\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 60, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/60\hfil Stable solitary waves]
{Stable solitary waves for one-dimensional Schr\"odinger-Poisson systems}

\author[G. Zhang, W. Zhang, S. Liu \hfil EJDE-2016/60\hfilneg]
{Guoqing Zhang, Weiguo Zhang, Sanyang Liu}

\address{Guoqing Zhang \newline
College of Sciences,
University of Shanghai for Science and Technology,
Shanghai 200093, China}
\email{shzhangguoqing@126.com}

\address{Weiguo Zhang \newline
College of Sciences,
University of Shanghai for Science and Technology,
Shanghai 200093, China}
\email{zwgzwm@126.com}

\address{Sanyang Liu \newline
College of Mathematics and Statistics,
Xidian University, Xi'an 710071 Shanxi, China}
\email{liusanyang@126.com}

\thanks{Submitted November 9, 2015. Published February 29, 2016.}
\subjclass[2010]{35J50, 35Q55, 37K45}
\keywords{Solitary waves; orbital stability; Schr\"odinger-Poisson system}

\begin{abstract}
 Based on the concentration compactness principle, we shoe the 
 existence of ground state solitary wave  solutions for one-dimensional 
 Schr\"odinger-Poisson systems with large $L^2$-norm in the energy space. 
 We also obtain orbital stability for ground state solitary waves.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks 

\section{Introduction}

 Consider the  one-dimensional Schr\"odinger-Poisson system
\begin{equation}
\begin{gathered}
i\partial_{t}\psi+\partial_{xx}\psi+W\psi+b|\psi|^{p-2}\psi=0,
\quad (t,x)\in \mathbb{R}^{1+1},\\
 -\partial_{xx}W=|\psi|^2,\quad  (t,x)\in \mathbb{R}^{1+1},\\
 \psi(0,x)=\psi_{0}(x),
\end{gathered}\label{e1}
\end{equation}
where $p>3$, $b$ is a real constant. The self-consistent Poisson potential
$W$ is explicitly given by
$$
W_{\psi}(t,x)=-\frac{1}{2}(|x|\ast|\psi(t,x)|^2)
=-\frac{1}{2}\int_{-\infty}^{+\infty}|x-y||\psi(t,y)|^2dy.
$$
Problem \eqref{e1} can  be reduced to the  nonlinear
nonlocal Schr\"odinger equation
\begin{equation}
\begin{gathered}
i\partial_{t}\psi+\partial_{xx}\psi-\frac{1}{2}(|x|\ast|\psi(t,x)|^2)
\psi+b|\psi|^{p-2}\psi=0,\quad (t,x)\in \mathbb{R}^{1+1},\\
\psi(0,x)=\psi_{0}(x).
\end{gathered}\label{e2}
\end{equation}
The model equation \eqref{e2} appears in various frameworks, such as
wave propagation in fibre optics to biophysics \cite{h1}, one-dimensional
 reduction of electron density in plasma physics \cite{b2}.

 Recently, one-dimensional (1D)
Schr\"odinger-Poisson system have been studied
extensively. In 2005, Stimming \cite{s1} obtained the global existence
result for \eqref{e2} by using the semi-group theory. In 2007,
De Leo, Rial \cite{d1} studied the global well-posedness and smoothing
effect of \eqref{e2}. In 2011, Masaki \cite{m1} proved that 
 \eqref{e2} is globally well-posed in the energy space, by means of
perturbation methods.

 We are interested in the search of solitary wave solutions
of \eqref{e2}, i.e., solutions to  \eqref{e2} in the form
$$
\psi(t,x)=e^{-i\lambda t}u(x),~\lambda \in \mathbb{R},
$$
and $u$ solving
\begin{equation}
-\partial_{xx}u+\frac{1}{2}(|x|\ast|u|^2)u-b|u|^{p-2}u=\lambda u,
\quad \lambda \in \mathbb{R}.\label{e3}
\end{equation}
  As $b=0$, based on the rearrangement inequality, Choquard, Stubble \cite{c4}
proved the existence and uniqueness result of ground states for
\eqref{e3}. Hartmann, Zakvzewski \cite{h2} obtained the analytic solitary wave
solutions which is approximated by a Gaussian, and soloved  \eqref{e3} numerically.

In this article, we look for solutions $u$ with a priori prescribed large 
$L^2$-norm by using the concentration compactness principle and
the constraint minimization method. Notice that the Schr\"odinger-Poisson 
system in three dimensional space, Catto, Dolbeault, Sanchez, Soler \cite{c1}
 reviewed some recent results and open problems concerning the existence of
 solitary wave solutions in the frame work of the concentration compactness principle.

This article is organized as follows. In Section 2, we give
some preliminary results and state our main theorems. In Section 3,
we prove the existence of ground state solitary wave solutions with
sufficiently large $L^2$-norm for  \eqref{e3}.

\section{Preliminary results and main theorems}\label{sec2}

 For any $1\leq q<+\infty$, $L^{q}(\mathbb{R})$ is the usual
Lebesgue space endowed with the norm
$|u|^{q}_{q}=\int^{+\infty}_{-\infty}|u|^{q}dx$. $H^1(\mathbb{R})$
is the usual Sobolev space with the norm
$\|u\|^2_{H^1(\mathbb{R})}
=\int^{+\infty}_{-\infty}(|\partial_{x}u|^2+|u|^2)dx$.
Consider the natural functional space 
$X=\{u: u\in H^1(\mathbb{R}),\sqrt{|x|}u\in L^2(\mathbb{R})\}$.
The energy space $X$ \cite{m1} is a Hilbert space with norm given by
$$
\|u\|^2_{X}=\|u\|^2_{H^1(\mathbb{R})}
+\int_{-\infty}^{+\infty}|x|u^2(x)dx=\|u\|^2_{H^1(\mathbb{R})}+|u|^2_{\ast},
$$
where $|u|^2_{\ast}=\int_{-\infty}^{+\infty}|x|u^2(x)dx$. 
By Rellich's criterion \cite{r1}, we have the following result.

\begin{lemma} \label{lem2.1}
 $X$ is compactly embedded in $L^{q}(\mathbb{R})$ for all $q\in [2,+\infty)$.
\end{lemma}

 Masaki \cite{m1} proved the following lemma in 2011.

\begin{lemma} \label{lem2.2} 
When $b>0$, $3\leq p<6$, $\psi_{0}(x)\in X$,  problem \eqref{e1} is globally
 well-posed in the energy space $X$.
\end{lemma}

We consider the  symmetric bilinear form
$$
(u,v)\mapsto B_{0}(u,v)=\int_{-\infty}^{+\infty}
\int_{-\infty}^{+\infty}|x-y|u(x)v(y)\,dx\,dy,
$$
and define the functional
$V:H^1(\mathbb{R})\to \mathbb{R}\cup \{+\infty\}$ as
\[
V(u)=B_{0}(u^2,u^2)=\int_{-\infty}^{+\infty}
\int_{-\infty}^{+\infty}|x-y|u^2(x)u^2(y)\,dx\,dy.
\]

\begin{lemma} \label{lem2.3} 
Let $\{u_{n}\}$ be a sequence in $L^2(\mathbb{R})$ such that $u_{n}\to u$ 
in $L^2(\mathbb{R})\setminus \{0\}$, $\{v_{n}\}$ be a bounded sequence in 
$L^2(\mathbb{R})$ and $\sup_{n\in \mathbb{N}}B_{0}(u_{n}^2,v_{n}^2)<\infty$.
Then there exist $n_{0}\in \mathbb{N}$, $C>0$ such that $|u_{n}|_{\ast}<C$ 
for $n\geq n_{0}$.
\end{lemma}

\begin{proof}
From the assumptions and Egorov's Theorem, 
that there exist $n_{0}\in \mathbb{N}$, $R$, $\delta>0$ and
$A\subset B_{R}(0)$ such that $|A|>0$ and $u_{n}^2(x)\geq\delta$
for all $n\geq n_{0}$. 
Since
$$
|x-y|\geq \frac{|y|}{2}\geq \sqrt{|y|}\quad \text{for all 
$x\in B_{R}(0)$ and } y\in \mathbb{R}\setminus B_{2R}(0),
$$
we have
\begin{align*}
B_{0}(u_{n},v_{n})
&\geq\int_{\mathbb{R}\setminus B_{2R}(0)}
\int_{A}|x-y||u_{n}(x)|^2|v_{n}(y)|^2\,dx\,dy \\
&\geq\frac{\delta |A|}{2}\int_{\mathbb{R}\setminus B_{2R}(0)}|y||v_{n}(y)|^2dy\\
&\geq\frac{\delta |A|}{2}(|v_{n}|_{\ast}^2-2R|v_{n}|_2^2).
\end{align*}
Hence, we have $|u_{n}|_{\ast}<C$ for all $n \geq n_{0}$ because 
$B_{0}(u_{n},v_{n})$ and $|v_{n}|_2^2$ are bounded.
\end{proof}

 \begin{remark} \label{rmk2.4}\rm
(1) From Lemma \ref{lem2.3}, we obtain that if $B_{0}(u_{n},v_{n})\to 0$ and 
$|v_{n}|_2\to 0$ as $n\to \infty$, then $|u_{n}|_{\ast}\to 0$ as $n\to \infty$.

(2) From Lemma \ref{lem2.1} and Lemma \ref{lem2.3}, it is easy to obtain that if 
$u_{n}\rightharpoonup u$ weakly in $X$, then we have 
$B_{0}(u_{n}^2, (u_{n}-u)u)\to 0$ as $n\to\infty$.
\end{remark}

  Now, for  problem \eqref{e3}, we consider the functionals 
$I,N:X\to \mathbb{R}$ defined by
\begin{align*}
I(u)
&=\frac{1}{2}\int_{-\infty}^{+\infty}|\partial_{x}u|^2dx
 +\frac{1}{4}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}|x-y|u^2(x)u^2(y)\,dx\,dy
 -\frac{b}{p}\int_{-\infty}^{+\infty}|u|^{p}dx\\
&=\frac{1}{2}\int_{-\infty}^{+\infty}|\partial_{x}u|^2dx+\frac{1}{4}V(u)
 -\frac{b}{p}\int_{-\infty}^{+\infty}|u|^{p}dx,
\end{align*}
and
$$
N(u)=\Big(\int_{-\infty}^{+\infty}|u|^2dx\Big)^{1/2}=|u|_2.
$$
From \cite{c4,r1}, we obtain that the functionals $I,N$ are well-defined on $X$.

\begin{lemma} \label{lem2.4} 
The functional $I$ is of class $C^1$ on $X$.
\end{lemma}

\begin{proof}
 Let $\{u_{n}\}$ be a sequence in $X$ converging to some $u\in X$, 
we obtain that $\{u_{n}\}$ is bounded and
\begin{align*}
&|V(u_{n})-V(u)|\\
&=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}
|x-y||u_{n}(x)|^2|u_{n}^2(y)-u^2(y)|\,dx\,dy\\
&\quad +\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}|x-y|
|u_{n}^2(x)-u^2(x)||u(y)|^2\,dx\,dy\\
&\leq \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}(|x|+|y|)
 |u_{n}(x)|^2|u_{n}(y)-u(y)||u_{n}(y)+u(y)|\,dx\,dy\\
&\quad +\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}(|x|+|y|)
 |u_{n}(x)-u(x)||u_{n}(x)+u(x)||u(y)|^2\,dx\,dy\\
&\leq |u_{n}(x)|_{\ast}^2|u_{n}-u|_2^2|u_{n}+u|_2^2
 +|u_{n}|_2^2|u_{n}-u|_{\ast}|u_{n}+u|_{\ast}\\
&\quad +|u_{n}|_2^2|u_{n}-u|_{\ast}|u_{n}+u|_{\ast}
+|u|_{\ast}^2|u_{n}-u|_2|u_{n}+u|_2\\
&\leq C\|u_{n}-u\|_{X}^2,
\end{align*}
for some $C>0$. So, we obtain that $V(u_{n})\to V(u)$ as $n\to \infty$. 
By a simple calculation, we have
$$
V'(u)v=4\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}|x-y|u(x)^2u(y)v(y)\,dx\,dy,
\quad \forall v\in X.
$$
When $u_{n}\to u$ in $X$, we can argue as before and obtain
\begin{align*}
&|V'(u_{n})v-V'(u)v|\\
&=4\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}|x-y|(u_{n}^2(x)u_{n}(y)
 -u^2(x)u(y))v(y)\,dx\,dy\\
&\leq 4[\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}(|x|+|y|)(u_{n}(x)
 -u(x))(u_{n}(x)+u(x))|u_{n}(y)||v(y)|\,dx\,dy\\
&\hskip0.38cm +\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}(|x|+|y|)
 (u_{n}(y)-u(y))|u(x)|^2|v(y)|\,dx\,dy]\\
&\leq 4(|u_{n}-u|_{\ast}|u_{n}+u|_{\ast}|u_{n}|_2^2|v|_2
 +|u_{n}-u|_2|u|_{\ast}^2|v|_2) \\
&\leq 4C\|u_{n}-u\|_{X}\|v\|_{X},\quad \forall v\in X.
\end{align*}
In conclusion, we obtain that $V(u)$ is $C^1$ on $X$. Since $3\leq p<6$, 
by Lemma \ref{lem2.1}, we obtain that $|u|^{p}_{p}$ is $C^1$ on $X$.
 Hence, the functional $I$ is of class $C^1$ on $X$.

  On the other hand, it is easy to obtain that $N(u)$ is $C^1$ on $X$ by 
Lemma \ref{lem2.1}.
\end{proof}

  Inspired by the papers \cite{c1,h1}, we look for the solution
of the problem \eqref{e3} with a priori prescribed $L^2$-norm. The natural
way is to consider the constrained critical points of the
functional $I$ on the set
$$
B_{M}=\{u\in X:~|u|_2=M\}.
$$
So by a solution of  \eqref{e3} we mean a couple
 $(\lambda_{M},u_{M})\in \mathbb{R}\times X$, where $\lambda_{M}$ 
is the Lagrange multiplier associated with the critical point $u_{M}$ on $B_{M}$.
  From a physical point of view, the most interesting case is
the existence of solutions for  \eqref{e3} with minimal energy 
(ground state solutions), that is the minimizers of 
\begin{equation}
I_{M}=\inf_{u\in B_{M}} I(u). \label{e4}
\end{equation}
Functionals $I$, $N$ are translation invariant, i.e., for
every $z\in \mathbb{R}$,
$$
I(u(\cdot +z))=I(u),\quad N(u(\cdot +z))=N(u).
$$
Therefore, the concentration compactness principle \cite{l1,l2} is the
natural framework for the study of the existence of a minimizer, and
for the analysis of the minimizing sequence of \eqref{e4}. It
is known that, in this kind of problems, the main difficulty is
the lack of compactness of the minimizing sequences $\{u_{n}\}$
in $B_{M}$; indeed, two possible bad scenarios are possible:
(1) (Vanishing)
\begin{equation}
u_{n}\rightharpoonup 0;\label{e5}
\end{equation}
(2) (Dichotomy)
\begin{equation}
u_{n} \rightharpoonup \bar{u}\neq 0\quad \text{and}\quad
0<|\bar{u}|_2<M.\label{e6}
\end{equation}
 By the concentration compactness principle, we obtain the
strict inequalities
$$
I_{M}<I_{M'}+I_{\sqrt{M^2-M'^2}}\quad\text{for all }M,M',\quad
\text{and}\quad 0<M'<M,
$$
as necessary and sufficient conditions for the precompactness of the
minimizing sequences $\{u_{n}\}$ in the problem \eqref{e4}.
Now, we state our main theorems in this paper.

 \begin{theorem} \label{thm2.5} 
Let $b>0$, $3<p<6$. Then the minimizing sequences for  \eqref{e4} is precompact 
in $X$ up to translations with prescribed large $L^2$-norm. 
In particular, there exists a couple $(\lambda_{M},u_{M})\in \mathbb{R}\times X$ 
solution of \eqref{e3}, i.e., problem \eqref{e1} has a ground state solitary
 wave solution.
\end{theorem}

 \begin{theorem} \label{thm2.6}
Let $b>0$, $3<p<6$. Then the set
$$
S_{M}=\{e^{i\theta}u,~\theta \in [0,2\pi),\, |u|_2=M,\, I(u)=I_{M}\},
$$
is orbitally stable, i.e., the ground state solitary wave solution of  \eqref{e1} 
is orbitally stable.
\end{theorem}

The definition of orbital stability is recalled in Definition \ref{def3.3} below.

\section{Existence and orbital stability}\label{sec3}

  In this case, our aim is to discuss the applicability of concentration 
compactness principle to the minimizing problem \eqref{e4} for proving 
the existence of ground state solutions of \eqref{e3}.
The next result is the Gagliardo-Nirenberg inequality in one-dimensional space, 
see \cite[p. 9]{c2}. 

\begin{lemma} \label{lem3.1} 
For all $u\in H^1(\mathbb{R})$, we have
\begin{equation}
|u|_{q}\leq C|u|_2^{(1-\delta)}|\partial_{x}u|_2^{\delta},\label{e7}
\end{equation}
where $2\leq q <\infty$, $\delta=\frac{1}{2}-\frac{1}{q}$, the constant $C$
only depends on $q$ and $\delta$.
\end{lemma}

  By Lemma \ref{lem3.1}, we obtain that for every $M>0$, the functional
$I$ is bounded from below on $B_{M}$. Indeed, from \eqref{e7} and positive
property of $V(u)$, we have
\begin{align*}
I(u)&\geq \frac{1}{2}\int_{-\infty}^{+\infty}|\partial_{x}u|^2dx
 -\frac{b}{p}\int_{-\infty}^{+\infty}|u|^{p}dx\\
&\geq \frac{1}{2}|\partial_{x}u|^2_2
 -\frac{b}{p}CM\frac{p(1-\delta)}{2}|\partial_{x}u|_2^{p\delta}.
\end{align*}
Since $3<p<6$, we have $p\delta=\frac{p}{2}-1<2$. Hence, we have
$$
I(u)\geq \frac{1}{2}\int_{-\infty}^{+\infty}|\partial_{x}u|^2dx+o(1),
$$
which concludes the proof. Moreover, we  also obtain that $I$ is coercive on $B_{M}$. 

Notice that if we set 
$u_{\theta}(x)=\theta^{(1-\frac{\alpha}{2})}u(\frac{x}{\theta^{\alpha}})$,
$\theta>0$, $\alpha$ is a real number, we have 
\begin{gather}
\int_{-\infty}^{+\infty}|u_{\theta}|^2dx=|u_{\theta}|_2^2=\theta^2|u|_2^2, \nonumber\\
\int_{-\infty}^{+\infty}|\partial_{x}u_{\theta}|^2dx
=\theta^{(2-2\alpha)}\int_{-\infty}^{+\infty}|\partial_{x}u|^2dx, \nonumber \\
\begin{aligned}
&\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}|x-y| 
 |u_{\theta}(x)|^2|u_{\theta}(y)|^2\,dx\,dy\\
&     =\theta^{(4+\alpha)}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}|x-y|
|u(x)|^2|u(y)|^2\,dx\,dy, 
\end{aligned} \label{e8} \\
\int_{-\infty}^{+\infty}|u_{\theta}|^{p}dx
=\theta^{(1-\frac{\alpha}{2})p+\alpha}\int_{-\infty}^{+\infty}|u|^{p}dx. \nonumber 
\end{gather}

\begin{lemma} \label{lem3.2} 
If $b>0$, $3<p<6$, then there exists $M_1>0$, such that
\begin{gather*}
I_{M'}<0,\quad \text{for all }M'\in (M_1,+\infty), \\
I_{M}<I_{M'}+I_{\sqrt{M^2-M'^2}},
\end{gather*}
for all $M>M_1$ and $0<M'<M$.
\end{lemma}

\begin{proof} 
By \eqref{e8}, we have
\begin{align*}
I(u_{\theta}) 
& =\frac{\theta^{(2-2\alpha)}}{2}\int_{-\infty}^{+\infty}|\partial_{x}u|^2dx
+\frac{\theta^{(4+\alpha)}}{4}\int_{-\infty}^{+\infty}
 \int_{-\infty}^{+\infty}|x-y||u(x)|^2|u(y)|^2\,dx\,dy\\
&\quad -\frac{b}{p}\theta^{(1-\frac{\alpha}{2})p+\alpha}
 \int_{-\infty}^{+\infty}|u|^{p}dx\\
&=\theta^2[I(u)+\frac{\theta^{(-2\alpha)}-1}{2}
 \int_{-\infty}^{+\infty}|\partial_{x}u|^2dx\\
&\quad +\frac{\theta^{(2+\alpha)}-1}{4}
 \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}|x-y||u(x)|^2|u(y)|^2\,dx\,dy\\
&\quad -\frac{b}{p}(\theta^{(1-\frac{\alpha}{2})p+\alpha-2}-1)
 \int_{-\infty}^{+\infty}|u|^{p}dx]\\
&=\theta^2(I(u)+g(\theta,u)),
\end{align*}
where
\begin{align*}
g(\theta,u)
&=\frac{\theta^{(-2\alpha)}-1}{2}\int_{-\infty}^{+\infty}|\partial_{x}u|^2dx\\
&\quad +\frac{\theta^{(2+\alpha)}-1}{4}\int_{-\infty}^{+\infty}
 \int_{-\infty}^{+\infty}|x-y||u(x)|^2|u(y)|^2\,dx\,dy\\
&\quad -\frac{b}{p}(\theta^{(1-\frac{\alpha}{2})p+\alpha-2}-1)
 \int_{-\infty}^{+\infty}|u|^{p}dx.
\end{align*}
Let $\alpha=-2/3$, we have
\begin{align*}
I(u_{\theta})
&=\frac{1}{2}\theta^{\frac{10}{3}}\int_{-\infty}^{+\infty}|\partial_{x}u|^2dx
+\frac{1}{4}\theta^{\frac{10}{3}}\int_{-\infty}^{+\infty}
 \int_{-\infty}^{+\infty}|x-y||u(x)|^2|u(y)|^2\,dx\,dy\\
&\quad  -\frac{b}{p}\theta^{(\frac{4}{3}p-\frac{2}{3})}
 \int_{-\infty}^{+\infty}|u|^{p}dx,\quad \forall u_{\theta}\in X.
\end{align*}
Hence, we obtain that $I(u_{\theta})<0$ for a sufficiently large $\theta$ 
which proves the first case because
$\frac{4}{3}p-\frac{2}{3}>\frac{10}{3}$ for $3<p<6$.
\smallskip

\noindent\textbf{Claim:} $I_{\theta M'}<\theta^2I_{M'}$
for $\theta$ sufficiently large. Indeed, let
$\{u_{n}\}$ be a minimizing sequence in $B_{M'}$ with $I_{M'}<0$.
Since $I_{M'}(u_{n})<0$, we have
$$
0<C_1<\int_{-\infty}^{+\infty}|\partial_{x}u_{n}|^2dx<C_2,\quad
0<C_3<\int_{-\infty}^{+\infty}|u_{n}|^{p}dx<C_4.
$$
When $\alpha=-2/3$, we have
\begin{align*}
g(\theta,u_{n})
&=\frac{(\theta^{\frac{4}{3}}-1)}{2}\int_{-\infty}^{+\infty}|\partial_{x}u_{n}|^2dx\\
&\quad +\frac{(\theta^{\frac{4}{3}}-1)}{4}\int_{-\infty}
 ^{+\infty}\int_{-\infty}^{+\infty}|x-y||u_{n}(x)|^2|u_{n}(y)|^2\,dx\,dy\\
&\quad -\frac{b}{p}(\theta^{(\frac{4}{3}p-\frac{8}{3})}-1)
 \int_{-\infty}^{+\infty}|u_{n}|^{p}dx,
\end{align*}
with $\frac{4}{3}p-\frac{8}{3}>\frac{4}{3}$, as
$3<p<6$. With a simple computation we obtain that
$$
\frac{d}{d\theta}g(\theta,u_{n})|_{\theta=1}<0\text{ and }
\frac{d^2}{d\theta^2}g(\theta,u_{n})<0,\quad \text{for all }\theta>1.
$$
In conclusion, we obtain that $g(\theta,u_{n})<0$ for all $\theta>1$ and 
$I_{\theta M'}<\theta^2I(u_{n})=\theta^2I_{M'}$.

  From the claim, we obtain that for $M'$ sufficiently large,
\begin{align*}
I_{M}
&=I_{\frac{M}{M'}M'}<\frac{M^2}{M'^2}I_{M'}\\
&=\frac{(M^2-M'^2+M'^2)}{M'^2}I_{M'}\\
&=I_{M'}+\frac{(M^2-M'^2)}{M'^2}I_{\frac{M'}{\sqrt{M^2-M'^2}}\sqrt{M^2-M'^2}}\\
&<I_{M'}+I_{\sqrt{M^2-M'^2}}.
\end{align*}
Hence, we complete the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2.5}]
 Since the functional $I$ is bounded below and coercive on $B_{M}$, 
we obtain that the minimizing sequence $\{u_{n}\}$ is bounded in 
$H^1(\mathbb{R}^2)$. By Lemma \ref{lem3.2}, we obtain that $I_{M}<0$ on $B_{M}$ 
for sufficiently large $M$. Hence, we have that
$$
\sup_{x\in \mathbb{R}}\int_{B_2(x)}|u_{n}|^2dx>0\quad text{and}\quad
u_{n}\rightharpoonup u\neq 0~~\text{in}~~H^1(\mathbb{R}^2).
$$
and the vanishing case does not hold. On the other hand, by Lemma \ref{lem3.2}, 
$I_{M}<I_{M'}+I_{\sqrt{M^2-M'^2}}$ for all $M'>M_1$ and $0<M'<M$, 
and the dichotomy case does not hold. Hence, from concentration compactness 
principle, we obtain that
\begin{equation}
\text{there exists $x_{n}\in \mathbb{R}$ such that }
\tilde{u}_{n}=u_{n}(y-x_{n})\in X,~n\in \mathbb{N},\label{e9}
\end{equation}
is precompact in $H^1(\mathbb{R})$ and converges strongly to some function
$u\in H^1(\mathbb{R})\backslash \{0\}$. We may also assume that
$\tilde{u}_{n}\to u$ pointwise almost everywhere in $\mathbb{R}$.
\smallskip

\noindent\textbf{Claim 1:} 
$|\tilde{u}_{n}|_{\ast}$ is bounded in $n$. Indeed, since 
$\{\tilde{u}_{n}\}$ is a minimizing sequence for $I_{M}$ on $B_{M}$,
 by the version of Ekeland Variational principle in [p.122]{w1}, we obtain 
that there exists $\lambda \in \mathbb{R}$ such that
\begin{equation}
I'(\tilde{u}_{n})-\lambda N'(\tilde{u}_{n})\to 0,\quad
I(\tilde{u}_{n})-\lambda N(\tilde{u}_{n})\to I_{M}-\lambda M^2\quad
\text{as } n\to \infty.\label{e10}
\end{equation}
Hence, we have
\[
B_{0}(\tilde{u}_{n}^2,\tilde{u}_{n}^2)=V(\tilde{u}_{n})=V(u_{n})
=o(1)+\lambda \|u_{n}\|^2_{L^2(\mathbb{R})}+b \|u_{n}\|^{p}_{L^{p}(\mathbb{R})}
-\|u_{n}\|^2_{H^1(\mathbb{R})}
\]
as $n\to\infty$,
and the right-hand side of this equality remains bounded in $n$.
So, we obtain that $B_{0}(\tilde{u}_{n}^2,\tilde{u}_{n}^2)$ is bounded in $n$.
By Lemma \ref{lem2.3} and Remark \ref{rmk2.4}, we have $|\tilde{u}_{n}|_{\ast}$ is bounded in $n$.
Hence, by the definition of the norm $\|\tilde{u}_{n}\|_{X}$, we obtain that
$\|\tilde{u}_{n}\|_{X}<\infty$.
\smallskip

\noindent\textbf{Claim 2:} 
$\|\tilde{u}_{n}-u\|_{X}\to 0$ as $n\to\infty$. Indeed, by \eqref{e10}, we have
$$
I'(\tilde{u}_{n})(\tilde{u}_{n}-u)-\lambda N'(\tilde{u}_{n})(\tilde{u}_{n}-u)\to 0
\quad \text{as } n\to\infty.
$$
By a simple calculation, we obtain that
\begin{align*}
&\|\tilde{u}_{n}\|^2_{H^1(\mathbb{R})}-\|u\|^2_{H^1(\mathbb{R})}+V(\tilde{u}_{n})(\tilde{u}_{n}-u)
-b\int_{-\infty}^{+\infty}|\tilde{u}_{n}|^{p-2}\tilde{u}_{n}(\tilde{u}_{n}-u)dx\\
&-\lambda\int_{-\infty}^{+\infty}|\tilde{u}_{n}|(\tilde{u}_{n}-u)dx
=o(1).
\end{align*}
Since $\|\tilde{u}_{n}\|_{X}$ is bounded in $X$, we obtain that 
$\tilde{u}_{n}\rightharpoonup u$ weakly in $X$. By Lemma \ref{lem2.1}, we obtain 
that $\tilde{u}_{n}\to u$ strongly in $L^{s}(\mathbb{R})$ for $s\in [2,\infty)$. 
Hence, we have
\[
V'(\tilde{u}_{n})(\tilde{u}_{n}-u)=B_{0}(\tilde{u}_{n}^2,\tilde{u}_{n}(\tilde{u}_{n}-u))
=B_{0}(\tilde{u}_{n}^2,(\tilde{u}_{n}-u)^2)+B_{0}
(\tilde{u}_{n}^2,u(\tilde{u}_{n}-u))\to 0
\]
as $n\to\infty$.
By Lemma \ref{lem2.3}, we obtain that $|\tilde{u}_{n}-u|_{\ast}\to 0$ as $n\to\infty$. 
Hence, we obtain that $\|\tilde{u}_{n}-u\|_{X}\to 0$ as $n\to\infty$.

  From Claim 2, we obtain that the minimizing sequence $\{\tilde{u}_{n}\}$ 
of \eqref{e4} is precompact in $X$ with prescribed large $L^2$-norm. 
So there exists a couple $(\lambda_{M},u_{M})\in \mathbb{R}\times X$ solution 
of  \eqref{e3}, and \eqref{e1} has a ground state solitary wave solution.
Let 
\[
S_{M}=\{e^{i\theta}u(x),\,\theta \in [0,2\pi),\, |u|_2=M,\, I(u)=I_{M}\}.
\]
\end{proof}

\begin{definition}[c3] \label{def3.3} \rm
We say $S_{M}$ is orbitally stable if for every $\varepsilon>0$, 
there exists $\delta(\varepsilon)>0$ such that if $\psi_{0}\in X$ satisfies
$\inf_{v\in S_{M}}\|v-\psi_{0}\|_{X}<\delta(\varepsilon)$,
then we have
$$
\sup_{t>0}\inf_{v\in S_{M}}\|\psi(t,x)-v(x)\|_{X}<\varepsilon,
$$
where $\psi(t,\cdot)$ is the solution of \eqref{e1} with initial datum $\psi_{0}$.
\end{definition}

\begin{proof}[Proof of Theorem \ref{thm2.6}]
 By [Theorem 1.5]{m1}, we obtain the solution of \eqref{e1} conserves 
$|\psi|^2_{L^2(\mathbb{R})}$ and the energy
\begin{align*}
E(t,\psi)&=\frac{1}{2}|\partial_{x}\psi|^2_2
 +\frac{1}{4}\int_{-\infty}^{+\infty}
 \int_{-\infty}^{+\infty}|x-y||\psi(t,x)|^2|\psi(t,y)|^2\,dx\,dy \\
&\quad -\frac{b}{p}\int_{-\infty}^{+\infty}|\psi(t,x)|^{p}dx,
\end{align*}
i.e.,
$$
\int_{-\infty}^{+\infty}|\psi(t,x)|^2dx
=\int_{-\infty}^{+\infty}|\psi_{0}(x)|^2dx,
$$
and
\begin{align*}
&\frac{1}{2}|\partial_{x}\psi|^2_2+\frac{1}{4}
 \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}|x-y||\psi(t,x)|^2|\psi(t,y)|^2\,dx\,dy
 -\frac{b}{p}\int_{-\infty}^{+\infty}|\psi|^{p}dx\\
&=\frac{1}{2}|\partial_{x}\psi_{0}|^2_2+\frac{1}{4}
 \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}|x-y||\psi_{0}(x)|^2|\psi_{0}(y)|^2\,dx\,dy
-\frac{b}{p}\int_{-\infty}^{+\infty}|\psi_{0}|^{p}dx.
\end{align*}
  Suppose by contradiction that there exists a $M$ such that $S_{M}$ is not 
orbitally stable. Hence, there exist a subsequence $\{\psi_{n}(0,x)\}$ and 
$\{t_{n}\}\in \mathbb{R}$ such that
$$
\inf_{v\in S_{M}}\|\psi_{n}(0,x)-v(x)\|_{X}\to 0,\quad
\inf_{v\in S_{M}}\|\psi_{n}(t_{n},x)-v(x)\|_{X}\geq \varepsilon
\quad \text{as } n\to \infty.
$$
Then, we obtain that there exists $u_{M}\in X$ minimizer of $I_{M}$ 
and $\theta\in [0,2\pi]$ such that $v=e^{i\theta}u_{M}$,
$$
|\psi_{n}(0,x)|_2\to |v|_2=M,\quad 
I(\psi_{n}(0,x))\to I(v)=I_{M}\quad \text{as }n\to \infty.
$$
  Actually, we can assume that  $\psi_{n}(0,x)\in B_M$ (there
exists $\alpha_n =\frac{M}{|\psi_{n}(0,x)|_2} \to 1$ so
that $\alpha_n \psi_{n}(0,x) \in B_M$ and 
$I(\alpha_n \psi_{n}(0,x))\to I_M$, i.e., we can replace $\psi_{n}(0,x)$ with
$\alpha_n \psi_{n}(0,x)$).
  So, we have $\{\psi_{n}(0,x)\}$ is a minimizing sequence for
$I_{M}$ and
$$
I(\psi_{n}(t_{n},x))=I(\psi_{n}(0,x)),
$$
and $\{\psi_{n}(t_{n},x)\}$ is a minimizing sequence for $I_{M}$.
Since we obtain that every minimizing sequence has a subsequence
converging in $X-$norm to a minimum on $B_{M}$, and it is a
contradiction.
\end{proof}

\subsection*{Acknowledgments}
  The authors would like to sincerely thank the referee for
valuable comments and suggestions.
 This research was supported by the Shanghai Natural
Science Foundation (No. 15ZR1429500).

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\end{document}