\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 109, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/109\hfil Schr\"odinger equations in photonic lattice] {Steady-state solutions for Schr\"odinger equations in photonic lattice} \author[W.-L. Li \hfil EJDE-2018/109\hfilneg] {Wen-Long Li} \address{Wen-Long Li \newline Department of Mathematics, Nanjing University, Nanjing 210093, China} \email{liwenlongchn@gmail.com} \dedicatory{Communicated by Vicentiu D. Radulescu} \thanks{Submitted July 18, 2017. Published May 8, 2018.} \subjclass[2010]{35J20, 35B09, 35J47} \keywords{Schr\"odinger equation; multiple solutions; positive solution; \hfill\break\indent nontrivial solution} \begin{abstract} In this article, we study a nonlinear Schr\"odinger equation arising in optics. Firstly we prove the existence of multiple solutions of this equation. Secondly, we consider a nonlinear Schr\"odinger system which is intimately related to the Schr\"odinger equation. We obtain the existence of nontrivial solutions to this system and we also get some results on its positive solutions. Finally, assuming Dirichlet or Neumann boundary condition, we show the existence and uniqueness of positive solution to the Schr\"odinger equation. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction and statement of main results}\label{intro} \subsection{Schr\"odinger equation} \label{singleequation} Schr\"odinger type equations have been studied extensively in the literature (see \cite{Bah, Cho, Gou, Hol, Wangzq} for Schr\"odinger equation, \cite{Col, Kri} for Schr\"odinger systems, i.e., two coupled equations, and comments at the end of sections \ref{singleequation} and \ref{coupled}). In recent years, many exciting phenomena were found by careful experiments on light waves propagating in nonlinear periodic lattices. These phenomena are governed by the following Schr\"odinger equation (cf. \cite{Are, YZ, Sch} and the references therein) \begin{equation}\label{equa1} i\frac{\partial \psi}{\partial z}+D\Delta \psi=g(x,|\psi|^2)\psi , \end{equation} where $D>0$ means the beam diffraction coefficient and $\Delta \psi=\frac{\partial^{2}\psi}{\partial {x_1}^2} +\frac{\partial^{2} \psi}{\partial {x_2}^2}$. We take $D=1$ for convenience. Steady wave beam propagate along $z$-axis direction and transversely spread along $x=(x_1,x_2)\in \mathbb{R}^2$. The functions of concern are all periodic with respect to $x$, so equation \eqref{equa1} is viewed as defined over a periodic spatial domain $\Omega$ in $\mathbb{R}^2$. Steady state solution of equation \eqref{equa1} is a solution with the form $\psi(x,z)=e^{i\lambda z}u(x)$. Here $\lambda\in \mathbb{R}$ is a constant and $u(x)$ is a real-valued function. If we insert $\psi(x,z)$ into equation \eqref{equa1} and take $g(x,|\psi|^2)$ (cf. \cite{Bar, Fle, Fle1, Nes, YZ, Sch}) as \[ %\label{equa2} g(x,|\psi|^2)=\frac{P}{1+V(x)+|\psi|^2}, \] we obtain \begin{equation}\label{equa3} \Delta u=\frac{Pu}{1+V+u^2}+ \lambda u. \end{equation} Here $V=V(x)\geq 0$ is potential function, which depends on the spatial variable $x$ periodically (modulo $\Omega$). In this paper, we set $V_0=\max_{x\in\bar{\Omega}}V(x)$, $v_0=\min_{x\in\bar{\Omega}}V(x)$. A special case is when $V(x)\equiv 0$. In this case, equation \eqref{equa3} has the following simple form \begin{equation}\label{equa30} \Delta u=\frac{Pu}{1+u^2}+ \lambda u. \end{equation} This equation will be heuristic when we study a system corresponding to equation \eqref{equa3}, as we will see later. Yang and Zhang \cite{YZ, YZ1} showed that \begin{theorem}\label{yangz} \begin{enumerate} \item For any $P$, equation \eqref{equa3} has a solution for some real value $\lambda =\lambda(P)$. \item If $P>0$, $\lambda <0$ and $|\lambda|$ is small enough, then equation \eqref{equa3} has a nontrivial solution. \item If $P<0$, $0<\lambda <|P|/(1+V_0)$, then equation \eqref{equa3} has a nontrivial solution.\label{yangzhang} \item If $P<0$, $|P|/(1+v_0)\leq \lambda$, then equation \eqref{equa3} has only the trivial solution.\label{yangzhang1} \end{enumerate} \end{theorem} Schechter \cite{Sch} studied equation \eqref{equa3} by the linking method. Before stating his results we need some notation. Let $\{\lambda _k; 0=\lambda_0 <\lambda_1 <\lambda_2 <\dots <\lambda_k <\dots\}$ be eigenvalues of the operator $-\Delta$ on functions in $L^2(\Omega)$ having the same periods as $\Omega$, then $\lambda_k$ has finite multiplicity. The corresponding eigenfunctions belong to $L^{\infty}(\Omega)$. Schechter \cite{Sch} proved the following result. \begin{theorem}\label{Sch} \begin{enumerate} \item If $P>0$, $\lambda <0$ and there is an $l\geq 0$ such that $\lambda_l +P/(1+v_0)\leq |\lambda| <\lambda_{l+1}+P/(1+V_0)$, $|\lambda|>\lambda_{l+1}$, then equation \eqref{equa3} has a nontrivial solution. \item If $P<0$, $\lambda <0$, and there is an $l\geq 0$ such that $\lambda_l +P/(1+V_0)< |\lambda| \leq\lambda_{l+1}+P/(1+v_0)$, $|\lambda|<\lambda_l$, then equation \eqref{equa3} has a nontrivial solution. \item If $P>0$, $\lambda <0$, and $0<|\lambda|

0$, and $0<\lambda<|P|/(1+V_0)$, then equation \eqref{equa3} has a nontrivial solution. \end{enumerate} \end{theorem} Note that Schechter's results hold in arbitrary dimensions. In this paper, we consider the existence of multiple solutions of equation \eqref{equa3} and we obtain the following result. \begin{theorem}\label{thm1} If $P<0$, $\lambda >0$ and there is an integer $k>0$ such that $\lambda +\lambda_k <\frac{|P|}{1+V_0}$, then equation \eqref{equa3} has at least $k$ pairs of distinct solutions. \end{theorem} The novelty of our result lies in establishing multiple solutions of equation \eqref{equa3} while in \cite{YZ, YZ1, Sch} the authors considered the existence of single solution under suitable conditions. There are also lots of studies on equation \eqref{equa1} when the function $g$ is defined in other forms. For instance, $g(x,|\psi|)=V(x)-\gamma |\psi|^{p-1}$ in \cite{Rabzamp, Bah, Cho, Wangzq} (in fact, their setting was abstract and thus their results covered a wider range), $g(x,|\psi|)=l(l+1)/|x|^2$ in \cite{Hol}. Other than steady-state solutions, blow-up solutions of Schr\"odinger equations have widely been investigated in the literature, too. In recent years, blow-up solutions of Schr\"odinger equation with defect have also attracted lots of attentions. We refer \cite{Gou} for references in this direction. \subsection{Schr\"odinger system: two coupled equations} \label{coupled} We also study the system \begin{equation}\label{sys} \begin{gathered} \Delta u= \frac{Pu}{1+u^2+v^2}+\lambda u , \\ \Delta v= \frac{Qv}{1+u^2+v^2}+\lambda v , \end{gathered} \end{equation} which is intimately related to equations \eqref{equa3} and \eqref{equa30}. In \eqref{sys} $P$, $Q$ and $\lambda$ are parameters and the functions $u$, $v$ are defined over a periodic bounded spatial domain $\Omega \subset \mathbb{R}^2$. System \eqref{sys} is also a nonlinear photonic lattic model and has been studied by many researchers (cf. \cite{CL, Fle1, Mal, Tro, Xie, Yang, Yang1, Sch1} and the references therein). A solution of system \eqref{sys} having the form $(u,v)$ is called nontrivial if $u\not\equiv 0$ and $v\not\equiv 0$. If $u\equiv0$ or $v\equiv0$ but not both, it will be called a semi-trivial solution. System \eqref{sys} was studied in \cite{CL, Sch1, Ren}. The authors proved that system \eqref{sys} had semi-trivial solutions when $P, Q, \lambda$ were suitably chosen. In particular, the authors of \cite{Sch1, Ren} studied equation \eqref{equa30} instead of \eqref{sys} after pointing out that the existence of nontrivial solutions of system \eqref{sys} had not been proved. In this article, we prove the existence of nontrivial solutions of system \eqref{sys} in two special cases. We also study the positive solutions of system \eqref{sys} in these two cases. Our results show that system \eqref{sys} has nontrivial solutions and we hope they will be helpful for understanding the physics of the nonlinear Schr\"odinger system \eqref{sys}. We give some comments on the recent work in \cite{Col, Kri}. In \cite{Col} the author considered the system of coupled nonlinear Schr\"odinger-Korteweg-de Vries equations, where the parameter $\lambda$ is different in each equation. We choose the same $\lambda$ because our analysis on system \eqref{sys} is based upon results of equations \eqref{equa3} and \eqref{equa30}. In \cite{Kri} the authors studied Schr\"odinger-Maxwell equations. They reduced this system to a single equation by ``elimination method''. More precisely, they represented one of the unknown functions by the other and then the system was equivalent to a single equation. Our system \eqref{sys} is not solved by this method because we do not know how the unknown functions rely on each other. \subsection{Schr\"odinger equation with zero Dirichlet or zero Neumann boundary condition} Finally we show the existence and uniqueness of positive solution of equation \eqref{equa3} with Dirichlet boundary condition $u|_{\partial\Omega}=0$ or Neumann boundary condition $\frac{\partial u}{\partial \nu}|_{\partial \Omega}=0$. Let $\varphi_1\in W^{1,2}_{0}(\Omega)$ be an eigenfunction of $-\Delta$ (with zero Dirichlet boundary condition) corresponding to the first eigenvalue $\mu_1 =\mu_1 (\Omega)$ with $\varphi_1 >0$ in $\Omega$ and set $M_1=\max_{\bar{\Omega}}{\varphi_1}$. Here we use $\mu_1$ to distinguish from $\lambda_1$ which correspond to $-\Delta$ but without boundary condition. We obtain the following theorems. \begin{theorem}\label{thm2} If $\lambda \geq0,P<0$ and $\mu_1 +\lambda <\frac{|P|}{1+V_0+M_1}$, then equation \eqref{equa3} with Dirichlet boundary condition $u|_{\partial\Omega}=0$ has a positive solution. \end{theorem} \begin{theorem}\label{thm3} If $P<0$, then equation \eqref{equa3} with Dirichlet boundary condition $u|_{\partial\Omega}=0$ has a unique positive solution. \end{theorem} For Neumann boundary condition we obtain the uniqueness of positive solution for small $\lambda >0$. \begin{theorem}\label{prop1} If $P<0$, $\lambda>0$ and $\lambda<\frac{|P|}{1+V_0}$, then equation \eqref{equa3} with Neumann boundary condition $\frac{\partial u}{\partial \nu}|_{\partial \Omega}=0$ has a unique positive solution. When $V(x)\equiv 0$, under the above conditions, the unique positive solution of equation \eqref{equa30} is $u=\sqrt{-1-\tfrac{P}{\lambda}}$. \end{theorem} The rest of this article is organized as follows. In section \ref{multiple}, we prove Theorem \ref{thm1} by verifying the conditions of Clark Theorem. In section \ref{soluofsystem}, we investigate nontrivial solutions and positive solutions of system \eqref{sys} in two special cases. Our discussions in this section are based on the existing results of equations \eqref{equa3} and \eqref{equa30}. In section \ref{positive}, we establish the existence and uniqueness of positive solution of equation \eqref{equa3} with zero Dirichlet (resp. Neumann) boundary condition, which proves Theorems \ref{thm2}-\ref{prop1}. \section{Multiple solutions} \label{multiple} In this section, we prove Theorem \ref{thm1}. We use the following result of Clark\cite[p.53, Theorem 9.1]{Rab}. \begin{theorem}\label{Clark} Let $E$ be a real Banach space, $I \in C^{1}{(E,\mathbb{R})}$ with $I$ even, bounded from below, and satisfying (PS) condition. Suppose $I(0)=0$, there is a set $K \subset E$ such that $K$ is homeomorphic to $S^{j-1}$ by an odd map, and $\sup_{K}I < 0$. Then $I$ possesses at least $j$ distinct pairs of critical points. \end{theorem} We will verify the conditions of Theorem \ref{Clark}. The corresponding energy functional of equation \eqref{equa3} is \[ I(u)=\frac{1}{2}\int_\Omega \Big\{|\nabla u|^2+\lambda u^2 +P[\log(1+V+u^2)-\log(1+V)]\Big\}\,dx \] for $u\in W^{1,2}(\Omega)$. It is easy to see that $I\in C^{1}(W^{1,2}(\Omega),\mathbb{R})$. $I$ is even, i.e., $I(-u)=I(u)$ and $I(0)=0$. \begin{lemma}\label{bdd} If $P<0$ and $\lambda >0$, then $I$ is bounded from below. \end{lemma} \begin{proof} For any $\epsilon >0$, we have \[ \log(1+V+u^2)-\log(1+V)\leq |u|\leq \epsilon |u|^2+1/(4\epsilon), \] and then \begin{equation}\label{fig3115} \begin{split} I(u)&=\frac{1}{2}\int_\Omega \Big\{|\nabla u|^2+\lambda u^2+P[\log(1+V+u^2) -\log(1+V)]\Big\}\,dx \\ &\geq \frac{P}{8\epsilon}|\Omega|+\frac{1}{2}\int_\Omega \Big(|\nabla u|^2 +\lambda u^2+P\epsilon u^2\Big)\,dx \\ &= \frac{P}{8\epsilon}|\Omega|+\frac{1}{2}\int_\Omega \Big(|\nabla u|^2 +\frac{\lambda u^2}{2}\Big)\,dx. \end{split} \end{equation} To conclude Lemma \ref{bdd}, we take $\epsilon =-\lambda/(2P)$ in the last equality. \end{proof} \begin{lemma}\label{ps} If $P<0$ and $\lambda >0$, then $I$ satisfies (PS) condition. In other words, if $\{u_n\}$ is a sequence in $W^{1,2}(\Omega)$ such that \begin{enumerate} \item[(i)] $I(u_n)$ is bounded, \item[(ii)] $I'(u_n)\to 0$ as $n\to \infty$, \end{enumerate} then $\{u_n\}$ contains a subsequence which converges in $W^{1,2}(\Omega)$. \end{lemma} \begin{proof} Suppose $\{u_n\}\subset W^{1,2}(\Omega)$ such that $|I(u_n)|\leq M$ and $I'(u_n)\to 0$, where $M>0$ is a fixed constant. We have \[ \big|\int_\Omega \Big\{|\nabla u_n|^2+\lambda u_{n}^{2} +P[\log(1+V+u^{2}_{n})-\log(1+V)]\Big\}\,dx\big|\leq 2M \] and \begin{equation}\label{fig3117} |\langle I'(u_n),v \rangle| =\big| \int_\Omega \Big\{\nabla u_n \nabla v+\lambda u_n v +P\frac{u_n v}{1+V+u_{n}^{2}}\Big\}\,dx\big| \leq \epsilon _n \|v\|_{W^{1,2}(\Omega)} , \end{equation} where $\epsilon_n>0$ approaches to $0$ as $n \to \infty$. Note that by \eqref{fig3115} it follows that \[ I(u_n)-\frac{P}{8\epsilon}|\Omega|\geq \frac{1}{2} \int_\Omega \Big\{|\nabla u_n|^2+\frac{\lambda}{2}u_{n}^{2}\Big\}\,dx. \] So $\{u_n\}$ is bounded in $W^{1,2}(\Omega)$. Without loss of generality we set $u_n$ convergence weakly to some $u$ in $W^{1,2}(\Omega)$. By Rellich-Kondrachov Theorem, we know that $u_n\to u$ and $\frac{u_n}{1+V+u_{n}^{2}}\to \frac{u}{1+V+u^2}$ in $L^2(\Omega)$. Letting $n \to \infty$ in \eqref{fig3117} we obtain \begin{equation}\label{fig3119} \int_{\Omega} \Big\{\nabla u \nabla v+\lambda u v +P\frac{uv}{1+V+u^2}\Big\}\,dx =0. \end{equation} Plugging \eqref{fig3119} into \eqref{fig3117} and taking $v=u_n -u$, we have \begin{align*} &\Big| \int_\Omega \Big\{|\nabla(u_n-u)|^2-P(u_n-u) \Big[\frac{u_n}{1+V+u_{n}^{2}}-\frac{u}{1+V+u^2} \Big] +\lambda(u_n-u)^2\Big\}\,dx \Big| \\ &\leq \epsilon_n \|u_n-u\|_{W^{1,2}(\Omega)} . \end{align*} Then \begin{align*} &\int_\Omega |\nabla(u_n-u)|^2\,dx \\ &\leq \epsilon_n \|u_n-u\|_{W^{1,2}(\Omega)} + |P| \int_\Omega |u_n-u| \big| \frac{u_n}{1+V+u_{n}^{2}} -\frac{u}{1+V+u^2} \big|\,dx \\ + & \lambda \int_\Omega |u_n-u|^2\,dx \to 0 . \end{align*} We obtain $u_n \to u$ in $W^{1,2}(\Omega)$. \end{proof} \begin{lemma}\label{subsetK} If $P<0$, $\lambda >0$ and $\lambda +\lambda_{k} <\frac{|P|}{1+V_0}$, then there is a set $K\subset W^{1,2}(\Omega)$ such that $K$ is homeomorphic to $S^{k-1}$ by an odd map and $\sup_{K}I < 0$. \end{lemma} \begin{proof} Indeed, let $\phi_k$ be one eigenfunction corresponding to $\lambda_k$. Without loss of generality, we can take $\|\phi_k\|_{W^{1,2}(\Omega)}=\|\nabla\phi_k\|_{L^2 (\Omega)} +\|\phi_k\|_{L^2 (\Omega)}=1$. Then $(1+\lambda _k) \int \phi^2_k\,dx=1$ since $-\Delta \phi_k +\phi_k =(\lambda_k +1)\phi_k$. Let \[ K=K(r)=\Big\{\sum_{i=1}^{k}\alpha _i \phi_i : \sum_{i=1}^{k}\alpha _{i}^{2}=r^2\Big\}. \] Clearly $K$ is homeomorphic to $S^{k-1}$ by an odd map for all $r>0$. We claim that $\sup_{K}I<0$ for $r>0$ small enough. In fact, we have \[ \log(1+V+u^2)-\log(1+V)\geq \frac{u^2}{1+V+u^2}\geq \frac{u^2}{1+V_0+u^2} \] provided $0\leq V(x) \leq V_0$. By Jensen's inequality \[ \varphi \Big( \frac{\int_{\Omega} f(x)p(x)dx}{\int_{\Omega} p(x)dx} \Big) \leq \frac{\int_{\Omega} \varphi (f(x))p(x)dx}{\int_{\Omega} p(x)dx}, \] if we take $\varphi(t)=\frac{1}{1+V_0+t}$ and $f(x)=p(x)=u^{2}(x)$, we have \[ \int_\Omega \frac{u^2}{1+V_0+u^2}\,dx \geq \int_\Omega \frac{u^2}{1+V_0 +\frac{\int_\Omega u^4\,dx}{\int_\Omega u^2\,dx}}\,dx . \] Using Gagliardo-Nirenberg inequality, we obtain \[ \int_\Omega u^4\,dx \leq C\|u\|_{W^{1,2}(\Omega)}^{2}\int_\Omega u^2\,dx , \] and then \[ \int_\Omega \frac{u^2}{1+V_0+u^2}\,dx \geq \frac{\int_\Omega u^2\,dx}{1+V_0+C\|u\|_{W^{1,2}(\Omega)}^{2}} . \] For $u\in K$ we have $\|u\|_{W^{1,2}(\Omega)}^{2}=r^2$, so we have arrived at the energy upper bound \begin{align*} I(u) &=\frac{1}{2}\int_\Omega \Big\{|\nabla u|^2+\lambda u^2+P[\log(1+V+u^2) -\log(1+V)]\Big\}\,dx \\ &\leq \frac{1}{2}\int_\Omega \Big\{|\nabla u|^2+\lambda u^2 +P\frac{u^2}{1+V_0+Cu^2}\Big\}\,dx \\ &\leq \frac{1}{2}\int_\Omega \Big\{|\nabla u|^2+\lambda u^2 +P\frac{u^2}{1+V_0+Cr^2}\Big\}\,dx \\ &=\frac{1}{2}r^2+\frac{1}{2}\Big(\lambda -1 +\frac{P}{1+V_0+Cr^2}\Big)\int_\Omega u^2\,dx \\ &=\frac{1}{2}r^2+\frac{1}{2}\Big(\lambda -1 +\frac{P}{1+V_0+Cr^2}\Big)\sum_{i=1}^{k} \Big(\frac{\alpha _{i}^{2}}{\lambda_i +1}\Big). \end{align*} But $\lambda-1+\frac{P}{1+V_0}<-1-\lambda_k<0$ provided $\lambda +\lambda _k <-\frac{P}{1+V_0}$. So there exists $r_1>0$, such that for all $r \in (0,r_1)$ we have $\lambda-1+\frac{P}{1+V_0+Cr^2}<-\lambda_k-1$ and \begin{align*} I(u)&\leq \frac{1}{2}r^2+\frac{1}{2} \Big(\lambda -1+\frac{P}{1+V_0+Cr^2}\Big) \frac{r^2}{\lambda_k +1} \\ &=\frac{1}{2}r^2\Big[1+\frac{1}{\lambda _k+1} (\lambda -1+\frac{P}{1+V_0+Cr^2})\Big] . \end{align*} Then there is $r_0 \in (0,r_1)$ such that $I(u)<0$ for all $u\in K(r_0)$. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1}] By Lemmas \ref{bdd}, \ref{ps}, \ref{subsetK} and Theorem \ref{Clark}, the functional $I$ has at least $k$ pairs distinct critical points and then equation \eqref{equa3} has at least $k$ pairs solutions. \end{proof} \section{Nontrivial solutions and positive solution of the system} \label{soluofsystem} In this section, we focus on the existence of the nontrivial solutions of system \eqref{sys}. As pointed out by Schechter \cite{Sch1} (see also \cite{Ren}), no one has showed the existence of nontrivial solutions of system \eqref{sys}. We consider two special cases in which system \eqref{sys} has nontrivial solutions. \subsection{Case $P=Q$} \label{p=q} System \eqref{sys} becomes \begin{equation}\label{syspq} \begin{gathered} \Delta u= \frac{Pu}{1+u^2+v^2}+\lambda u , \\ \Delta v= \frac{Pv}{1+u^2+v^2}+\lambda v . \end{gathered} \end{equation} Observing this system is closely connected to equation \eqref{equa3}, more precisely to equation \eqref{equa30}, we have the following theorem. \begin{theorem}\label{thmsys} If $\lambda >0$, $P<0$, and $\lambda <|P|$, then system \eqref{syspq} has infinitely many solutions. \end{theorem} \begin{proof} The following equation has a nontrivial solution, say $w$, by Theorem \ref{yangz} \[ \Delta w=\frac{Pw}{1+w^2}+\lambda w . \] Then $(w\cos{\theta},w\sin{\theta})$, for all $\theta \in (-\pi, +\pi)\setminus \{0,\pm\frac{\pi}{2}\}$, will be nontrivial solutions of system \eqref{syspq}. \end{proof} \begin{remark}\label{aaa} \rm Combining our method and Theorems \ref{yangz}, \ref{Sch} and results mentioned in section \ref{coupled}, one has similar results of Theorem \ref{thmsys}. \end{remark} \begin{remark}\label{aaaa} \rm If system \eqref{syspq} has positive solution $(u,v)$ with $u>0$ and $v>0$, then $(u,v)$ has the form $(u,cu)$, i.e., $v$ is proportional to $u$. In fact, $u>0$ and $v>0$ are solutions of linear equation \[ \Delta w +f(x)w=0 , \] where $f(x)=|P|/[1+u^{2}(x)+v^{2}(x)]-\lambda$. Then $u$ and $v$ are eigenfunctions corresponding to eigenvalue $0$. So $v=cu$ for some constant $c>0$ because $0$ is the smallest eigenvalue which is simple. (If $0$ is not smallest, then there is some eigenvalue, say $\lambda_0<0$. Denote $\varphi_0$ is a positive eigenfunction corresponding to $\lambda_0$. This will lead to a contradiction since different eigenfunction corresponding to different eigenvalue should be orthogonal in $L^2(\Omega)$. But $u$ and $\varphi_0$ are positive, so they cannot be orthogonal in $L^2 (\Omega)$.) \end{remark} \begin{remark}\label{posi} \rm Under the assumption of Theorem \ref{thmsys}, system \eqref{syspq} has infinitely positive solutions. In fact, our proof of Theorem \ref{thmsys} is based on Theorem \ref{yangz} \eqref{yangzhang}, which is proved by minimizing method. When $w$ is a global minimizer so is $|w|$ because energy functional $I$ satisfies $I(w) \geq I(|w|)$. Regularity theory and maximum principle ensure that $|w|$ is a positive solution. Thus $(w\cos \theta,w\sin \theta)$, for all $\theta \in (0, \pi /2)$, will be positive solutions of system \eqref{syspq}. \end{remark} \subsection{Case $Q=0$} System \eqref{sys} will be ``decoupled'' into \begin{equation}\label{sysq0} \begin{gathered} \Delta u= \frac{Pu}{1+u^2+v^2}+\lambda u , \\ \Delta v= \lambda v . \end{gathered} \end{equation} \begin{theorem}\label{thmq00} \begin{enumerate} \item If $-\lambda =\lambda_k >0$, $P>0$, and there is an integer $l\in (0,k)$ such that $\lambda_l +P/(1+\phi_{k}^{0})\leq \lambda_k <\lambda_{l+1}+P/(1+\phi_{k}^{1})$, then system \eqref{sysq0} has a nontrivial solution. \item If $-\lambda =\lambda_k >0$, $P<0$, and there is an integer $l>k$ such that $\lambda_l +P/(1+\phi_{k}^{1})< \lambda_k \leq \lambda_{l+1}+P/(1+\phi_{k}^{0})$, then system \eqref{sysq0} has a nontrivial solution. \end{enumerate} Where $\phi_{k}$ is the eigenfunction corresponding to $\lambda_k$ and $\phi_{k}^{0}=\min_{\bar{\Omega}}(\phi_{k})^2$, $\phi_{k}^{1}=\max_{\bar{\Omega}}(\phi_{k})^2$. \end{theorem} \begin{proof} Clearly $v=\phi_k$ is a nontrivial solution of the second equation of system \eqref{sysq0}. For this fixed function $v$, we conclude this theorem by Theorem \ref{Sch}. \end{proof} \begin{remark} \rm If $P<0$ and system \eqref{sysq0} has positive solution $(u,v)$ with $u>0$ and $v>0$, we find that $\lambda$ must be $0$ and then $v$ is constant. We claim: all of the positive solutions of system \eqref{sysq0} have the form $(Cu_{0}^{c}, c)$, where $C$, $c$ are positive constants and $u_{0}^{c}$ is a positive solution of \[ \Delta u=\frac{(Pu)/1+c^2}{1+u^2}(=:\frac{P_c u}{1+u^2}). \] In fact, if $(u,v)$ are positive solution of system \eqref{sysq0}, then $\lambda=0$ and $v=c$ for some constant $c$. The first equation of system \eqref{sysq0} becomes $\Delta u=\frac{Pu}{1+c^2+u^2}$, i.e., $\Delta w=\frac{P_c w}{1+w^2}$, where $w=u/\sqrt{1+c^2}$. Using the same argument as in Remark \ref{aaaa} we obtain our claim. \end{remark} We complete this section with a remark on more general cases. \begin{remark} \rm If $\lambda>0$, $P<0$, $Q<0$ and $\lambda\geq\min\{|P|,|Q|\}$, then equation \eqref{sys} has no positive solution $(u,v)$ with $u>0$ and $v>0$. This follows from Remark \ref{aab} of sec. \ref{positive} and $\lambda\geq\min\{|P|,|Q|\}>\min\{\frac{|P|}{1+v^{2}_{0}},\frac{|Q|}{1+u^{2}_{0}}\}$, where $u_0=\min_{x\in \Omega}u(x)$ and $v_0=\min_{x\in \Omega}v(x)$. Please compare this remark with Remark \ref{aaaa}. \end{remark} \section{Positive solution with zero Dirichlet or Neumann boundary condition} \label{positive} In this section, we prove Theorems \ref{thm2}, \ref{thm3} and \ref{prop1}. We set the energy functional corresponding to equation \eqref{equa3} with zero Dirichlet boundary condition as \[ \tilde{I}(u)=\frac{1}{2}\int_\Omega \Big\{|\nabla u|^2+\lambda u^2\Big\}\,dx + \frac{1}{2}\int_\Omega P\Big[\log(1+V+u^{2}_{+})-\log(1+V)\Big]\,dx \] for $u \in W^{1,2}_{0}(\Omega)$, where $u_{+}=\max\{u,0\}$. We first give some lemmas. \begin{lemma}\label{bdd1} If $P<0$ and $\lambda \geq 0$ then $\tilde{I} $ is bounded from below. \end{lemma} \begin{proof} Since \[ \log(1+V+u^{2}_{+})-\log(1+V)\leq |u_{+}|\leq \epsilon |u_{+}|^2 +\frac{1}{4\epsilon}, \] we have \begin{align*} \tilde{I}(u) \geq & \frac{1}{2}\int_\Omega \Big(|\nabla u|^2+\lambda u^2 +P\epsilon |u_{+}|^2\Big)\,dx +\frac{P}{8\epsilon}|\Omega| \\ \geq & \frac{P}{8\epsilon}|\Omega| + \frac{1}{2}\int_\Omega \Big(\mu_1+\lambda +P\epsilon \Big)u^2\,dx \\ \geq & \frac{P}{8\epsilon}|\Omega|, \end{align*} where $\epsilon > 0 $ is taken small enough such that $\mu_1 +\lambda +P\epsilon > 0$. \end{proof} \begin{lemma}\label{global} If $P<0$ and $\lambda \geq 0$, then $\tilde{I} $ has a global minimizer, say $w$, in $W^{1,2}_{0}(\Omega) $. \end{lemma} \begin{proof} Let $\{u_{n}\}$ be a minimizing sequence satisfying $\tilde{I}(u_n) \to \inf_{W^{1,2}_{0}(\Omega)}\tilde{I}$. Then there exists $N \in \mathbb{N}$ such that for all $n>N$ we have $|\tilde{I}(u_n)|\leq |\inf_{W^{1,2}_{0}(\Omega)}\tilde{I}|+1$. So for $\epsilon>0$, we obtain \begin{align*} &\frac{1}{2}\int_\Omega \Big(|\nabla u_n|^2 + \lambda u_{n}^{2}\Big)\,dx \\ &\leq |\inf_{W_{0}^{1,2}(\Omega)}\tilde{I}|+1-\frac{1}{2}P\int_\Omega \Big[\log(1+V+u_{n}^{2})-\log(1+V)\Big]\,dx\\ &\leq |\inf_{W_{0}^{1,2}(\Omega)}\tilde{I}|+1-\frac{P}{8\epsilon}|\Omega| -\frac{P\epsilon}{2}\int_\Omega u_{n}^{2}\,dx . \end{align*} Then $\{u_n\}$ is bounded in $W^{1,2}_{0}(\Omega)$. Without loss of generality $u_{n} \to w$ weakly in $W^{1,2}_{0}(\Omega)$. Since $\tilde{I}$ is weakly lower continuity, $\tilde{I}$ has a global minimizer $w$ in $W^{1,2}_{0}(\Omega)$. \end{proof} \begin{lemma}\label{lessthan0} If $P<0$, $\lambda>0$ and $\mu_1 +\lambda <\frac{|P|}{1+V_0 +M_1}$, then $\tilde{I}(w)<0$, where $w$ is defined in Lemma \ref{global}. \end{lemma} \begin{proof} It is well known that $\mu_1>0$. By the definition of $w$, we obtain $\tilde{I}(w)\leq \tilde{I}(\varphi_1) $. Noticing \[ \log(1+V+u_{+}^{2})-\log(1+V)\geq \frac{u_{+}^{2}}{1+V+u_{+}^{2}}, \] we obtain \begin{align*} \tilde{I}(\varphi_1) =&\frac{1}{2}\int_\Omega \Big(|\nabla \varphi_1|^2+\lambda \varphi_{1}^{2}\Big)\,dx +\frac{1}{2}\int_\Omega P\Big[\log(1+V+\varphi_{1}^{2})-\log(1+V)\Big]\,dx \\ \leq &\frac{1}{2}\int_\Omega \Big(\mu_1 + \lambda\Big)\varphi_{1}^{2}\,dx +\frac{1}{2}\int_\Omega \frac{P\varphi_{1}^{2}}{1+V+\varphi_{1}^{2}}\,dx \\ \leq & \Big[ \frac{1}{2}(\mu_1+\lambda)+\frac{1}{2}\frac{P}{1+V_0+M_1} \Big] \int_\Omega \varphi_{1}^{2}\,dx < 0 \end{align*} provided that $\mu_1 +\lambda <\frac{-P}{1+V_0+M_1}$. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2}] Firstly $w_{+}\not\equiv 0 $: if $w_{+}\equiv0$ then $\tilde{I}(w)=\frac{1}{2}\int_\Omega (|\nabla w|^2+\lambda w^2)\,dx \geq 0$, which contradicts to Lemma \ref{lessthan0}. Secondly $w\geq 0$. $w$ is a weak solution of \begin{equation}\label{eqn1897} \begin{gathered} \Delta w= \frac{Pw_{+}}{1+w_{+}^{2}+V}+\lambda w \quad x \in \Omega, \\ w = 0 \quad x \in \partial \Omega. \end{gathered} \end{equation} Multiplying the first equation in \eqref{eqn1897} by $w_{-} $ and integrating by parts on $\Omega$, we obtain \[ \int_\Omega \big(|\nabla w_{-}|^2+\lambda w_{-}^{2}\big)\,dx= 0 . \] So $w_{-} = 0$ and then $w\geq 0$, i.e., $w $ is a weak solution of equation \eqref{equa3} with zero Dirichlet boundary condition. Note that $w $ is a classical solution by elliptic regularity theory, and $w>0$ follows from the strong maximum principle. \end{proof} To prove Theorem \ref{thm3}, we first prove the following lemma. \begin{lemma}\label{uniqu} If $P<0$, then equation \eqref{equa3} with Dirichlet boundary condition $u|_{\partial \Omega}=0$ has at most one positive solution. \end{lemma} \begin{proof} If $u$, $w$ are positive solutions of equation \eqref{equa3}, then $u$, $w > 0 $ in $\Omega $ and \begin{gather}\label{eqna101} \Delta u= \frac{Pu}{1+V+u^2}+\lambda u , \\ \label{eqna102} \Delta w= \frac{Pw}{1+V+w^2}+\lambda w . \end{gather} Set $\Omega_{+}=\{x \in \Omega; u(x)-w(x)>0 \}$ then $\Omega_{+}$ is a piecewise $C^1$ smooth domain. We claim that $\Omega_{+} = \emptyset $. In fact, if $\Omega_{+} \neq \emptyset $, multiplying equation \eqref{eqna101} by $w$ and subtracting equation \eqref{eqna102} which is multiplied by $u$, we have \begin{align*} &\int _{\partial \Omega_{+}}\Big(w\frac{\partial u}{\partial \nu} -u\frac{\partial w}{\partial \nu}\Big)\,dx \\ &= \int _{\Omega_{+}}Puw \Big[ \frac{1}{1+V+u^2}-\frac{1}{1+V+w^2} \Big]\,dx \\ &= \int _{\Omega_{+}}Puw\frac{w^2-u^2}{(1+V+w^2)(1+V+u^2)}\,dx > 0 . \end{align*} However, \[ \int _{\partial \Omega_{+}}\Big(w\frac{\partial u}{\partial \nu} -u\frac{\partial w}{\partial \nu}\Big)\,dx = \int _{\partial \Omega_{+} \setminus \partial \Omega} \Big(w\frac{\partial u}{\partial \nu}-u\frac{\partial w}{\partial \nu}\Big)\,dx = \int _{\partial \Omega_{+} \setminus \partial \Omega}w \frac{\partial (u-w)}{\partial \nu}\,dx \leq 0 . \] This contradiction shows that $\Omega _{+}= \emptyset$. Similarly $\Omega_{-}=\{ x \in \Omega ;u(x)-w(x)<0\}=\emptyset $. Therefore we obtain $ u \equiv w $ in $\Omega $. \end{proof} \begin{remark}\label{c} \rm The method in the proof of Lemma \ref{uniqu} is modified from \cite{WY}. It is easy to see that the same conclusion holds if boundary condition $u|_{\partial\Omega}=0$ is replaced by $\frac{\partial u}{\partial \nu}|_{\partial\Omega}=0$. \end{remark} Theorem \ref{thm3} now follows from Theorem \ref{thm2} and Lemma \ref{uniqu}. \begin{proof}[Proof of Theorem \ref{prop1}] We need only to prove that equation \eqref{equa3} has a positive solution because of Remark \ref{c}. But this is a direct consequence of Theorem \ref{yangz} \eqref{yangzhang} and Remark \ref{posi}. Notice that in Theorem \ref{yangz} there is no boundary condition while we add Neumann boundary in our case, but we can use their proof since we can still integrate by parts in the proof. \end{proof} \begin{remark}\label{aab} \rm When $P<0$, $\lambda>0$ and $\lambda \geq |P|/(1+v_0)$, equation \eqref{equa3} with zero Neumann boundary condition has no positive solution because the only solution is $0$ as pointed out by Yang and Zhang in Theorem \ref{yangz} \eqref{yangzhang1}. In particular this is a sharp conclusion for equation \eqref{equa30}. \end{remark} \subsection*{Acknowledgments} The author would like to thank Professor Zhi-Qiang Wang (Utah State University) for helpful discussions and suggestions. The author would also like to thank Xiaojun Cui (Nanjing University) for reading the manuscript and providing useful criticism. The author would also like to thank the anonymous referee for the useful comments and suggestions which improve the quality of the paper. The author is partially supported by the National Natural Science Foundation of China (Grants 11571166, 11631006, 11790272), the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD) and the Fundamental Research Funds for the Central Universities. \begin{thebibliography}{99} \bibitem{Are} Arecchi, F. T.; Boccaletti, S.; Ramazza, P.; \emph{Pattern formation and competition in nonlinear optics} Physics Reports, 318 (1999), no. 1-2, 1-83. \bibitem{Bah} Bahrouni, A.; Ounaies, H.; R\u{a}dulescu, V.; \emph{Infinitely many solutions for a class of sublinear Schr\"odinger equations with indefinite potentials}. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 145 (2015), no. 3, 445-465. \bibitem{Bar} Bartal, G.; Manela, O., Cohen, O.; Fleischer, J. W.; Segev, M.; \emph{Observation of second-band vortex solitons in 2D photonic lattices}. Physical review letters, 95 (2005), no. 5, 053904. \bibitem{CL} Chen, S.; Lei, Y.; \emph{Existence of steady-state solutions in a nonlinear photonic lattice model}. J. Math. Phys., 52 (2011), no. 6, 063508. \bibitem{Yang} Chen, Z.; Bezryadina, A.; Makasyuk, I.; Yang, J.; \emph{Observation of two-dimensional lattice vector solitons}. Optics letters, 29 (2004), no. 14, 1656-1658. \bibitem{Cho} Chorfi, N.; R\u{a}dulescu, V.; \emph{Standing wave solutions of a quasilinear degenerate Schr\"odinger equation with unbounded potential}. Electronic Journal of Qualitative Theory of Differential Equations. 2016, Paper No. 37, 12 pp. \bibitem{Col} Colorado, E.; \emph{On the existence of bound and ground states for some coupled nonlinear Schr\"odinger-Korteweg-de Vries equations}. Advances in Nonlinear Analysis, 6 (2017), no. 4, 407-426. \bibitem{Fle} Fleischer, J. W.; Bartal, G.; Cohen, O.; Manela, O.; Segev, M.; Hudock, J.; Christodoulides, D. N.; \emph{Observation of vortex-ring `discrete' solitons in 2D photonic lattices}. Physical review letters, 92 (2004), no. 12, 123904. \bibitem{Fle1} Fleischer, J. W.; Segev, M.; Efremidis, N. K.; Christodoulides, D. N.; \emph{Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices}. Nature (London), 422 (2003), no. 6928, 147-149. \bibitem{Gou} Goubet, O.; Hamraoui E.; \emph{Blow-up of solutions to cubic nonlinear Schr\"odinger equations with defect: The radial case}. Advances in Nonlinear Analysis, 6 (2017), no. 2, 183-197. \bibitem{Hol} Holzleitner, M.; Kostenko A.; Teschl, G.; \emph{Dispersion Estimates for Spherical Schr\" odinger Equations: The Effect of Boundary Conditions}. Opuscula Math. 36 (2016), no. 6, 769-786. \bibitem{Kri} Krist\'aly, A.; Repov\v{s}, D.; \emph{On the Schr\"odinger-Maxwell system involving sublinear terms}. Nonlinear Analysis: Real World Applications, 13 (2012), no. 1, 213-223. \bibitem{Wangzq} Li, Y., Wang, Z.-Q.; Zeng, J.; \emph{Ground states of nonlinear Schr\"odinger equations with potentials}. Annales de l'Institut Henri Poincar\'e (C) Non Linear Analysis. Elsevier Masson., 23 (2006), no. 6, 829-837. \bibitem{Ren} Liu, C; Ren, Q.; \emph{On the steady-state solutions of a nonlinear photonic lattice model}. J. Math. Phys., 56 (2015), no. 3, 031501. \bibitem{Mal} Malomed, B. A.; Kevrekidis, P. G.; \emph{Discrete vortex solitons}. Physical Review E 64 (2001) no. 2, 026601. \bibitem{Nes} Neshev, D. N.; Alexander, T. J.; Ostrovskaya, E. A.; Kivshar, Y. S.; Martin, H.; Makasyuk. I.; Chen. Z.; \emph{Observation of discrete vortex solitons in optically induced photonic lattices}. Physical review letters, 92 (2004), no. 12, 123903. \bibitem{Rab} Rabinowitz, P. H.; \emph{Minimax methods in critial point theory with applications to differential equations}. CBMS 65, AMS, Providence RI 1986. \bibitem{Rabzamp} Rabinowitz, P. H.; \emph{On a class of nonlinear Schr\"odinger equations}. Zeitschrift f\"ur Angewandte Mathematik und Physik (ZAMP), 43 (1992), no. 2, 270-291. \bibitem{Sch} Schechter, M.; \emph{Steady stet solutions for Schr\"odinger equations governing nonlinear optics}. J. Math. Phys., 53 (2012), no. 4, 043504. \bibitem{Sch1} Schechter, M.; \emph{Photonic lattices}. J. Math. Phys., 54 (2013), no. 6, 061502. \bibitem{Tro} Trombettoni, A.; Smerzi, A.; \emph{Discrete solitons and breathers with dilute Bose-Einstein condensates.} Physical Review Letters, 86 (2001), no. 11, 2353. \bibitem{WY} Wei, J.; Yao, W.; \emph{Uniqueness of positive solutions to some coupled nonlinear Schr\"odinger equations}. Commun. Pure Appl. Anal, 11 (2012), no. 3, 1003-1011. \bibitem{Xie} Xie, A.; vander Meer, L.; Hoff, W.; Austin, R. H.; \emph{Long-lived amide I vibrational modes in myoglobin}. Physical review letters, 84 (2000), no. 23, 5435. \bibitem{Yang1} Yang, J.; Makasyuk, I.; Bezryadina, A.; Chen, Z.; \emph{Dipole and quadrupole solitons in optically induced two-dimensional photonic lattices: theory and experiment}. Studies in applied mathematics 113 (2004), no. 4, 389-412. \bibitem{YZ} Yang, Y.; Zhang, R.; \emph{Steady state solutions for nonlinear Schr\"odinger equation arising in optics}. J. Math. Phys., 50 (2009), no. 5, 053501. \bibitem{YZ1} Yang, Y.; Zhang, R.; \emph{Erratum to ``Steady state solutions for nonlinear Schr\"odinger equation arising in optics''[J. Math. Phys. 50 (2009), no. 5, 053501]}, J. Math. Phys. 51 (2010), no. 4, 049902. \end{thebibliography} \end{document}