\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 $0
0$ 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^*}}