!|}_{H^{s}}\geq\varepsilon,$
(28)

for some sequence $t_{n}\subset\mathbb{R}$, where $\phi^{n}(t_{n},.)$ is the
solution of the Cauchy problem $\mathscr{S}$ corresponding to the initial
condition $\phi^{n}_{0}$. Now let $w_{n}=\phi^{n}(t_{n},.)$, since
${\mathcal{J}}(w)=\hat{\mathcal{I}}_{\lambda}$, it follows from the continuity
of the $L^{2}$ norm and $\mathcal{J}$ in $H^{s}$ that
${|\\!|}\phi^{n}_{0}{|\\!|}_{2}\xrightarrow[n\to+\infty]{}\sqrt{\lambda}$ and
$\mathcal{J}(w_{n})=\mathcal{J}(\phi^{n}_{0})=\hat{\mathcal{I}}_{\lambda}$.
With the conservation of mass and energy associated with the dynamics of the
system $\mathscr{S}$, we deduce that

$\displaystyle{|\\!|}w_{n}{|\\!|}_{2}={|\\!|}\phi^{n}_{0}{|\\!|}_{2}\xrightarrow[n\to+\infty]{}\sqrt{\lambda}\quad\text{and}\quad\mathcal{J}(w_{n})=\mathcal{J}(\phi_{0}^{n})\xrightarrow[n\to+\infty]{}\hat{\mathcal{I}}_{\lambda}.$

Therefore if $(w_{n})_{n\in\mathbb{N}}$ has a subsequence converging to an
element $w\in H^{s}$: ${|\\!|}w{|\\!|}_{2}=\sqrt{\lambda}$ and
$\mathcal{J}(w)=\hat{\mathcal{I}}_{c}$. This shows that
$w\in\hat{\mathcal{O}}_{\lambda}$, but

$\inf_{z\in\hat{\mathcal{O}}_{\lambda}}{|\\!|}\phi^{n}(t_{n},.)-z{|\\!|}_{H^{s}}\leq{|\\!|}w_{n}-w{|\\!|}_{H^{s}}$

contradicting (28).

In summary, to show the orbital stability of $\hat{\mathcal{O}}_{\lambda}$,
one has to prove that $\hat{\mathcal{O}}_{\lambda}$ is not empty and that any
sequence $(w_{n})_{n\in\mathbb{N}}\subset H^{s}$ such that

${|\\!|}w_{n}{|\\!|}_{2}\xrightarrow[n\to+\infty]{}\sqrt{\lambda}\quad\mbox{
and
}\quad\mathcal{J}(w_{n})\xrightarrow[n\to+\infty]{}\hat{\mathcal{I}}_{\lambda},$
(29)

is relatively compact in $H^{s}$ (up to a translation).

From now on, we consider a sequence $(w_{n})_{n\in\mathbb{N}}$ satisfying
(29). Our aim is to prove that it admits a convergent subsequence to an
element $w\in H^{s}$. If $(w_{n})_{n\in\mathbb{N}}\subset H^{s}$, it is easy
to see that

$(|w_{n}|)_{n\in\mathbb{N}}\subset H^{s}\,;\quad w_{n}=(u_{n},v_{n}).$

Thanks to $\mathcal{A}_{0}$, we have that $(w_{n})_{n\in\mathbb{N}}$ is
bounded in $H^{s}$ and hence by passing to a subsequence, there exists
$w=(u,v)\in H^{s}$ such that

$\left\\{\begin{array}[]{l}u_{n}\mbox{ converges weakly to }u\mbox{ in
}H^{s},\\\ \\\ v_{n}\mbox{ converges weakly to }v\mbox{ in }H^{s},\\\ \\\
\mbox{ the limit when}\>n\>\mbox{goes to
}+\infty\>\mbox{of}\>{|\\!|}\nabla_{s}u_{n}{|\\!|}_{2}+{|\\!|}\nabla_{s}v_{n}{|\\!|}_{2}\>\mbox{
exists }.\end{array}\right.$ (30)

Now, a straightforward calculation shows that

$\mathcal{J}(w_{n})-\mathcal{E}(|w_{n}|)=\frac{1}{2}{|\\!|}\nabla_{s}w_{n}{|\\!|}^{2}_{2}-\frac{1}{2}{|\\!|}\nabla_{s}|w_{n}|{|\\!|}^{2}_{2}\geq
0.$ (31)

Thus we have

$\hat{\mathcal{I}}=\lim_{n\to+\infty}\mathcal{J}(w_{n})\geq\limsup_{n\to+\infty}\mathcal{E}(|w_{n}|).$
(32)

But

${|\\!|}|w_{n}|{|\\!|}^{2}_{2}={|\\!|}w_{n}{|\\!|}^{2}_{2}=\lambda_{n}\xrightarrow[n\to+\infty]{}\lambda.$
(33)

By the continuity of the mapping $\lambda\mapsto\mathcal{I}_{\lambda}$ (see
Proposition 3.3), we obtain

$\lim_{n\to+\infty}\mathcal{J}(w_{n})\geq\liminf_{n\to+\infty}\mathcal{I}_{\lambda_{n}}=\mathcal{I}_{\lambda}\geq\hat{\mathcal{I}}_{\lambda}.$
(34)

Hence

$\lim_{n\rightarrow+\infty}\mathcal{J}(w_{n})=\lim_{n\rightarrow+\infty}\mathcal{E}(|w_{n}|)=\mathcal{I}_{\lambda}=\hat{\mathcal{I}}_{\lambda}.$

The properties (30) and the inequalities (31) and (34) imply that

$\lim_{n\rightarrow+\infty}{|\\!|}\nabla_{s}u_{n}{|\\!|}^{2}_{2}-{|\\!|}\nabla_{s}v_{n}{|\\!|}^{2}_{2}-{|\\!|}\nabla_{s}(u^{2}_{n}+v_{n}^{2})^{1/2}{|\\!|}^{2}_{2}=0,$
(35)

which is equivalent to say that

$\lim_{n\rightarrow+\infty}{|\\!|}\nabla_{s}w_{n}{|\\!|}^{2}=\lim_{n\rightarrow+\infty}{|\\!|}\nabla_{s}|w_{n}|{|\\!|}^{2}_{2}.$
(36)

The convergence (33), the inequality (34) and Theorem 1.2 imply that $|w_{n}|$
is relatively compact in $H^{s}$ (up to a translation). Therefore, there
exists $\varphi\in H^{s}$ such that

$(u^{2}_{n}+v^{2}_{n})^{1/2}\rightarrow\varphi\mbox{ in}\>\>H^{s}\>\>\mbox{
and
}\>\>{|\\!|}\varphi{|\\!|}_{2}=\sqrt{\lambda}\>\>\mbox{with}\>\>\mathcal{E}(\varphi)=I_{\lambda}.$

Let us prove that $\varphi=|w|=(u^{2}+v^{2})^{1/2}$. Using (30), it follows
that $u_{n}\xrightarrow[n\to+\infty]{}u$ and
$v_{n}\xrightarrow[n\to+\infty]{}v$ in $L^{2}(B(0,R))$

$\displaystyle|(u^{2}_{n}+v^{2}_{n})^{1/2}-(u^{2}+v^{2})^{1/2}|\leq|u_{n}-u|^{2}+|v_{n}-v|^{2},$
$\displaystyle(u^{2}_{n}+v^{2}_{n})^{1/2}\xrightarrow[n\to+\infty]{}(u^{2}+v^{2})^{1/2}\quad\mbox{
in }L^{2}(B(0,R)).$

Thus we certainly have that $(u^{2}+v^{2})^{1/2}=|w|=\varphi$. On the other
hand
${|\\!|}|w_{n}|{|\\!|}_{2}={|\\!|}w_{n}{|\\!|}_{2}\xrightarrow[n\to+\infty]{}\sqrt{\lambda}={|\\!|}w{|\\!|}_{2}={|\\!|}|w|{|\\!|}_{2}$.
Therefore, we are done if we prove that
$\lim_{n\to\infty}{|\\!|}\nabla_{s}w_{n}{|\\!|}^{2}_{2}={|\\!|}\nabla_{s}w{|\\!|}^{2}_{2}$.
From (36), we have that
$\lim_{n\to+\infty}{|\\!|}\nabla_{s}w_{n}{|\\!|}^{2}_{2}=\lim_{n\to+\infty}|\nabla_{s}|w_{n}|{|\\!|}^{2}_{2}$
and
$\lim_{n\to+\infty}{|\\!|}\nabla_{s}|w_{n}|{|\\!|}^{2}_{2}={|\\!|}\nabla_{s}|w|{|\\!|}^{2}_{2}$.
Hence by the lower semi-continuity of ${|\\!|}\nabla_{s}\cdot{|\\!|}_{2}$, we
obtain

${|\\!|}\nabla_{s}w{|\\!|}^{2}_{2}\leq\lim_{n\to+\infty}{|\\!|}\nabla_{s}|w_{n}|{|\\!|}^{2}_{2}={|\\!|}\nabla_{s}|w|{|\\!|}^{2}_{2}.$
(37)

Eventually, using (31), it follows that

${|\\!|}\nabla_{s}w{|\\!|}^{2}_{2}\geq{|\\!|}\nabla_{s}|w|{|\\!|}^{2}_{2}.$

Since by (30), we know that $w_{n}$ converges weakly to $w$ in $H^{s}$, it
follows that $w_{n}\xrightarrow[n\to+\infty]{}w$ in $H^{s}$, which completes
the proof.
Now, we turn to the characterization of the Orbit
$\hat{\mathcal{O}}_{\lambda}$. We show the following

######  Proposition 4.1.

With the same assumptions of Theorem 1.4, we have

$\hat{\mathcal{O}}_{\lambda}=\left\\{e^{i\sigma}w(.+y),\quad\sigma\in\mathbb{R},y\in\mathbb{R}^{N}\right\\},$

$w$ is a minimizer of (4).

###### Proof.

Let $z=(u,v)\in\hat{\mathcal{O}}_{\lambda}$ and set
$\varphi=(u^{2}+v^{2})^{1/2}$. By the previous section, we know that
$\mathcal{E}(\varphi)=I_{\lambda}$, thus $\varphi$ satisfies the partial
differential equation :

$(-\Delta)^{s}\varphi+\kappa\varphi=V\star G(|\varphi|)G^{\prime}(\varphi),$
(38)

where $\kappa$ is a Lagrange multiplier. Furthermore the equality
$\|\nabla_{s}w\|_{2}=\|\nabla_{s}|w|\|_{2}$ implies that

$u(x)v(y)-v(x)u(y)=0.$ (39)

By Proposition 4.2, it is plain that $\varphi\in C(\mathbb{R}^{N})$ and
$V\star G(|\varphi|)\in C(\mathbb{R}^{N})$. We can write
$(-\Delta)^{s}\varphi+\kappa\varphi=V\star
G(|\varphi|)\frac{G^{\prime}(\varphi)}{\varphi}\chi_{\\{\varphi\neq
0\\}}\varphi$, with $\chi_{A}$ being the characteristic function of the set
$A$. Since $\varphi$ is nontrivial and $V\star
G(|\varphi|)\frac{G^{\prime}(\varphi)}{\varphi}\chi_{\\{\varphi\neq 0\\}}\in
L^{\infty}_{loc}({\mathbb{R}^{N}})$, we conclude that $\varphi>0$ in
${\mathbb{R}^{N}}$ by the Harnack inequality (see Lemma 4.9 in [2]) and a
standard argument of intersecting balls.

Case 1 : $u\equiv 0$

Case 2 : $v\equiv 0$

Case 3 : $u\neq 0$ and $v\neq 0$ everywhere.

Then (39) implies that

$\displaystyle\frac{u(x)}{v(x)}=\frac{u(y)}{v(y)}\;\quad\forall\;x,y\in\mathbb{N}^{N},$
$\displaystyle\Rightarrow\frac{u(x)}{v(x)}=\alpha\Rightarrow u(x)=\alpha
v(x)\quad\forall\;x\in\mathbb{R}^{N},$ $\displaystyle z=(\alpha+i)v\Rightarrow
z=e^{i\sigma}w,w=|z|.$

Let us now prove (39). By the fact that
$\mathcal{J}(z)=\hat{\mathcal{I}}_{\lambda}$, we can find a Lagrange
multiplier $\alpha\in\mathbb{C}$ such that
$\mathcal{J}^{\prime}(z)(\xi)=\displaystyle{\frac{\alpha}{2}\int_{\mathbb{R}^{N}}}z\bar{\xi}+\xi\bar{z}$
for all $\xi\in H^{s}$. Putting $\xi=z$, it follows immediately that
$\alpha\in\mathbb{R}$ and

$\left\\{\begin{array}[]{l}\displaystyle{\int}_{{\mathbb{R}^{N}}}\nabla_{s}u\nabla_{s}f-\displaystyle{\int}_{{\mathbb{R}^{N}}\times{\mathbb{R}^{N}}}G(u^{2}+v^{2})^{1/2}(y)V(|x-y|)dyG^{\prime}(f(x))dx\\\
\\\ \hskip 215.0pt=\alpha\displaystyle{\int}_{{\mathbb{R}^{N}}}u(x)f(x)dx,\\\
\\\
\displaystyle{\int}_{{\mathbb{R}^{N}}}\nabla_{s}v\nabla_{s}f-\displaystyle{\int}_{{\mathbb{R}^{N}}\times{\mathbb{R}^{N}}}G(u^{2}+v^{2})^{1/2}(y)V(|x-y|)dyG^{\prime}(f(x))dx\\\
\\\ \hskip
215.0pt=\alpha\displaystyle{\int}_{{\mathbb{R}^{N}}}v(x)f(x)dx,\end{array}\right.$

$\nabla_{s}$ denotes the fractional gradient, for all $f\in H^{s}$. It follows
that $u$ and $v$ solve the following system

$\left\\{\begin{array}[]{ll}(-\Delta)^{s}\,u+\displaystyle{\int}G(u^{2}+v^{2})^{1/2}(y)V(|x-y|)dy\,G^{\prime}(u(x))+\alpha
u(x)=0,\\\
(-\Delta)^{s}\,v+\displaystyle{\int}G(u^{2}+v^{2})^{1/2}(y)V(|x-y|)dy\,G^{\prime}(v(x))+\alpha
v(x)=0.\end{array}\right.$

By Proposition 4.2, we have that $u$ and $v\in C(\mathbb{R}^{N})$ because
$(u^{2}+v^{2})^{1/2}\in H^{s}(\mathbb{R}^{N})$. Let
$\Omega=\\{x\in\mathbb{R}^{N}:u(x)=0\\}$, obviously $\Omega$ is closed since
$u$ is continuous. Let us prove that it is also open. Suppose that
$x_{0}\in\Omega$. Knowing that $\varphi(x_{0})>0$, we can find a ball $B$
centered in $x_{0}$ such that $v(x)\neq 0$ for any $x\in B$. Replacing $u$ and
$v$ in (35), we certainly have that

$u(x)v(y)-v(x)u(y)=0\quad\forall\;x,y\in B.$

This proves the result. ∎

## Appendix

In this appendix, we prove the following

######  Proposition 4.2.

Let $s\in(0,1),N-2s\leq\beta<N,\beta>0,u,\varphi\in H^{s}(\mathbb{R}^{N})$,
$G$ such that $\mathcal{A}_{0}$ holds and $\kappa$ is a real number such that

$(-\Delta)^{s}u-\kappa u=[V\star G(\varphi)]G^{\prime}(u).$ (40)

Then, there exists $\alpha\in(0,1)$ depending only on $N,\kappa,s,\beta$ such
that $u\in C^{0,\alpha}_{loc}({\mathbb{R}^{N}})$. Moreover, if $\varphi\in
L^{\infty}_{loc}({\mathbb{R}^{N}})$, then $u\in
C^{0,\alpha}_{loc}({\mathbb{R}^{N}})$ if $\beta\leq 1$ and $u\in
C^{1,\alpha}_{loc}({\mathbb{R}^{N}})$ if $\beta>1$ and in addition $V\star
G(\varphi)\in C^{0,\alpha}_{loc}({\mathbb{R}^{N}})$.

###### Proof.

We start by recalling the Gagliardo-Nirenberg inequality

${|\\!|}\varphi{|\\!|}_{L^{p}({\mathbb{R}^{N}})}\leq
c_{N,s,p}{|\\!|}\varphi{|\\!|}_{H^{s}({\mathbb{R}^{N}})}\quad\textrm{ for all
$\varphi\in H^{s}(\mathbb{R}^{N})$ },$

for $p\in\left[2,\frac{2N}{N-2s}\right]$ if $N>2s$ and for all
$p\in\left[\left.2,\frac{2N}{N-2s}\right)\right.$ and $2s\geq N$ (here we put
$\frac{2N}{N-2s}\equiv+\infty$). Also we recall the Hardy-Littlewood-Sobolev
inequality:

$\|V\star g\|_{L^{\frac{qN}{N-q\beta}}({\mathbb{R}^{N}})}\leq
C_{N,\beta,q}\|g\|_{L^{q}({\mathbb{R}^{N}})}\quad\textrm{ for every $g\in
L^{q}({\mathbb{R}^{N}})$,}$

for $N-q\beta>0$. First of all we focus on the case $N>2s$. Thus, we have

${|\\!|}G(\varphi){|\\!|}_{L^{q}({\mathbb{R}^{N}})}\leq{|\\!|}\varphi^{2}{|\\!|}_{L^{q}({\mathbb{R}^{N}})}+{|\\!|}|\varphi|^{\mu}{|\\!|}_{L^{q}({\mathbb{R}^{N}})}={|\\!|}\varphi{|\\!|}_{L^{2q}({\mathbb{R}^{N}})}^{2}+{|\\!|}\varphi{|\\!|}_{L^{\mu
q}({\mathbb{R}^{N}})}^{\mu}.$

Hence, since $\varphi\in H^{s}(\mathbb{R}^{N})$, we infer that $G(\varphi)\in
L^{q}({\mathbb{R}^{N}})$ provided that $1\leq q\leq\frac{N}{N-2s}$ and
$\frac{2}{\mu}\leq q\leq\frac{1}{\mu}\frac{2N}{N-2s}$, that is $1\leq
q\leq\frac{N}{N-2s}$ and $1\leq q\leq\frac{2N^{2}}{(N-2s)(N+2s+\beta)}$. Now,
thanks to the fact that $N-2s\leq\beta<N$, we get
$1<\frac{N}{\beta}\leq\frac{N}{N-2s}$ and
$1<\frac{N}{\beta}\leq\frac{2N^{2}}{(N-2s)(N+2s+\beta)}$. In particular, we
deduce that $G(\varphi)\in L^{q}({\mathbb{R}^{N}})$ for all
$q\in\left[1,\frac{N}{\beta}\right]$. Now, for all $\epsilon>0$ we let
$q_{\epsilon}=\frac{N}{\beta}-\epsilon>1$. Using the Hardy-Littelwood-Sobolev
inequality, we get $V\star G(\varphi)\in
L^{\frac{Nq_{\epsilon}}{\epsilon\beta}}({\mathbb{R}^{N}})$ which in turns with
the fact that $\beta\geq N-2s$ shows that $V\star G(\varphi)\in
L^{r}({\mathbb{R}^{N}})$ for all $r>\frac{N}{N-\beta}\geq\frac{N}{2s}$. Now,
using the notation $b(x)=\frac{G^{\prime}(u)}{1+|u|}$ and
$sign(u)=\frac{u}{|u|}$, we reformulate the equation (40) as follows

$\displaystyle(-\Delta)^{s}u(x)-\kappa u(x)$ $\displaystyle=$
$\displaystyle[V\star G(\varphi)]\,b(x)\,(1+|u|)),$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{R}^{N}}}V(|x-y|)G(\varphi(y))dy\,b(x)\,(1+sign(u)\,u).$

Observing that $\mu-2<\frac{2N}{N-2s}-2=\frac{4s}{N-2s}$, then for all
$r>\frac{N}{2s}$, we can write

$\displaystyle{|\\!|}[V\star G(\varphi)]b{|\\!|}_{L^{r}({\mathbb{R}^{N}})}$
$\displaystyle={|\\!|}[V\star
G(\varphi)]\,\frac{G^{\prime}(u)}{1+|u|}{|\\!|}_{L^{r}({\mathbb{R}^{N}})},$
$\displaystyle\leq c\,{|\\!|}[V\star
G(\varphi)]\frac{|u|+|u|^{\mu-1}}{1+|u|}{|\\!|}_{L^{r}({\mathbb{R}^{N}})},$
$\displaystyle\leq c\,{|\\!|}V\star
G(\varphi){|\\!|}_{L^{r}({\mathbb{R}^{N}})}+c\,{|\\!|}[V\star
G(\varphi)]|u|^{\mu-2}{|\\!|}_{L^{r}({\mathbb{R}^{N}})}.$

In order to deduce that the right hand side of this estimate is finite, we use
Hölder’s inequality to get

$\displaystyle{|\\!|}[V\star
G(\varphi)]\,|u|^{\mu-2}{|\\!|}^{r}_{L^{r}({\mathbb{R}^{N}})}$
$\displaystyle\leq{|\\!|}V\star
G(\varphi)]{|\\!|}_{L^{\frac{r\,\theta}{\theta-1}}({\mathbb{R}^{N}})}\,{|\\!|}|u|{|\\!|}^{\mu-2}_{L^{r\,(\mu-2)\,\theta}({\mathbb{R}^{N}})},$

for all $\theta>1$. Therefore, we can choose $r>\frac{N}{2s}$ and $\theta>1$
respectively close to $\frac{N}{2s}$ and $1$ so that
$1<r\,(\mu-2)\,\theta<\frac{2N}{N-2s}$. Hence using the Gagliardo-Nirenberg
inequality and the fact that $V\star G(\varphi)\in L^{r}({\mathbb{R}^{N}})$
for all $r>\frac{N}{N-\beta}\geq\frac{N}{2s}$ and $u\in
H^{s}(\mathbb{R}^{N})$, we end up with $[V\star G(\varphi)]b\in L^{r}$ for
some $r>\frac{N}{2s}$, hence $u\in C^{0,\alpha}_{loc}({\mathbb{R}^{N}})$. Now,
we write the equation (40) as follows

$\displaystyle(-\Delta)^{s}u(x)=c(x)u(x)+d(x):$ $\displaystyle
c(x)=\kappa+[V\star G(\varphi)]\,b(x)\,sign(u)\in L^{r}({\mathbb{R}^{N}}),$
$\displaystyle d(x)=[V\star G(u)]\,b(x)\in L^{r}({\mathbb{R}^{N}}),$

for some $r>\frac{N}{2s}$. Thus, using the regularity result of Ref. [20], we
conclude that $u\in C^{0,\alpha}_{loc}({\mathbb{R}^{N}})$ for some
$\alpha\in(0,1)$ provided $\frac{N}{2s}>1$. If $N=1$ and $s>\frac{1}{2}$, then
it is well-known that $H^{s}(\mathbb{R}^{N})$ is embedded in
$C^{0,\alpha}_{loc}({\mathbb{R}^{N}})$ with
$\alpha=s-\frac{1}{2}-\left[s-\frac{1}{2}\right]$ so that $u\in
C^{0,\alpha}_{loc}({\mathbb{R}^{N}})$. Moreover, if $N=1$ and $s=\frac{1}{2}$,
we have obviously $u\in L^{p}({\mathbb{R}^{N}})$ for every $p\geq 2$ and
classical elliptic regularity yields $u\in
C^{0,\alpha}_{loc}({\mathbb{R}^{N}})$ for some $\alpha\in(0,1)$. In the
following, $[\cdot]$ stands for the integer part of $\cdot$. Let us introduce
a cutoff function $\eta\in C_{c}^{\infty}({\mathbb{R}^{N}})$ such that
$\eta\equiv 1$ in the closed ball $B_{R}$ of center $0$ and radius $R>0$ and
$\eta\equiv 0$ in ${\mathbb{R}^{N}}\setminus B_{2R}$. To alleviate the
notation, we denote $f=G(\varphi)$ which belongs to
$L^{\infty}_{loc}({\mathbb{R}^{N}})\cap L^{q}({\mathbb{R}^{N}})$ with
$1<q\leq\frac{N}{\beta}$. We define $V_{1}(\varphi):=V\star(\eta f)$ and
$V_{2}(\varphi):=V\star((1-\eta)f)$. Then using Fourier transform, we get
$(-\Delta)^{\frac{\beta}{2}}V_{1}(\varphi)=f$ in the sense of distributions.
Now, if $\frac{\beta}{2}\in\mathbb{N}^{\star}$, then it is rather easy to show
using classical regularity theory that $V\star G(\varphi)\in
C^{\beta}({\mathbb{R}^{N}})$. Next, if $0<\frac{\beta}{2}<1$, then we apply
Proposition 2.1.9 of Ref. [19] to show that $V_{1}(\varphi)\in
C^{0,\alpha}({\mathbb{R}^{N}})$ for $\beta\leq 1$ and $V_{1}(\varphi)\in
C^{[\beta],\alpha}({\mathbb{R}^{N}})$ for $\beta>1$ and some $\alpha\in(0,1)$.
Now, $V_{2}(\varphi)$ is smooth on $B_{R}$ since it is
$\frac{\beta}{2}-$harmonic in such a ball, see Ref. [1]. Hence, $V\star
G(\varphi)\in C^{0,\alpha}_{loc}({\mathbb{R}^{N}})$ for $\beta\leq 1$ and
$V\star G(\varphi)\in C^{[\beta],\alpha}_{loc}({\mathbb{R}^{N}})$ for
$\beta>1$ and some $\alpha\in(0,1)$. Let us now turn to the case of
$\frac{\beta}{2}>1$ and $\frac{\beta}{2}\not\in\mathbb{N}$. we let
$\sigma=\frac{\beta}{2}-\left[\frac{\beta}{2}\right]$. Using Fourier
transform, we have

$(-\Delta)^{\left[\frac{\beta}{2}\right]}V_{1}(\varphi)=(-\Delta)^{\left[\frac{\beta}{2}\right]}\left((-\Delta)^{\sigma}V_{1}(\varphi)\right)=\eta\,f$

in the sense of distributions. Again, classical regularity theory arguments
implies that $(-\Delta)^{\sigma}V_{1}(\varphi)\in
C^{[\beta]}({\mathbb{R}^{N}})$ and so $V_{1}(\varphi)\in
C^{[\beta]}({\mathbb{R}^{N}})$. Similarly, we have

$(-\Delta)^{\frac{\beta}{2}}V_{2}(\varphi)=(-\Delta)^{\sigma}\left((-\Delta)^{\left[\frac{\beta}{2}\right]}V_{2}(\varphi)\right)=(1-\eta)\,f$

in the sense of distributions. Therefore the function
$g:=(-\Delta)^{\left[\frac{\beta}{2}\right]}V_{2}(\varphi)$ is given by

$(-\Delta)^{\left[\frac{\beta}{2}\right]}V_{2}(\varphi)(x)=\int_{{\mathbb{R}^{N}}}\frac{(1-\eta(y))\,f(y)}{|x-y|^{N-\sigma}}\,dy.$

Also, using the Hardy-Littelwood-Sobolev inequality, it is rather
straightforward to see that $g\in L^{p}({\mathbb{R}^{N}})$ for some $p>1$.
Thus, $g$ belongs to the set
$\left\\{u,\>\int_{{\mathbb{R}^{N}}}\frac{|u(x)|}{1+|x|^{N+2\sigma}}\,dx<+\infty\right\\}$.
Again, since $g$ is $\sigma-$harmonic in $B_{R}$, we deduce that $g$ is smooth
on $B_{R}$ by Ref. [1]. The radius $R$ being arbitrary, it follows that
$V_{2}(\varphi)$ is smooth on ${\mathbb{R}^{N}}$. In particular, we have
$V_{2}(\varphi)\in C^{[\beta]}({\mathbb{R}^{N}})$ because
$\left[\frac{\beta}{2}\right]$ is a positive integer. Recalling that we showed
$V_{1}(\varphi)\in C^{[\beta]}({\mathbb{R}^{N}})$, we conclude $V\star
G(\varphi)\in C^{[\beta]}({\mathbb{R}^{N}})$. Let us now summarize and
conclude the proof. We considered the partial differential equation (40) and
proved that for some $\alpha\in(0,1)$, we have $V\star G(\varphi)\in
C^{0,\alpha}_{loc}({\mathbb{R}^{N}})$ for $\beta\leq 1$ and $V\star
G(\varphi)\in C^{[\beta],\alpha}_{loc}({\mathbb{R}^{N}})$ for $\beta>1$. Since
$G^{\prime}$ is locally Lipschitz, we deduce that $u\in
C^{0,\alpha}_{loc}({\mathbb{R}^{N}})$ for $\beta\leq 1$ and $u\in
C^{1,\alpha}_{loc}({\mathbb{R}^{N}})$ for $\beta>1$ by adapting the proof of
Lemma 3.3 of Ref. [8] for $N>2s$. If $N=1$ and $2s\geq 1$, we have that
$[V\star G(\varphi)]\,G^{\prime}(u)\in C^{0,\gamma}_{loc}({\mathbb{R}^{N}})$
for some $\gamma\in(0,1)$, thus using Proposition 2.1.8 of Ref. [19], we get
$u\in C^{1,\alpha}_{loc}({\mathbb{R}^{N}})$. ∎

## Acknowledgments

Y. Cho was supported by NRF grant 2010-0007550 (Republic of Korea). M. M. Fall
is supported by the Alexander von Humboldt foundation.

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