\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 38, pp. 1--9.\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-2018/38\hfil Singular quasilinear Schr\"odinger equations] {Existence and uniqueness of solutions to singular quasilinear Schr\"odinger equations} \author[L.-L. Wang \hfil EJDE-2018/38\hfilneg] {Li-Li Wang} \address{Li-Li Wang \newline School of Mathematics, Tonghua Normal University, 134002 Tonghua, Jilin, China} \email{lili\_wang@aliyun.com, 4120369@qq.com} \dedicatory{Communicated by Vicentiu D. Radulescu} \thanks{Submitted June 23, 2017. Published January 30, 2018.} \subjclass[2010]{35J20, 35A15, 35J75, 35J62} \keywords{Quasilinear Schr\"odinger equation; singularity; uniqueness} \begin{abstract} In this article we study a quasilinear Schr\"{o}dinger equations with singularity. We obtain a unique and positive solution by using the minimax method and some analysis techniques. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction and statement of main results} This article concerns the singular quasilinear Schr\"odinger equation with the Dirichlet boundary value condition \begin{equation}\label{11} \begin{gathered} -\Delta u-\Delta(u^2)u=g(x)u^{-r}-u^{p-1}\quad \text{in }\Omega,\\ u>0\quad \text{in }\Omega,\\ u=0\quad \text{on }\partial\Omega, \end{gathered} \end{equation} where $\Omega\subset\mathbb{R}^N~(N\ge3)$ is a bounded smooth domain with boundary $\partial\Omega$, $r\in(0,1)$ and $p\in[2,22^*]$ are constants. The coefficient $g\in L^{\frac{22^*}{22^*-1+r}}(\Omega)$ with $g(x)>0$ for almost every $x\in\Omega$ and $2^*=\frac{2N}{N-2}$ denotes the critical Sobolev exponent for the embedding $H_0^1(\Omega)\hookrightarrow L^q(\Omega)$ for every $q\in[1,2^*]$. Solutions of \eqref{11} are related to standing wave solutions for the quasilinear Schr\"odinger equations \begin{equation}\label{121} i\partial_t\psi=-\Delta \psi+\psi+\eta(|\psi|^2)\psi -k\Delta \rho(|\psi|^2)\rho'(|\psi|^2)\psi, \end{equation} where $\psi=\psi(t,x),~\psi:\mathbb R\times\Omega\to\mathbb C$, $k>0$ is a constant. The quasilinear equations of the form \eqref{121} play an important role in several areas of physics in correspondence to different type of functions $\rho$. For example, it models the superfluid film equation in plasma physics for $\rho(s)=s$ (see \cite{Kurihara1981Large}), while for $\rho(s)=(1+s)^{1/2}$ it models the self-channeling of a high-power ultra short laser pulse in matter (see \cite{Chen1993Necessary,Bouard1997Global,Ritchie1994Relativistic}). For further physical motivations and developing the physical aspects we refer to \cite{Hasse1980A,Laedke1983Evolution,Lange1995Time,Poppenberg2002On} and the references therein. Motivated by the above mentioned physical aspects, equation \eqref{121} has received a lot of attention. Indeed, up to our knowledge, the first existence results for the subcritical quasilinear equations have been discussed in \cite{Poppenberg2002On} using constraint minimization arguments. Subsequently, many authors in \cite{Colin2004Solutions,liu2003soliton,Moameni2006Existence} were interested in the existence results of standing wave solutions for \eqref{121} by using a change of variable and reducing the quasilinear equations into the semilinear ones in an appropriate Orlicz space. For critical case, we can refer to \cite{Silva2010Quasilinear,Jo2010Soliton,Jo2007Soliton,Moameni2006Existence}. It is worth noticing that up to now there are only one paper \cite{MR2610258} investigating the singular case, where they established the singular quasilinear Schr\"odinger equation \begin{equation*} -\Delta u-\frac{1}{2}\Delta(u^2)u=\lambda u^3-u-u^{-\alpha},\quad u>0,\; x\in\Omega, \end{equation*} where $\Omega$ is a ball in $\mathbb R^N$ $(N\ge2)$ centered at the origin, $0<\alpha<1$. And they proved the existence of radially symmetric positive solutions by employing Nehari manifold and some techniques related to implicit function theorem when $\lambda$ belongs to a certain neighborhood of the first eigenvalue $\lambda_1$ of the eigenvalue problem \begin{equation*} -\Delta u-\frac{1}{2}\Delta(u^2)u=\lambda u^3. \end{equation*} The singular problems are much more complicated than the regular one and they require some hard analysis. For singular elliptic problems, there are many authors (see e.g. \cite{Ghergu2003Sublinear,Crandall1977On,Coclite1993On,sun2011An, Pino1992A,Ghergu2008Singular,Radulescu2007Singular}) have studied. Especially, Ghergu and R\u{a}dulescu in \cite{Ghergu2003Sublinear} established several existence and nonexistence results for the boundary value problem \begin{equation}\label{1212} \begin{gathered} -\Delta u+K(x)g(u)=\lambda f(x,u)+\mu h(x)\quad \text{in }\Omega,\\ u>0\quad \text{in }\Omega,\\ u=0\quad \text{on }\partial\Omega, \end{gathered} \end{equation} where $\Omega$ is a smooth bounded domain in $\mathbb R^N~(N\ge2)$, $\lambda$ and $\mu$ are positive parameters, $h$ is a positive function, $f$ has a sublinear growth and the function $g$ satisfies the condition \begin{equation*} \lim_{s\to\infty}g(s)=+\infty. \end{equation*} Obviously, $g(s)=s^{-r},r\in(0,1)$ satisfies the above assumption. When $K(x)\equiv-1,~f(x,u)=u^p$ and $g(s)=s^{-r}$ in \eqref{1212}, where $r\in(0,1),p\ge 0$, Coclite and Palmieri in \cite{Coclite1993On} proved that there is at least one solution for all $\lambda\ge 0$ if $00$ and no solution for large $\lambda>0$ if $p\ge1$. For Second-Order Differential Equations, such as Sturm-Liouville operator, Dirac Operators etc., there are many authors being interested, we can refer to \cite{Nursultanov2017Eigenvalue,Levitan1991Sturm} and the references therein. The main purpose of this article is to study the singular quasilinear Schr\"odinger equation \eqref{11} and introduce a uniqueness result of solutions for \eqref{11}, which is the first work on this subject up to our knowledge. \smallskip \noindent\textbf{Notation.} $C$ is a positive constant whose value can be different. The domain of an integral is $\Omega$ unless otherwise indicated. $\int f(x)dx$ is abbreviated to $\int f(x)$. $L^p(\Omega)$, $1\le p\le\infty$, denotes the Lebesgue space with the norms $\|u\|_p=(\int|u|^p)^{\frac{1}{p}}$, for $1\le p<\infty$, $\|u\|_\infty=\inf\{C>0:|u(x)|\le C \text{ almost everywhere in }\Omega\}$. $X=H_0^1(\Omega)$ denotes the Hilbert space equipped with the norm $\|u\|=(\int|\nabla u|^2)^{1/2}$. The main result is described as follows. \begin{theorem}\label{Th12} Suppose that $r\in(0,1)$, $p\in[2,22^*]$ and $g\in L^{\frac{22^*}{22^*-1+r}}(\Omega)$ with $g(x)>0$ for almost every $x\in\Omega$. Then problem \eqref{11} has a unique positive solution in $X$. Moreover, this solution is the global minimizer solution. \end{theorem} %\begin{remark}\label{R1}\rm The classic semilinear singular equation \begin{gather*} -\Delta u=g(x)u^{-r}+\lambda u^{p-1},\quad \text{in }\Omega,\\ u=0,\quad \text{on } \partial\Omega, \end{gather*} where $p=2^*$, has been studied for $\lambda>0$ in \cite{sun2011An} and also in \cite{Pino1992A} for $\lambda=0$ under the condition $g(x)\in L^\infty(\Omega)$. We point out that the condition $g\in L^{\frac{22^*}{22^*-1+r}}(\Omega)$ is more general than the condition $g(x)\in L^\infty(\Omega)$. To the best of our knowledge, the existence and uniqueness of solutions for the quasilinear Schr\"odinger equation \eqref{11} has not been discussed up to now. This article is organized as follows: Some preliminaries are given in the next section. In Section 3, we give the proof of Theorem \ref{Th12}. \section{Preliminary results} We observe that the energy functional corresponding to \eqref{11} given by \begin{align*} J(u) :=\frac{1}{2}\int (1+2u^{2})|\nabla u|^{2} -\frac{1}{1-r}\int g(x)|u|^{1-r}+\frac{1}{p}\int|u|^p \end{align*} is not well defined in $X$. To overcome this problem, we use the change of variable $v:=f^{-1}(u)$ introduced in \cite{liu2003soliton}, where $f$ is defined by \[ f'(t)=\frac{1}{\sqrt{1+2f^{2}(t)}} \text{ on } [0,+\infty), \quad\text{and}\quad f(t)=-f(-t) \text{ on } (-\infty ,0]. \] We list some properties of $f$, whose proofs can be found in \cite{Colin2004Solutions,severo2010solitary}. \begin{lemma}\label{L21} The function $f$ satisfies the following properties: \begin{itemize} \item[(1)] $f$ is uniquely defined, $C^{\infty}$ and invertible; \item[(2)] $|f'(t)|\leq 1$ for all $t\in \mathbb{R}$; \item[(3)] $|f(t)|\leq |t|$ for all $t\in \mathbb{R}$; \item[(4)] $f(t)/t\rightarrow 1$ as $t\rightarrow 0$; \item[(5)] $|f(t)f'(t)| < 1/\sqrt{2}$,~$\forall t\in \mathbb{R}$; \item[(6)] $f(t)/2\leq tf'(t)\leq f(t)$ for all $t\ge0$; \item[(7)] $|f(t)|\leq 2^{1/4}|t|^{1/2}$ for all $t\in \mathbb{R}$; \item[(8)] the function $f^{-r}(t)f'(t)$ is decreasing for all $t>0$; \item[(9)] the function $f^{p-1}(t)f'(t)$ is increasing for all $t>0$. \end{itemize} \end{lemma} \begin{proof} We only prove (8) and (9). By $f''(t)=-2f(t)[f'(t)]^4$, for all $t\in\mathbb R$, $p\ge2$ and $(5)$, with simple computation we obtain $$ \frac{d[f^{-r}(t)f'(t)]}{dt}=-rf^{-r-1}(t)[f'(t)]^2-2f^{1-r}(t)[f'(t)]^4<0, \quad \forall t>0 $$ and $$ \frac{d[f^{p-1}(t)f'(t)]}{dt}=f^{p-2}(t)[f'(t)]^2[p-1-2f^{2}(t)[f'(t)]^2]>0, \quad \forall t>0, $$ which imply that $f^{-r}(t)f'(t)$ is decreasing and $f^{p-1}(t)f'(t)$ is increasing for all $t>0$. \end{proof} By exploiting the change of variable, we can rewrite the functional in the form \begin{align*} I(v): =\frac{1}{2}\int |\nabla v|^{2}-\frac{1}{1-r}\int g(x)|f(v)|^{1-r} +\frac{1}{p}\int|f(v)|^p,\quad v\in X. \end{align*} By Lemma \ref{L21}-(7), the H\"older inequality and the Sobolev inequality we have \begin{align}\label{12} \int g(x)|f(v)|^{1-r}\le C\|g\|_{\frac{22^*}{22^*-1+r}}\|v\|^{\frac{1-r}{2}}. \end{align} Then $I$ is well-defined but only continuous on $X$. Also equation \eqref{11} can be rewritten as \begin{align}\label{a} -\Delta v=g(x)f^{-r}(v)f'(v)-f^{p-1}(v)f'(v),~v>0,~x\in\Omega. \end{align} In general, a function $v\in X$ is called a weak solution of \eqref{a} with $v>0$ in $\Omega$ if it holds \begin{align}\label{13} \int \nabla v \nabla w- g(x)f^{-r}(v)f'(v)w+f^{p-1}(v)f'(v)w=0,\quad \forall w\in X. \end{align} We observe that if $v\in X$ is a weak solution of \eqref{a}, the function $u=f(v)\in X$ is a solution of \eqref{11} (cf:\cite{Colin2004Solutions}). \section{Proof of Theorem \ref{Th12}} In this section, we shall show that there exists a unique positive solution $v_0$ of \eqref{a}, which is the global minimizer of the functional $I$ in $X$, and then $u_0=f(v_0)\in X$ is the unique positive solution of \eqref{11}. \begin{lemma}\label{L31} The functional $I$ attains the global minimizer in $X$; that is, there exists $v_0\in X\setminus\{0\}$ such that $I(v_0)=m:=\inf_X I<0$. \end{lemma} \begin{proof} For $v\in X$, from \eqref{12} it follows that \begin{align}\label{21} I(v)\ge \frac{1}{2}\|v\|^2-\frac{C}{1-r}\|g\|_{\frac{22^*}{22^*-1+r}} \|v\|^{\frac{1-r}{2}}. \end{align} Since $r\in(0,1)$, $I$ is coercive and bounded from below on $X$. Thus $m:=\inf_X I$ is well defined. For $t>0$ and given $v\in X\setminus\{0\}$ by Lemma \ref{L21}-(7) one gets \begin{align*} I(tv) &=\frac{t^2}{2}\|v\|^2-\frac{1}{1-r}\int g(x)|f(tv)|^{1-r}+\frac{1}{p} \int |f(tv)|^p\\ &\le\frac{t^2}{2}\|v\|^2-\frac{1}{1-r}\int g(x)|f(tv)|^{1-r} +\frac{C}{p}t^{\frac{p}{2}}\int |v|^{\frac{p}{2}}. \end{align*} Note that the function $|\frac{f(tv)}{tv}|^{1-r}$ is non-increasing for $t>0$. By Lemma \ref{L21}-(4) and Beppo-Levi Monotone Convergence Theorem, we can see \begin{align*} \lim_{t\to 0^+}\frac{I(tv)}{t^{1-r}}=-\frac{1}{1-r}\int g(x)|v|^{1-r}<0. \end{align*} So we have $I(tv)<0$ for all $v\not\equiv0$ and $t>0$ small enough. Hence, we obtain $m<0$. According to the definition of $m$, there exists a minimizing sequence $\{v_n\}\subset X$ such that $\lim_{n\to\infty}I(v_n)=m<0$. Since $I(v_n)=I(|v_n|)$, we may assume that $v_n\ge0$. It follows from \eqref{21} that there exists a constant $C>0$ such that $\|v_n\|\le C$. Passing if necessary to a subsequence, we can assume that there exists $v_0\in X$ such that \begin{gather*} v_n\rightharpoonup v_0\quad \text{in }X, \\ v_n\to v_0\quad \text{in }L^p(\Omega),\; p\in[1,2^*),\\ v_n(x)\to v_0(x)\quad \text{a.e. in } \Omega, \end{gather*} there exists a function $k\in L^p(\Omega)$, $p\in[1,2^*)$, such that \begin{equation} \label{b} |u_n(x)|\le k(x)\quad \text{a.e. in }\Omega. \end{equation} By Vitali's theorem (see \cite{Rudin1966Real}), we claim that \begin{align}\label{22} \lim_{n\to\infty}\int g(x)f^{1-r}(v_n)=\int g(x)f^{1-r}(v_0). \end{align} Indeed, we only need prove that $\{\int g(x)f^{1-r}(v_n),~n\in \mathbb N\}$ is equi-absolutely-continuous. For all $\varepsilon>0$, by the absolutely-continuity of $\int|g(x)|^{\frac{22^*}{22^*-1+r}}$, there exists $\delta>0$ such that $\int_E|g(x)|^{\frac{22^*}{22^*-1+r}}<\varepsilon^{\frac{22^*}{22^*-1+r}}$ for all $E\subset\Omega$ with $\operatorname{meas}E<\delta$. Consequently, by \eqref{12} and the fact that $\|v_n\|\le C$, we have \begin{align*} \int_Eg(x)f^{1-r}(v_n) \le C\|v_n\|^{\frac{1-r}{2}}\Big(\int_E|g(x)|^{\frac{22^*}{22^*-1+r}} \Big)^{\frac{22^*-1+r}{22^*}}0$ in $\Omega$. Fix $\phi\in X$ with $\phi\ge0$, let $t>0$, one has \begin{align*} 0&\le I(v_0+t\phi)-I(v_0)\\ &=\frac{1}{2}\|v_0+t\phi\|^2-\frac{1}{2}\|v_0\|^2 -\frac{1}{1-r}\int g(x)[f^{1-r}(v_0+t\phi)-f^{1-r}(v_0)]\\ &\quad +\frac{1}{p}\int f^p(v_0+t\phi)-f^p(v_0). \end{align*} Dividing by $t>0$ and passing to the limit as $t\to0^+$ in the above inequality, we have \begin{equation}\label{14} \begin{aligned} &\frac{1}{1-r}\liminf_{t\to0^+}\int g(x)\frac{f^{1-r}(v_0+t\phi)-f^{1-r}(v_0)}{t}\\ &\le\int\nabla v_0\nabla\phi+f^{p-1}(v_0)f'(v_0)\phi. \end{aligned} \end{equation} Note that \begin{equation*} \int g(x)\frac{f^{1-r}(v_0+t\phi)-f^{1-r}(v_0)}{t} =(1-r)\int g(x) f^{-r}(v_0+t\theta\phi)f'(v_0+t\theta\phi)\phi, \end{equation*} where $\theta(x)\in(0,1)$. For any $x\in \Omega$, we denote $$ h(t)=g(x)f^{-r}(v_0+t\theta\phi)f'(v_0+t\theta\phi)\phi,\quad t>0. $$ It follows from $g(x)>0$ a.e. $x\in\Omega$ and Lemma \ref{L21}-(8) that $h(t)$ is non-increasing for $t>0$. Moreover, $$ \lim_{t\to0^+}h(t)= g(x) f^{-r}(v_0(x))f'(v_0(x))\phi(x) $$ for every $x\in\Omega$, which may be $+\infty$ when $v_0(x)=0$. Consequently, by the Beppo-Levi Monotone Convergence Theorem, we obtain \begin{equation*} \liminf_{t\to0^+}\frac{1}{1-r} \int g(x)\frac{f^{1-r}(v_0+t\phi)-f^{1-r}(v_0)}{t} =\int g(x) f^{-r}(v_0)f'(v_0)\phi, \end{equation*} which together with \eqref{14} implies that \begin{align}\label{15} \int\nabla v_0\nabla\phi- g(x) f^{-r}(v_0)f'(v_0)\phi +f^{p-1}(v_0)f'(v_0)\phi\ge0,\quad \phi\in X,\; \phi\ge0. \end{align} Therefore, $$ -\Delta v_0+f^{p-1}(v_0)f'(v_0)\ge0 $$ in the weak sense. Hence the maximum principle implies that $v_0>0$ in $\Omega$. Secondly, we show that $v_0$ is a solution of \eqref{a}, that is, we prove $v_0$ satisfies \eqref{13}. For given $\delta>0$, define $H:[-\delta,\delta]\to \mathbb R$ by $H(t)=I((1+t)v_0)$, then $H$ attains its minimum at $t=0$ by Lemma \ref{L31}, namely \begin{align}\label{16} H'(0)=\|v_0\|^2-\int g(x)f^{-r}(v_0)f'(v_0)v_0-f^{p-1}(v_0)f'(v_0)v_0=0. \end{align} Choose $\varphi\in X\setminus\{0\},\varepsilon>0$. Define $\phi\in X$ by $\phi=(v_0+\varepsilon\varphi)^+$. Let $$ \Omega_1=\{x\in\Omega:v_0(x)+\varepsilon\varphi(x)>0\},\quad \Omega_2=\{x\in\Omega:v_0(x)+\varepsilon\varphi(x)\le0\}. $$ Easily, we see $\phi|_{\Omega_1}=v_0+\varepsilon\varphi$ and $\phi|_{\Omega_2}=0$. Inserting $\phi$ into \eqref{15} and applying with \eqref{16}, one obtains \begin{equation} \label{17} \begin{aligned} 0&\le\int\nabla v_0\nabla\phi- g(x) f^{-r}(v_0)f'(v_0)\phi +f^{p-1}(v_0)f'(v_0)\phi \\ &=\int_{\Omega_1}\nabla v_0\nabla (v_0+\varepsilon\varphi) - g(x) f^{-r}(v_0)f'(v_0)(v_0+\varepsilon\varphi) \\ &\quad +f^{p-1}(v_0)f'(v_0)(v_0+\varepsilon\varphi) \\ &=\int_{\Omega\setminus\Omega_2}\nabla v_0\nabla (v_0+\varepsilon\varphi) - g(x) f^{-r}(v_0)f'(v_0)(v_0+\varepsilon\varphi) \\ &\quad +f^{p-1}(v_0)f'(v_0)(v_0+\varepsilon\varphi) \\ &=\varepsilon\int\nabla v_0\nabla\varphi-g(x) f^{-r}(v_0)f'(v_0)\varphi +f^{p-1}(v_0)f'(v_0)\varphi \\ &\quad -\int_{\Omega_2}\nabla v_0\nabla(v_0+\varepsilon\varphi) -g(x) f^{-r}(v_0)f'(v_0)(v_0+\varepsilon\varphi) \\ &\quad +f^{p-1}(v_0)f'(v_0)(v_0+\varepsilon\varphi) \\ &\le \varepsilon\int\nabla v_0\nabla\varphi-g(x) f^{-r}(v_0)f'(v_0)\varphi +f^{p-1}(v_0)f'(v_0)\varphi\\ &\quad -\varepsilon\int_{\Omega_2}\nabla v_0\nabla\varphi +f^{p-1}(v_0)f'(v_0)\varphi . \end{aligned} \end{equation} From $\operatorname{meas}\Omega_2\to 0$ as $\varepsilon\to0$, it follows that $$ \int_{\Omega_2}\nabla u_0\nabla \varphi+f^{p-1} (v_0)f'(v_0)\varphi\to0\quad \text{as }\varepsilon\to0. $$ Then dividing by $\varepsilon>0$ and letting $\varepsilon\to0$ in \eqref{17}, we conclude that $$ \int\nabla v_0\nabla\varphi-g(x) f^{-r}(v_0)f'(v_0)\varphi+f^{p-1}(v_0) f'(v_0)\varphi\ge0. $$ By the arbitrariness of $\varphi$, the above inequality also holds for $-\varphi$, so we get that $v_0$ solves \eqref{13}. Hence $v_0\in X$ is a positive solution of \eqref{a} with $I(v_0)=m<0$, that is, $v_0$ is the global minimizer solution. Finally, we show that $v_0\in X$ is the unique solution of \eqref{a}. Assume that $v\in X$ is also a solution of \eqref{a}, it follows from \eqref{13} that \begin{equation}\label{18} \int\nabla v_0\nabla(v_0-v)-g(x) f^{-r}(v_0)f'(v_0)(v_0-v) +f^{p-1}(v_0)f'(v_0)(v_0-v)=0 \end{equation} and \begin{equation}\label{19} \int\nabla v\nabla(v_0-v)-g(x) f^{-r}(v)f'(v)(v_0-v)+f^{p-1}(v)f'(v)(v_0-v)=0. \end{equation} Subtracting \eqref{18} and \eqref{19}, since $g(x)>0$ a.e. $x\in\Omega$, by Lemma \ref{L21}-(8),~(9) we get \begin{align*} \|v_0-v\|^2 &=\int g(x) [f^{-r}(v_0)f'(v_0)-f^{-r}(v)f'(v)](v_0-v)\\ &\quad -\int[f^{p-1}(v_0)f'(v_0)-f^{p-1}(v)f'(v)](v_0-v)\le 0, \end{align*} which implies that $\|v_0-v\|=0$, that is $v_0=v$. Therefore, $v_0\in X$ is the unique solution of \eqref{a}, and then $u_0=f(v_0)\in X$ is the unique solution of \eqref{11}. 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