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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 317, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/317\hfil Sign-changing solutions]
{Sign-changing solutions for asymptotically linear Schr\"odinger equation
in bounded domains}

\author[S. Chen, Y. Li, X. Tang \hfil EJDE-2016/317\hfilneg]
{Sitong Chen, Yinbin Li, Xianhua Tang}

\address{Sitong Chen \newline
School of Mathematics and Statistics,
Central South University,
Changsha, 410083 Hunan, China}
\email{mathsitongchen@163.com}

\address{Yinbin Li \newline
School of Mathematics and Statistics,
Central South University,
Changsha, 410083 Hunan, China}
\email{liyinbin1991@163.com}

\address{Xianhua Tang \newline
School of Mathematics and Statistics,
Central South University,
Changsha, 410083 Hunan, China}
\email{tangxh@mail.csu.edu.cn}


\thanks{Submitted June 22, 2016. Published December 14, 2016.}
\subjclass[2010]{35J10, 35J20}
\keywords{Schr\"odinger equation; sign-changing solutions;
 asymptotically linear}

\begin{abstract}
 In this article we study the Schr\"odinger equation
 $$
 -\Delta u=f(x,u),\quad x\in\Omega, \quad u\in H_0^1(\Omega),
 $$
 where $\Omega$ is a bounded domain in $\mathbb{R}^N$ and $f(x,u)$ is
 asymptotically  linear at infinity with respect to $u$. 
 Inspired by the works of Salvatore \cite{B3} on sign-changing solutions, 
 in which $f(x,u)$ is asymptotically linear at zero with respect  to $u$,
 we prove, via the  constraint variational method and the quantitative 
 deformation lemma,  that the equation possesses one sign-changing solution 
 with exactly two nodal domains.
\end{abstract}

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

\section{Introduction and statement of main results}

 In this article, we consider the Schr\"odinger equation
 \begin{equation} \label{ps}
\begin{gathered}
 -\Delta u=f(x,u),\quad x\in\Omega, \\
 u\in H_0^1(\Omega),
 \end{gathered}
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^N$ and $f:\Omega\times\mathbb{R}\to\mathbb{R}$
is continuous. The main aim of this paper is to find sign-changing solutions 
of \eqref{ps} when $f$ is asymptotically linear. Precisely, we assume that 
$f$ satisfies the following assumptions:
\begin{itemize}
\item[(A1)] $f\in C(\Omega\times\mathbb{R})$,
 $F(x,t):=\int_0^tf(x,s)\mathrm{d}s\geq0$ and 
$f(x,t)=o(|t|)$ as $|t|\to 0$, uniformly in $x\in\Omega$;

\item[(A2)] $f(x,t)=V_\infty(x)t+f_1(x,t)$, $V_\infty\in C(\Omega)$, and 
$f_1(x,t)=o(|t|)$ as $|t|\to+\infty$, uniformly in $x\in\Omega$;

\item[(A3)] $t\mapsto f(x,t)/|t|$ is strictly increasing on 
$(-\infty,0)\cup(0,\infty)$ for every $x\in\Omega$;


\item[(A4)] $\widetilde{F}(x,t):=\frac{1}{2}f(x,t)t-F(x,t)\to+\infty $ as 
$t\to+\infty$ uniformly in $x\in\Omega$.

\end{itemize}

 The nonlinear Schr\"odinger equation is of interest in many branches of physics. 
As we know, the solutions of problems like \eqref{ps} are related to the 
existence of standing wave solutions for nonlinear Schr\"odinger equation like
\begin{equation}\label{p2}
 i\hbar\frac{\partial \Psi}{\partial t}=-\hbar^2\triangle \Psi+V(x)\Psi-f(x,\Psi)
\quad\text{for all } x\in\Omega,
\end{equation}
where $\Omega$ is a domain in $\mathbb{R}^N$, $\hbar>0$ and $\Psi$ is the amplitude
of the wave. Equation \eqref{p2} is one of the main objects of quantum physics, 
for it appears in problems involving nonlinear optics, plasma physics and condensed 
matter physics, see Anderson and Bonnedal \cite{AB}, Chen \cite{Ch}, Chiao et al
 \cite{CGT}, Gatz and Herrmann \cite{GH},  Karlsson \cite{Ka},
Sodha et al \cite{SGT},  Stuart \cite{Stu} and the references therein.

In recent years, problems like \eqref{ps} have been widely studied under 
variant assumptions on $f$, and the existence of positive solutions, 
ground state solutions, multiple solutions and semiclassical states were 
obtained in many papers, see for example 
\cite{S2,CT,QT,T6, R5,T2,T5,ZTZ1,ZTZ2} 
and the references therein. When $f$ is superlinear at infinity in $u$, 
the existence of sign-changing solutions of \eqref{ps} was established 
by Bartsch, Liu and Weth in \cite{B2}. For more discussions on the existence
of sign-changing solutions of \eqref{ps}, in this case, we refer the readers 
to \cite{BW,CCN,NW,Zo} and the references therein. When $f$ is asymptotically 
linear at zero in $u$, that is, $f$ satisfies the condition:
\begin{equation} \label{SA}
\mu_1<\liminf_{t\to0}\frac{f(x,t)}{t}\leq\limsup_{t\to0}\frac{f(x,t)}{t}<\mu_k
\quad\text{uniformly for }x\in\Omega,
\end{equation}
where $\{\mu_j\}$ is the sequence of eigenvalues of the Schr\"odinger operator
$-\Delta+V(x)$ and $V$ is a linear potential, Salvatore \cite{B3} proved the
existence of sign-changing solutions. Note that conditions (A1) and \eqref{SA}
are quite different and were considered in different situations.
 To the best of our knowledge, there are no works concerning the least energy
sign-changing solutions for Problem \eqref{ps} with asymptotically linear
case at infinity, and it is an interesting problem.

Let $H^{1}(\Omega)$ be the usual Sobolev space with the standard scalar product 
and norm
 $$
 (u, v)=\int_{\Omega}(\nabla u \nabla v+uv)\mathrm{d}x, \quad
\|u\|^2=\int_{\Omega}\left(|\nabla u|^2+u^2\right)\mathrm{d}x.
 $$
 Define the energy functional $\Phi:H_0^1(\Omega)\to\mathbb{R}$ by
\begin{equation}\label{a1}
 \Phi(u)=\frac{1}{2}\int_\Omega|\nabla u|^2\mathrm{d}x
-\int_\Omega F(x,u)\mathrm{d}x.
\end{equation}
Conditions (A1) and (A2) imply that $\Phi$ is a well-defined of class
 $C^{1}$ functional, and that
 \begin{equation}\label{a2}
 \langle\Phi'(u),\varphi\rangle=\int_\Omega \nabla u\nabla \varphi\mathrm{d}x
-\int_\Omega f(x,u)\varphi\mathrm{d}x, \quad \forall u,\varphi\in H_0^1(\Omega).
 \end{equation}

Clearly, critical points of $\Phi$ are the weak solutions of \eqref{ps}. 
Furthermore, if $u\in H_0^1(\Omega)$ is a solution of \eqref{ps} and $u^\pm\neq0$, 
then $u$ is a sign-changing solution of \eqref{ps}, where
 $$
 u^+(x):=\max \{u(x),0\}, \quad u^-(x):=\min \{u(x),0\}.
 $$
 Using \eqref{a1} and \eqref{a2}, it is obvious that
 \begin{gather*}
 \Phi(u)= \Phi(u^+)+\Phi(u^-), \\
\langle\Phi'(u),u^+\rangle=\langle\Phi'(u^+),u^+\rangle,\quad
\langle\Phi'(u),u^-\rangle=\langle\Phi'(u^-),u^-\rangle\,.
 \end{gather*}
To obtain a sign-changing solution of \eqref{ps}, we first seek a minimizer
 of the energy functional $\Phi$ under the  constraint
\[
 \mathcal{M}=\{u\in H_0^1(\Omega): u^\pm\neq 0,\; 
\langle\Phi'(u),u^+\rangle=\langle\Phi'(u),u^-\rangle=0\},
\]
then show that the minimizer is a sign-changing solution of \eqref{ps}.

 To state our results, we make the following assumption:
\begin{itemize}
\item[(A5)]  $\inf_{x\in\Omega}V_\infty(x)>\mu:=\inf_{u\in\Pi}
\max\{\|\nabla u^+\|_2^2,\ \|\nabla u^-\|_2^2\}$, where 
\[
\Pi:=\{u\in H_0^1(\Omega): u^\pm\neq0,\; \int_\Omega|u^\pm|^2\mathrm{d}x=1\}.
\]
\end{itemize}
Let $\lambda_1$ be the first eigenvalue of $-\Delta$, then for any 
$u\in H_0^1(\Omega)$ and $u\ne0$,
\begin{align*}
\lambda_1
&\le\frac{\|\nabla u\|_2^2}{\|u\|_2^2} \\
&=\frac{\|\nabla u^+\|_2^2+\|\nabla u^-\|_2^2}{\|u^+\|_2^2+\|u^-\|_2^2} \\
&\le \frac{2}{\|u^+\|_2^2+\|u^-\|_2^2}
 \max\{\|\nabla u^+\|_2^2, \|\nabla u^-\|_2^2\}.
\end{align*}
By the definition of $\mu$ and (A5), one has $\lambda_1\leq\mu<+\infty$.


\begin{remark} \label{rmk1.1}\rm
 Using (A1) and (A2), it is obvious that for any $\varepsilon>0$, there exists 
$C_\varepsilon>0$ such that
 \begin{equation}\label{a6}
 |f(x,t)|\leq \varepsilon|t|+C_\varepsilon|t|^{p-1}\quad \text{and}\quad
 |F(x,t)|\leq \varepsilon|t|^2+C_\varepsilon|t|^p
 \end{equation}
for all $(x,t)\in\Omega\times\mathbb{R}$, where $2<p<2^*=\frac{2N}{N-2}$.
Furthermore, (A1) and (A3) imply 
 \begin{equation}\label{yb}
 \frac{1}{2}f(x,t)t>F(x,t)>0, \quad \forall t\neq0,\; x\in\Omega.
 \end{equation}
It follows from (A1)--(A3) and (A5) that
 \begin{gather*}
 \frac{f_1(x,t)}{|t|}\to-V_\infty(x)<0\quad \text{as } |t|\to0, \\
 t\mapsto\frac{f_1(x,t)}{|t|} \text{ is negative, strictly increasing on }
(-\infty,0)\cup(0,\infty),
 \end{gather*}
 which, together with $f_1(x,t)=o(|t|)$ as $|t|\to\infty$ uniform in $x$, yields
 \begin{equation}\label{yb1}
 tf_1 (x,t)<0,\quad \forall t\neq0.
 \end{equation}
\end{remark}

\begin{theorem} \label{thm1.2}
Assume {\rm (A1)--(A5)} are satisfied.
Then  \eqref{ps} has a sign-changing solution $u\in \mathcal{M}$ such that 
$\Phi(u)=\inf_\mathcal{M}\Phi>0$, which has precisely two nodal domains.
\end{theorem}

Now, we give an example to illustrate the feasibility of assumptions (A1)--(A5). 
Let
\[
 F(x,t)=\frac{V_\infty(x)}{2}t^2\big(1-\frac{1}{1+|t|^\alpha}\big),
\forall\ x\in\Omega,\; t\in\mathbb{R},
\]
where $\alpha\in(0,2)$, $V_\infty\in C(\Omega)$, $\inf_{\Omega} V_\infty>\mu$.
By elementary computations, it is easy to check that $f$ satisfies (A1)--(A5).

The main tools this article are the minimization argument and the quantitative 
deformation lemma. We must point out that the difficulty in proving Theorem 
\ref{thm1.2} is to show that $\mathcal{M}\neq\emptyset$ and the minimizer 
is a critical point of $\Phi$.

This article organized as follows. 
In Section 2, we prove several preliminary lemmas.
 The proof of Theorem \ref{thm1.2} will be given in the last section.

\section{Preliminaries}

 In this section, we prove that the minimizer of the energy functional
 $\Phi$ under the constraint $\mathcal{M}$ is a critical point. To this end,
 we show $\mathcal{M}\ne\emptyset$ with the aid of an important behavior 
of strictly increasing functions.

\begin{lemma}[{\cite[Lemma 2.3]{R5}}] \label{lem2.1} 
Suppose that $h(x,t)$ is strictly increasing in $t\in\mathbb{R}$ and $h(x,0)=0$ 
for any $x\in \mathbb{R}^N$. Then
\[
 \frac{1-\theta^2}{2}h(x,\tau)\tau|\tau|>\int^\tau_{\theta\tau}h(x,s)|s|\mathrm{d}s,
\quad \forall \theta\in[0,1)\cup(1,\infty),\; \tau\in\mathbb{R}\backslash\{0\}.
\]
\end{lemma}

\begin{lemma} \label{lem2.2} 
 Suppose that {(A1)--(A3)} are satisfied. Then for any 
$u=u^++u^-\in H_0^1(\Omega)$ with $u^\pm\neq0$, $s$, $t\geq0$ and 
$(s-1)^2+(t-1)^2\neq0$,
 \begin{equation}\label{b2}
 \Phi(u)>\Phi(su^++tu^-)+\frac{1-s^2}{2}\langle\Phi'(u),u^+\rangle
+\frac{1-t^2}{2}\langle\Phi'(u),u^-\rangle.
 \end{equation}
\end{lemma}

\begin{proof} For any $x\in\Omega$,  from (A3) and Lemma \ref{lem2.1} it follows that
 \begin{equation}\label{b21}
 \frac{1-\theta^2}{2}f(x,\tau)\tau>\int^\tau_{\theta\tau}f(x,\xi)\mathrm{d}\xi,\quad
\forall \theta\in[0,1)\cup(1,\infty),\; \tau\in\mathbb{R}\backslash\{0\}.
 \end{equation}
By  \eqref{a1}, \eqref{a2} and \eqref{b21}, for any $u=u^++u^-\in H_0^1(\Omega)$ 
with $u^\pm\neq0$, $s$, $t\geq0$ and $(s-1)^2+(t-1)^2\neq0$, we have
 \begin{align*}
&\Phi(u)-\Phi(su^++tu^-) \\
&=\frac{1}{2}\int_\Omega|\nabla u|^2\mathrm{d}x
 -\int_\Omega F(x,u)\mathrm{d}x+\int_\Omega F(x,su^++tu^-)\mathrm{d}x
 -\frac{1}{2}\int_\Omega|\nabla (su^++tu^-)|^2\mathrm{d}x\\
&=\frac{1-s^2}{2}\langle\Phi'(u),u^+\rangle+\frac{1-t^2}{2}\langle
 \Phi'(u),u^-\rangle
+\int_\Omega\Big[\frac{1-t^2}{2}f(x,u^-)u^- \\
&\quad -\int^{u^-}_{tu^-}f(x,\xi)\mathrm{d}\xi \Big]\mathrm{d}x
 +\int_\Omega\Big[\frac{1-s^2}{2}f(x,u^+)u^+
 -\int^{u^+}_{su^+}f(x,\xi)\mathrm{d}\xi\Big]\mathrm{d}x\\
 &>\frac{1-s^2}{2}\langle\Phi'(u),u^+\rangle+\frac{1-t^2}{2}\langle\Phi'(u),u^-\rangle.
 \end{align*}
This shows that \eqref{b2} holds.
\end{proof}

From Lemma \ref{lem2.2}, we have the following two corollaries.


 \begin{corollary} \label{coro2.3} 
 Suppose that {\rm (A1)--(A3)} are satisfied. Then for any 
$u=u^++u^-\in \mathcal{M}$, 
\[
 \Phi(u)\geq\Phi(su^++tu^-),\quad \forall s,t\geq0.
\]
\end{corollary}

\begin{corollary} \label{coro2.4} 
Suppose that {\rm (A1)--(A3)} are satisfied. Then for any 
$u=u^++u^-\in \mathcal{M}$,
\[
 \Phi(u^++u^-)=\max_{s,t\geq0}\Phi(su^++tu^-).
\]
\end{corollary}

Define the set 
\begin{equation}\label{gj}
 E_0=\big\{u\in H_0^1(\Omega):\|\nabla u^\pm\|_2^2
 -\int_\Omega V_\infty(x)|u^\pm|^2\mathrm{d}x<0 \big\}.
\end{equation}


\begin{lemma} \label{lem2.5} 
 Suppose that {\rm (A1)--(A3), (A5)}  are satisfied. 
Then $E_0\ne\emptyset$ and $\mathcal{M}\subset E_0$.
\end{lemma}

\begin{proof} 
In view of (A5), the definition of $\mu$ implies that there exists 
$v\in\Pi$ such that
$$
 \max\{\|\nabla v^+\|_2^2, \|\nabla v^-\|_2^2\}
\le \mu+\frac{\inf_{\Omega}V_\infty-\mu}{2}
=\frac{\inf_{\Omega}V_\infty+\mu}{2}.
$$
It follows that
\begin{align*}
\|\nabla v^\pm\|_2^2-\int_\Omega V_\infty(x)|v^\pm|^2\mathrm{d}x
& \le \max\{\|\nabla v^+\|_2^2, \|\nabla v^-\|_2^2\}-\inf_{\Omega}V_\infty \\
&\le\frac{\mu-\inf_{\Omega}V_\infty}{2}<0.
\end{align*}
Hence, we have $v\in E_0$. This shows that $E_0\ne\emptyset$ because of (A5). 
Moreover, by \eqref{a2} and \eqref{yb1}, we can easily derive that for any 
$u\in\mathcal {M}$,
 \[
 \|\nabla u^\pm\|_2^2-\int_\Omega V_\infty(x)|u^\pm|^2\mathrm{d}x
=\int_\Omega f_1(x,u^\pm)u^\pm\mathrm{d}x<0.
 \]
 This shows that $\mathcal{M}\subset E_0$.
\end{proof}

\begin{lemma} \label{lem2.6} 
Suppose that {\rm (A1)--(A3), (A5)} are satisfied. 
If $u\in E_0$, then there exists a unique pair $(s_u,t_u)$ of positive 
numbers such that $s_uu^++t_uu^-\in \mathcal{M}$.
\end{lemma}

\begin{proof} 
 Let
 \begin{gather}\label{b51}
 g_1(s)=s^2\int_\Omega|\nabla u^+|^2\mathrm{d}x
-\int_\Omega f(x,su^+)su^+\mathrm{d}x, \\
\label{b52}
 g_2(t)=t^2\int_\Omega|\nabla u^-|^2\mathrm{d}x
-\int_\Omega f(x,tu^-)tu^-\mathrm{d}x.
 \end{gather}
Clearly, $g_1(0)=g_2(0)=0$. Using (A1), (A2), \eqref{a6} and \eqref{gj}, 
we  conclude that $g_1(s)>0$  for $s>0$ small, and
\begin{align*}
 g_1(s)&=s^2\int_\Omega|\nabla u^+|^2\mathrm{d}x
 -\int_\Omega f(x,su^+)su^+\mathrm{d}x\\
 &=s^2\int_\Omega[|\nabla u^+|^2-V_\infty(x)|u^+|^2]\mathrm{d}x
 -\int_\Omega\frac{f_1(x,su^+)}{su^+}(su^+)^2\mathrm{d}x<0
 \end{align*}
for $s$ large.
 From the continuity of $g_1(\cdot)$, there is a $s_u>0$ such that 
$g_1(s_u)=0$. Using (A3), it is easy to verify that $s_u$ is unique.
 Then it follows from \eqref{a2} and \eqref{b51} that 
$\langle\Phi'({s}_uu^+),u^+\rangle=0$. Similarly, there is a unique 
$t_u>0$ such that $g_2(t_u)=0$, and so $\langle\Phi'({t}_uu^-),u^-\rangle=0$.
\end{proof}

\begin{lemma} \label{lem2.7} 
 Suppose that {\rm (A1)--(A3), (A5)}  satisfied. Then
 \[
 \inf_{u\in\mathcal{M}}\Phi(u)=m=\inf_{u\in E_0}\max_{s,t\geq0}\Phi(su^++tu^-).
 \]
\end{lemma}

 Combining Corollary \ref{coro2.4}, Lemmas \ref{lem2.5} and \ref{lem2.6}, 
we obtain the proof of the above lemma.

\begin{lemma} \label{lem2.8} 
 Suppose that {\rm (A1)--(A5)} are satisfied. Then $m>0$ is achieved.
\end{lemma}

\begin{proof}
 Let $\{u_n\}\subset\mathcal{M}$ be such that $\Phi(u_n)\to m$. 
Next, we prove that $\{u_n\}$ is bounded in $H_0^1(\Omega)$.
 Arguing by contradiction, suppose that $\|u_n\|\to\infty$.
 Let $v_n=u_n/\|u_n\|$, then $\|v_n\|=1$. By Sobolev imbedding theorem,
 passing to a subsequence, we may assume that there exists $v\in H_0^1(\Omega)$ 
such that $v_n\rightharpoonup v$ weakly in $H_0^1(\Omega)$, $v_n\to v$ 
strongly in $L^s(\Omega)$, $2\leq s<2^*$. If $v=0$, then $v_n\to 0$ in $L^s(\Omega)$, 
$2\leq s<2^*$. Fix $R>[2(1+m)]^{1/2}$, by \eqref{a6}, one has
\begin{equation}\label{b71}
 \limsup_{n\to\infty}\int_\Omega F(x,Rv_n)\mathrm{d}x
\leq R^2\varepsilon\lim_{n\to\infty}\|v_n\|_2^2
+R^pC_\varepsilon\lim_{n\to\infty}\|v_n\|_p^p=0.
 \end{equation}
Let $t_n=R/\|u_n\|$. Then by \eqref{b71} and Corollary \ref{coro2.3}, one has
\begin{align*}
 m&=\Phi(u_n)+o(1)\geq\Phi(t_nu_n)+o(1) \\
&=\frac{t_n^2}{2}\|u_n\|^2-\int_\Omega F(x,t_nu_n)\mathrm{d}x+o(1)\\
 &=\frac{R^2}{2}-\int_\Omega F(x,Rv_n)\mathrm{d}x+o(1) \\
&=\frac{R^2}{2}+o(1)>m+1+o(1),
 \end{align*}
which is a contradiction. Thus $v\neq 0$.

 For $x\in\Omega_0:=\{y\in\Omega:\ v(y)\neq 0\}$, we have 
$\lim_{n\to\infty}|u_n(x)|=\infty$. Thus, 
 from \eqref{a1}, \eqref{a2}, (A3), (A4) and Fatou's lemma it follows that
\[
 m+1\geq\lim_{n\to\infty}\big[\Phi(u_n)
-\frac{1}{2}\langle\Phi'(u_n),u_n\rangle\big] 
\geq\liminf_{n\to\infty}\int_{\Omega_0}\widetilde{F}(x,u_n)\mathrm{d}x
 =+\infty.
\]
This contradiction shows that $\{\|u_n\|\}$ is bounded. 
Hence, passing to a subsequence, there exists $\widetilde{u}\in H_0^1(\Omega)$ 
such that $u_n^\pm\rightharpoonup\widetilde{u}^\pm$ weakly in $H_0^1(\Omega)$, 
$u_n^\pm\to \widetilde{u}^\pm$ strongly in $L^s(\Omega)$, $2\leq s<2^*$. 
Since $u_n\in \mathcal{M}$, we have $\langle\Phi'(u_n),u_n^\pm\rangle=0$. 
In view of \eqref{a6} and Sobolev embedding theorem, there exists $C_1>0$ 
such that
 \[
 \|u_n^\pm\|^2=\int_\Omega f(x,u_n^\pm)u_n^\pm\mathrm{d}x
 \leq \frac{1}{2}\|u_n^\pm\|^2+C_1\|u_n^\pm\|^2\|u_n^\pm\|_p^{p-2},
 \]
 which implies
 \[
 \int_\Omega|u_n^\pm|^p\mathrm{d}x\geq(\frac{1}{2C_1})^\frac{p}{p-2}.
 \]
 By the compactness of the embedding $H_0^1(\Omega)\hookrightarrow L^s(\Omega)$ 
for $2\leq s<2^*$, we obtain
 \[
 \int_\Omega|\widetilde u^\pm|^p\mathrm{d}x\geq(\frac{1}{2C_1})^\frac{p}{p-2}.
 \]
 Thus, $\widetilde u^\pm\neq0$. Moreover, (A1), (A2) and \cite[A.2]{R1} imply
 \begin{gather}\label{b73}
 \lim_{n\to\infty}\int_\Omega f(x,u_n^\pm)u_n^\pm\mathrm{d}x
=\int_\Omega f(x,\widetilde u^\pm)\widetilde u^\pm\mathrm{d}x,\\
 \lim_{n\to\infty}\int_\Omega F(x,u_n^\pm)\mathrm{d}x
=\int_\Omega F(x,\widetilde u^\pm)\mathrm{d}x , \\
\label{bbb}
 \lim_{n\to\infty}\int_\Omega f_1(x,u_n^\pm)u_n^\pm\mathrm{d}x
=\int_\Omega f_1(x,\widetilde u^\pm)\widetilde u^\pm\mathrm{d}x.
 \end{gather}
 From \eqref{yb1}, \eqref{bbb} and the weak semicontinuity of norm, we have
\begin{align*}
\|\nabla\widetilde u^\pm\|_2^2
 -\int_\Omega V_\infty(x)|\widetilde u^\pm|^2\mathrm{d}x 
 &\leq\liminf_{n\to\infty}\big\{\|\nabla u_n^\pm\|_2^2
 -\int_\Omega V_\infty(x)| u_n^\pm|^2\mathrm{d}x\big\}\\
 &=\liminf_{n\to\infty}\int_\Omega f_1(x,u_n^\pm)u_n^\pm\mathrm{d}x \\
&=\int_\Omega f_1(x,\widetilde u^\pm)\widetilde u^\pm\mathrm{d}x<0,
 \end{align*}
which shows that $\widetilde u\in E_0$. By Lemma \ref{lem2.6}, there exist $s_0>0$ and
 $t_0>0$ such that $s_0\widetilde u^++t_0\widetilde u^-\in\mathcal{M}$ and 
$\Phi(s_0\widetilde u^++t_0\widetilde u^-)\geq m$.
 By \eqref{a2}, \eqref{b73} and the weak semicontinuity of norm, we have
 \begin{equation}\label{bb}
 \begin{aligned}
 \langle\Phi'(\widetilde u),\widetilde u^\pm\rangle
 &=\|\widetilde u^\pm\|^2-\int_\Omega f(x,\widetilde u^\pm)
 \widetilde u^\pm\mathrm{d}x\\
 &\leq\liminf_{n\to\infty}\big\{\|u_n^\pm\|^2
 -\int_\Omega f(x,u_n^\pm)u_n^\pm\mathrm{d}x\big\}=0.
 \end{aligned}
 \end{equation}
 From \eqref{a1}, \eqref{a2}, \eqref{b2}, \eqref{b73}, \eqref{bb},
 Fatou's Lemma and Lemma \ref{lem2.7} it follows that
\begin{align*}
 m&=\lim_{n\to\infty}\big[\Phi(u_n)-\frac{1}{2}\langle\Phi'(u_n),u_n\rangle\big]
 =\lim_{n\to\infty}\int_{\Omega}
 \big[\frac{1}{2}f(x,u_n)u_n-F(x,u_n)\big]\mathrm{d}x\\
 &=\lim_{n\to\infty}\int_{\Omega}
 \big[\frac{1}{2}f(x,\widetilde u)\widetilde u-F(x,\widetilde u)\big]\mathrm{d}x
 =\Phi(\widetilde u)-\frac{1}{2}\langle\Phi'(\widetilde u),\widetilde u\rangle\\
 &\geq\Phi(s_0\widetilde u^++t_0\widetilde u^-)+\frac{1-s_0^2}{2}
 \langle\Phi'(\widetilde u),\widetilde u^+\rangle+\frac{1-t_0^2}{2}
 \langle\Phi'(\widetilde u),\widetilde u^-\rangle
  -\frac{1}{2}\langle\Phi'(\widetilde u),\widetilde u\rangle\\
 &\geq m-\frac{s_0^2}{2}\langle\Phi'(\widetilde u),\widetilde u^+\rangle
 -\frac{t_0^2}{2}\langle\Phi'(\widetilde u),\widetilde u^-\rangle.
 \end{align*}
This implies that $\widetilde u\in\mathcal{M}$ and $\Phi(\widetilde u)=m$.
\end{proof}

\begin{lemma} \label{lem2.9} 
Suppose that {\rm (A1)--(A5)} are satisfied. 
If $\hat u\in\mathcal{M}$ and $\Phi(\hat u)=m$,
 then $\hat u$ is a critical point of $\Phi$.
\end{lemma}

\begin{proof}
 Assume that $\hat u=\hat u^++\hat u^-\in\mathcal{M}$, $\Phi(\hat u)=m$ and 
$\Phi'(\hat u)\neq0$. Then there exist $\delta>0$ and  $\lambda>0$ such that
 \[
 \|\Phi'(u)\|\geq\lambda,\quad \text{for all } \|u-\hat u\|\leq3\delta
 \text{ and } u\in H_0^1(\Omega).
 \]
 Let $D=(1/2,3/2)\times(1/2,3/2)$. It follows from Lemma \ref{lem2.2} that
 \begin{equation}\label{be}
 \chi:=\max_{(s,t)\in \partial D}\Phi(s\hat u^++t\hat u^-)<m.
 \end{equation}

For $\varepsilon:=\min\{(m-\chi)/3,\lambda\delta/8\}$, $S:=B(\hat u,\delta)$, 
\cite[Lemma 2.3]{R1} yields a deformation $\eta\in C([0,1]\times H_0^1(\Omega))$ 
such that
\begin{itemize}
\item[(i)]  $\eta(1,u)=u$ if $\Phi(u)<m-2\varepsilon$ or
$ \Phi(u)>m+2\varepsilon$;

\item[(ii)] $\eta(1,\Phi^{m+\varepsilon}\cap B(\hat u,\delta))
 \subset\Phi^{m-\varepsilon}$;

\item[(iii)] $\Phi(\eta(1,u))\leq\Phi(u)$ for all $u\in H_0^1(\Omega)$.
\end{itemize}

 We claim that
 \begin{equation}\label{b83}
 \max_{(s,t)\in\overline D}\Phi(\eta(1,s\hat u^++t\hat u^-))<m.
 \end{equation}
Indeed, by Lemma \ref{lem2.2} and (iii), we have
 \begin{equation}\label{zz1}
 \Phi(\eta(1,s\hat u^++t\hat u^-))\leq\Phi(s\hat u^++t\hat u^-)<\Phi(\hat u)= m,
 \end{equation}
for all $s,t\geq 0$, $|s-1|^2+|t-1|^2\geq\delta^2/\|\hat u\|^2$.

 On the other hand, by Corollary \ref{coro2.4}, we have 
$\Phi(s\hat u^++t\hat u^-)\leq\Phi(\hat u)=m$ for $s,t\geq0$, then it follows 
from (ii) that
 \begin{equation}\label{zz2}
 \Phi(\eta(1,s\hat u^++t\hat u^-))\leq m-\varepsilon,\quad \forall s,t\geq 0,\;
 |s-1|^2+|t-1|^2<\delta^2/\|\hat u\|^2.
 \end{equation}
 Both \eqref{zz1} and \eqref{zz2} imply that \eqref{b83} holds. 
Define $h(s,t)=s\hat{u}^{+}+t\hat{u}^{-}$. We now prove that 
$\eta(1,h(D))\cap \mathcal{M}\ne\emptyset$, contradicting to the definition of $m$. 
We adopt the idea from \cite{R2}. Let $\beta(s,t):=\eta(1,h(s,t))$ and
\begin{gather*}
\Psi_0(s,t):=\big(\Phi'(h(s,t))\hat{u}^+, \Phi'(h(s,t))\hat{u}^-\big), \\
\Psi_1(s,t):=\big(\frac{1}{s}\Phi'(\beta(s,t))(\beta(s,t))^+, 
\frac{1}{t}\Phi'(\beta(s,t))(\beta(s,t))^-\big).
\end{gather*}
 By Lemma \ref{lem2.6} and  degree theory, we can derive that 
$\deg (\Psi_0,D, 0) = 1$. From \eqref{be} and (i) it follows that $\beta=h$ on
$\partial D$. 
Consequently, $\deg (\Psi_1,D, 0)=\deg (\Psi_0,D, 0) = 1$, and so,
 $\Psi_1(s_0, t_0) = 0$ for some
$(s_0, t_0)\in D$, that is $\eta(1, h(s_0, t_0)) = \beta(s_0, t_0) \in \mathcal{M}$, 
which contradicts \eqref{b83}. From this, we
conclude that $\hat{u}$ is a critical point of $\Phi$.
\end{proof}

\section{Sign-changing solutions}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
In view of Lemmas \ref{lem2.8} and \ref{lem2.9}, there exists a 
$u\in\mathcal{M}$ such that 
$\Phi(u)=m$ and $\Phi'(u)=0$.  Now, we show that $u$ has exactly two nodal domains. 
Set $u=u_1+u_2+u_3$ and $\langle\Phi'(u),u_i\rangle=0$ $(i=1,2,3)$, where
 \begin{equation}\label{c1}
\begin{gathered}
 u_1\geq0,\quad u_2\leq0,\quad \Omega_1\cap\Omega_2=\emptyset,\quad
 u_3|_{\Omega_1\cup\Omega_2}=0, \\
\Omega_1:=\{x\in\Omega:u_1(x)>0\},\quad
\Omega_2:=\{x\in\Omega:u_2(x)<0\},
 \end{gathered}
\end{equation}
and $\Omega_1$, $\Omega_2$ are connected open subsets of $\Omega$.

Let $v=u_1+u_2$, then $v^+=u_1$, $v^-=u_2$, $v^\pm\neq0$ and 
$\langle\Phi'(v),v^\pm\rangle=0$. By \eqref{a1}, \eqref{a2}, \eqref{yb}, \eqref{b2},
 \eqref{c1} and Lemma \ref{lem2.7}, we have
\begin{align*}
 m&=\Phi(u)=\Phi(u)-\frac{1}{2}\langle\Phi'(u),u\rangle\\
 &=\Phi(v)+\Phi(u_3)
 -\frac{1}{2}[\langle\Phi'(v),v\rangle+\langle\Phi'(u_3),u_3\rangle]\\
 &\geq\sup_{s,t\geq0}\big\{\Phi(sv^++tv^-)+\frac{1-s^2}{2}\langle\Phi'(v),
 v^+\rangle+\frac{1-t^2}{2}\langle\Phi'(v),v^-\rangle\big\}\\
 &\quad +\Phi(u_3)-\frac{1}{2}[\langle\Phi'(v),v\rangle+\langle\Phi'(u_3),u_3]\\
 &=\sup_{s,t\geq0}\Phi(sv^++tv^-)
+\int_\Omega\big[\frac{1}{2}f(x,u_3)u_3-F(x,u_3)\big]\mathrm{d}x\geq m,
 \end{align*}
which shows that $u_3=0$. Therefore, $u$ has exactly two nodal domains.
\end{proof}

\subsection*{Acknowledgements}
This work is partially supported by the National Natural Science Foundation 
of China (No: 11571370).
The authors thank the anonymous referees for their valuable suggestions and comments.

\begin{thebibliography}{00}

\bibitem{AB} D. Anderson, M. Bonnedal;
\emph{Variational approach to nonlinear self-focusing of Gaussian laser beams}, 
Physics of Fluids (1958-1988) 22(1) (1979), 105-109.

\bibitem{B2} T. Bartsch, Z. L. Liu, T. Weth;
\emph{Sign changing solutions of superlinear Schr\"odinger equations}, 
Commun. Partial Differential Equations, 29 (2004), 25-42.

\bibitem{S2} T. Bartsch, Z. Q. Wang;
\emph{Existence and multiplicity results for some superlinear elliptic problemson
$\mathbb{R}^N$}, Comm. PDE., 20 (1995), 1725-1741.

\bibitem{BW} T. Bartsch, T. Weth;
\emph{Three nodal solutions of singularly perturbed elliptic equations on 
domains without topology}. Ann. Inst. H. Poincar\'{e} Anal. Non Lin\'{e}aire, 
22 (2005), 259-281.

\bibitem{CCN} A. Castro, J. Cossio, J. M. Neuberger;
\emph{A sign-changing solution for a superlinear Dirichlet problem}, 1997.

\bibitem{Ch} Y. Chen;
\emph{Self-trapped light in saturable nonlinear media}, Optics letters,
 16(1) (1991), 4-6.

\bibitem{CT} S. T. Chen, X. H. Tang;
\emph{Semiclassical solutions for linearly coupled Schrodinger equations},
Electron. J. Differ Eq., 251 (2014), 1-11.

\bibitem{CGT} R. Y. Chiao, E. Garmire, C. H. Townes;
\emph{Self-trapping of optical beams}, Phys. Rev. Lett., 13 (1964), 479-482.

\bibitem{GH} S. Gatz, J. Herrmann;
\emph{Soliton propagation in materials with saturable nonlinearity}, 
JOSA B, 8(11) (1991), 2296-2302.

\bibitem{Ka} M. Karlsson;
\emph{Optical beams in saturable self-focusing media}, 
Physical Review A, 46(5) (1992), 2726.

\bibitem{NW} E. S. Noussair, J. Wei;
\emph{On the effect of the domain geometry on the existence and profile of 
nodal solution  of some singularly perturbed semilinear Dirichlet problem}, 
Indiana Univ. Math. J., 46 (1997), 1321-1332.

\bibitem{QT} D. D. Qin, X. H. Tang;
\emph{Asymptotically linear Schr\"odinger equation with zero on the boundary
of the spectrum}, Electronic J. Differ. Equ., 213 (2015), 1-15.

\bibitem{T6} D. D. Qin, X. H. Tang, J. Zhang;
\emph{Multiple solutions for semilinear elliptic equations with sign-changing 
potential and nonlinearity}, Electron. J. Differential Equations, 207 (2013), 1-9.

\bibitem{B3} A. Salvatore;
\emph{Sign-changing solutions for an asympotically linear Schr\"odinger equation}, 
Discrete Contin Dyn Syst Suppl., (2009), 669-677.

\bibitem{S3} A. Szulkin, T. Weth;
\emph{Ground state solutions for some indefinite variational problems}, 
J. Func. Anal., 257 (2009), 3802-3822.

\bibitem{R2} W. Shuai;
\emph{Sign-changing solutions for a class of Kirchhoff-type problem in 
bounded domains}, J. Differential Equations, 259.4 (2015), 1256-1274.

\bibitem{SGT} M. S. Sodha, A. K. Ghatak, V. K. Tripathi;
\emph{Self Focusing of Laser Beams in Plasmas and Semiconductors}, 
Progress in optics, 13 (1976), 169-265.

\bibitem{R3} M. Struwe;
\emph{Variational methods: applications to nonlinear partial differential
 equations and Hamiltonian systems}, Berlin, Springer, 1990.

\bibitem{Stu} C. A. Stuart;
\emph{Guidance properties of nonlinear planar waveguides}, 
Archive for rational mechanics and analysis, 125(2) (1993), 145-200.

\bibitem{R5} X. H. Tang;
\emph{Non-Nehari manifold method for asymptotically linear Schr\"odinger equation}, 
J. Aust. Math. Soc., 98 (2015), 104-116.

\bibitem{T2} X. H. Tang;
\emph{Infinitely many solutins for semilinear Schr\"odinger equation with 
sign-changing potential and nonlinearity}, J. Math. Anal. Appl., 401 (2013), 407-415.

\bibitem{T5} X. H. Tang;
\emph{Non-Nehari manifold method for asymptotically periodic Schr\"odinger equations},
 Science China Math., 58 (2015), 715-728.

\bibitem{R4} T. Weth;
\emph{Energy bounds for entire nodal solutions of autonomous superlinear equations},
 Calc. Var. Partial Differential Equations, 27 (2006), 421-437.

\bibitem{R1} M. Willem;
\emph{Minimax Theorems}, Birkh\"auser, Boston, 1996.

\bibitem{ZTZ1} J. Zhang X. H. Tang, W. Zhang;
\emph{Existence and multiplicity of stationary solutions for a class of 
Maxwell-Dirac system},  Nonlinear Anal., 127 (2015) 298-311.

\bibitem{ZTZ2} J. Zhang X. H. Tang, W. Zhang;
\emph{Ground state solutions for a class of nonlinear Maxwell-Dirac system}, 
Topol. Meth. Nonl. Anal., 46 (2015), 785-798.

\bibitem{Zo} W. M. Zou;
\emph{Sign-Changing Critical Point Theory}, Springer, New York, 2008.

\end{thebibliography}

\end{document}