\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 295, pp. 1--12.\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-2016/295\hfil Perturbed subcritical Dirichlet problems] {Perturbed subcritical Dirichlet problems with variable exponents} \author[R. Alsaedi \hfil EJDE-2016/295\hfilneg] {Ramzi Alsaedi} \address{Ramzi Alsaedi \newline Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia} \email{ramzialsaedi@yahoo.co.uk} \thanks{Submitted September 8, 2016. Published November 16, 2016.} \subjclass[2010]{35J60, 58E05} \keywords{Nonhomogeneous elliptic problem; variable exponent; \hfill\break\indent Dirichlet boundary condition; mountain pass theorem} \begin{abstract} We study a class of nonhomogeneous elliptic problems with Dirichlet boundary condition and involving the $p(x)$-Laplace operator and power-type nonlinear terms with variable exponent. The main results of this articles establish sufficient conditions for the existence of nontrivial weak solutions, in relationship with the values of certain real parameters. The proofs combine the Ekeland variational principle, the mountain pass theorem and energy arguments. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} Let $\Omega\subset\mathbb{R}$ be a bounded domain with smooth boundary. In a pioneering paper, Ambrosetti and Rabinowitz \cite{ambrab} consider the subcritical elliptic problem \begin{equation}\label{ambro} \begin{gathered} -\Delta u=|u|^{p-2}u,\quad x\in\Omega\\ u=0,\quad x\in\partial\Omega, \end{gathered} \end{equation} where $1
1
\text{ for all } x\in\overline\Omega\}.
$$
For all $h\in C_+(\overline\Omega)$ we define
$$
h^+=\sup_{x\in\Omega}h(x)\quad\text{and}\quad
h^-=\inf_{x\in\Omega}h(x).
$$
The real numbers $h^+$ and $h^-$ will play a crucial role in our arguments
and usually the gap between these quantities produces new results,
which are no longer valid for constant exponents.
For any $p\in C_+(\overline\Omega)$, we define the \emph{variable exponent
Lebesgue space}
$$
L^{p(x)}(\Omega)=\{u: u \text{ is measurable and }
\int_\Omega|u(x)|^{p(x)}\,dx<\infty\}.
$$
This vector space is a Banach space if it is endowed with the
\emph{Luxemburg norm}, which is defined by
$$
|u|_{p(x)}=\inf\big\{\mu>0;\;\int_\Omega|
\frac{u(x)}{\mu}|^{p(x)}\,dx\leq 1\big\}.
$$
Then $L^{p(x)}(\Omega)$ is reflexive if and only if $1 < p^-\leq p^+<\infty$
and continuous functions with compact support
are dense in $L^{p(x)}(\Omega)$ if $p^+<\infty$.
The inclusion between Lebesgue spaces with variable exponent generalizes
the classical framework, namely if
$0 <|\Omega|<\infty$ and $p_1$, $p_2$ are variable exponents so that
$p_1\leq p_2$ in $\Omega$ then there exists the continuous embedding
$L^{p_2(x)}(\Omega)\hookrightarrow L^{p_1(x)}(\Omega)$.
Let $L^{p'(x)}(\Omega)$ be the conjugate space
of $L^{p(x)}(\Omega)$, where $1/p(x)+1/p'(x)=1$. For any
$u\in L^{p(x)}(\Omega)$ and $v\in L^{p'(x)}(\Omega)$ the following
H\"older-type inequality holds:
\begin{equation}\label{Hol}
\big|\int_\Omega uv\,dx\big|\leq\Big(\frac{1}{p^-}+
\frac{1}{p'^-}\Big)|u|_{p(x)}|v|_{p'(x)}\,.
\end{equation}
The \emph{modular} of $L^{p(x)}(\Omega)$ is the mapping
$\rho_{p(x)}:L^{p(x)}(\Omega)\to\mathbb{R}$ defined by
$$
\rho_{p(x)}(u)=\int_\Omega|u|^{p(x)}\,dx.
$$
If $(u_n)$, $u\in L^{p(x)}(\Omega)$ and $p^+<\infty$ then the following
relations are true:
\begin{gather}\label{L4}
|u|_{p(x)}>1\;\Rightarrow\;|u|_{p(x)}^{p^-}\leq\rho_{p(x)}(u)
\leq|u|_{p(x)}^{p^+}, \\
\label{L5}
|u|_{p(x)}<1\;\Rightarrow\;|u|_{p(x)}^{p^+}\leq
\rho_{p(x)}(u)\leq|u|_{p(x)}^{p^-}, \\
\label{L6}
|u_n-u|_{p(x)} \to 0\; \Leftrightarrow\;\rho_{p(x)}
(u_n-u)\to 0.
\end{gather}
We define the variable exponent Sobolev space by
$$
W^{1,p(x)}(\Omega)=\{u\in L^{p(x)}(\Omega):|\nabla u|\in L^{p(x)} (\Omega) \}.
$$
On $W^{1,p(x)}(\Omega)$ we may consider one of the following
equivalent norms
$$
\|u\|_{p(x)}=|u|_{p(x)}+|\nabla u|_{p(x)}
$$
or
$$\|u\|=\inf\big\{\mu>0;\;\int_\Omega\Big(| \frac{\nabla
u(x)}{\mu}|^{p(x)}+|\frac{u(x)}{\mu}|^{p(x)}\Big)\,dx\leq 1\big\}\,.
$$
We define $W_0^{1,p(x)}(\Omega)$ as the closure of the set of compactly
supported $W^{1,p(x)}$-functions with respect to the norm $\|u\|_{p(x)}$.
When smooth functions are dense, we can also use the closure of
$C_0^\infty(\Omega)$ in $W^{1,p(x)}(\Omega)$. Using the Poincar\'e inequality,
the space $W_0^{1,p(x)}(\Omega)$ can be defined, in an equivalent manner,
as the closure of $C_0^\infty(\Omega)$ with respect to the norm
$$
\|u\|_{p(x)}=|\nabla u|_{p(x)}.
$$
The space $(W^{1,p(x)}_0(\Omega),\|\cdot\|)$ is a separable and
reflexive Banach space.
Moreover, if $0 <|\Omega|<\infty$ and $p_1$, $p_2$ are variable exponents
so that $p_1\leq p_2$ in $\Omega$ then there exists the continuous embedding
$W^{1,p_2(x)}_0(\Omega)\hookrightarrow W^{1,p_1(x)}_0(\Omega)$.
Set
\begin{equation}\label{rho2}
\varrho_{p(x)}(u)=\int_\Omega |\nabla u(x)|^{p(x)}\,dx.
\end{equation}
If $(u_n)$, $u\in W^{1,p(x)}_0(\Omega)$ then the following
properties are true:
\begin{gather}\label{M4}
\|u\|>1\;\Rightarrow\;\|u\|^{p^-}\leq \varrho_{p(x)}(u) \leq\|u\|^{p^+}\,, \\
\label{M5}
\|u\|<1\;\Rightarrow\;\|u\|^{p^+}\leq \varrho_{p(x)}(u) \leq\|u\|^{p^-}\,, \\
\label{M6}
\|u_n-u\|\to 0\;\Leftrightarrow\;\varrho_{p(x)} (u_n-u)\to 0\,.
\end{gather}
Set
$$
p^*(x)=\begin{cases}
\frac{Np(x)}{N-p(x)} &\text{if $p(x) 0$ such that for all $\mu<\mu^*$ problem \eqref{prob1}
has at least one solution.
\end{theorem}
The key ingredient in the proof of Theorem \ref{t2} is the Ekeland variational
principle, which asserts the existence of almost critical points of $\mathcal{E}$.
The subcritical framework of the problem yields a nontrivial critical point
of $\mathcal{E}$, hence a weak solution of problem \eqref{prob1}.
We point out that the Ekeland variational principle can be viewed as the
nonlinear version of the Bishop-Phelps theorem \cite{bis1}.
The arguments developed in this paper show that a similar result
holds if the $p(x)$-Laplace operator is replaced with other nonhomogeneous
differential operators with variable exponent, for instance the
\emph{generalized mean curvature operator} defined by
$$
\operatorname{div} \Big((1+|\nabla u|^2)^{[p(x)-2]/2}\nabla u \Big).
$$
\section{Proof of Theorem \ref{t1}}
The proof strongly relies on the mountain pass theorem in relationship
with some ideas developed in
\cite{mora} and \cite{pucci}.
We start with the verification of the geometric hypotheses of the mountain pass
theorem. We observe that $\mathcal{E}(0)=0$ and we show the existence of a mountain
near the origin, namely there exist positive numbers $r$ and $\eta$ such that
$\mathcal{E}(u)\geq\eta$ for all $u\in W^{1,p(x)}_0(\Omega)$ with $\|u\|=r$. We first observe that
the definition of $\lambda^*$ combined with the fact that $\lambda<\lambda^*$
imply that there exists $\delta>0$ such that
$$
\int_\Omega\frac{1}{p(x)}(|\nabla u|^{p(x)}-\lambda |u|^{p(x)})dx
\geq \delta |\nabla u|_{p(x)},\quad\text{for all}\ u\in W^{1,p(x)}_0(\Omega).
$$
But
$$
\int_\Omega\frac{|u|^{q(x)}}{q(x)}dx\leq\frac{1}{q^-}|u|_{q(x)},
\quad\text{for all } u\in W^{1,p(x)}_0(\Omega).
$$
Combining these inequalities, we deduce that
\begin{equation} \label{enough}
\mathcal{E}(u)\geq \delta\,|\nabla u|_{p(x)}-\frac{1}{q^-}\,|u|_{q(x)}\,.
\end{equation}
Fix $r\in(0,1)$ and $u\in W^{1,p(x)}_0(\Omega)$ with $\|u\|=r$.
Then relations \eqref{M5}, \eqref{enough} and the Sobolev embedding
$W^{1,p(x)}_0(\Omega)\hookrightarrow L^{q(x)}(\Omega)$ yield
$$
\mathcal{E}(u)\geq\delta\,\|u\|^{p^+}-\frac{C}{q^-}\,\|u\|^{q^-}.
$$
Choosing eventually $r\in(0,1)$ smaller if necessary, we conclude that
there exists $\eta>0$ such that $\mathcal{E}(u)\geq\eta$ for all $u\in W^{1,p(x)}_0(\Omega)$ with
$\|u\|=r$.
Next, we argue the existence of a valley over the chain of mountains.
For this purpose, we fix $s>1$ and $w\in W^{1,p(x)}_0(\Omega)\setminus\{0\}$. It follows that
\begin{equation} \label{abcc}
\begin{aligned}
\mathcal{E} (sw)
& =\int_\Omega \frac{s^{p(x)}}{p(x)}\left( |\nabla w|^{p(x)}
-\lambda |w|^{q(x)}\right)dx-\int_\Omega\frac{s^{q(x)}}{q(x)}\,|w|^{q(x)}dx\\
&\leq A \frac{s^{p^+}}{p^-}-B \frac{s^{q^-}}{q^+},
\end{aligned}
\end{equation}
where
$$
A=\int_\Omega \left( |\nabla w|^{p(x)}-\lambda |w|^{q(x)}\right)dx\quad\text{and}\quad
B=\int_\Omega |w|^{q(x)}dx.
$$
Using hypothesis \eqref{pq}, relation \eqref{abcc} yields $\mathcal{E} (sw)<0$ for $s$
large enough.
To apply Theorem \ref{th1} to our problem \eqref{prob} it remains to check
that the energy functional $\mathcal{E}$ satisfies the Palais-Smale compactness condition.
Let $(u_n)\subset W^{1,p(x)}_0(\Omega)$ be an arbitrary Palais-Smale sequence for $\mathcal{E}$, namely
\begin{equation}\label{ps1}
\mathcal{E} (u_n)=O(1)\quad\text{as } n\to\infty
\end{equation}
and
\begin{equation} \label{ps2}
\|\mathcal{E}' (u_n)\|_{W^{-1,p'(x)}(\Omega)}=o(1)\quad\text{as } n\to\infty\,.
\end{equation}
We claim that
\begin{equation} \label{claim}
\text{the sequence $(u_n)$ is bounded in } W^{1,p(x)}_0(\Omega) .
\end{equation}
Relations \eqref{ps1} and \eqref{ps2} yield
\begin{equation} \label{ps3}
\int_\Omega\frac{1}{p(x)}\Big( |\nabla u_n|^{p(x)}-\lambda |u_n|^{p(x)}\Big)dx
-\int_\Omega\frac{1}{q(x)} |u_n|^{q(x)}dx=O(1)\quad\text{as } n\to\infty
\end{equation}
and for all $v\in W^{1,p(x)}_0(\Omega)$,
\begin{equation} \label{ps4}
\begin{aligned}
&\int_\Omega\left( |\nabla u_n|^{p(x)-2}\nabla u_n\cdot\nabla v
-\lambda |u_n|^{p(x)-2}u_nv\right)dx-\int_\Omega |u_n|^{q(x)-2}u_nv\,dx \\
&= o(1)\|v\|\quad\text{as } n\to\infty\,.
\end{aligned}
\end{equation}
Choosing $v=u_n$ in \eqref{ps4} we deduce that
\begin{equation} \label{ps5}
\int_\Omega\Big( |\nabla u_n|^{p(x)}-\lambda |u_n|^{p(x)}\Big)dx
-\int_\Omega |u_n|^{q(x)}dx=o(1)\|u_n\|\quad\text{as } n\to\infty\,.
\end{equation}
On the other hand, relation \eqref{ps3} implies
\begin{align*}
& O(1)+\frac{1}{p^+}\int_\Omega\big( |\nabla u_n|^{p(x)}-\lambda |u_n|^{p(x)}\big)dx \\
&\leq \int_\Omega\frac{1}{q(x)}\, |u_n|^{q(x)}dx \\
&\leq O(1)+\frac{1}{p^-}\int_\Omega\big( |\nabla u_n|^{p(x)}
-\lambda |u_n|^{p(x)}\big)dx.
\end{align*}
Using now relation \eqref{ps5} we deduce that
\begin{align*}
& O(1)+o(1)\|u_n\|+\frac{1}{p^+}\int_\Omega |u_n|^{q(x)}dx \\
&\leq \int_\Omega\frac{1}{q(x)}\, |u_n|^{q(x)}dx \\
&\leq O(1)+o(1)\|u_n\|+\frac{1}{p^-}\int_\Omega |u_n|^{q(x)}dx.
\end{align*}
It follows that
\begin{equation} \label{ps6}
\int_\Omega |u_n|^{q(x)}dx=O(1)+o(1)\|u_n\|\quad\text{as } n\to\infty\,.
\end{equation}
Returning to \eqref{ps3} and using relation \eqref{ps6} we deduce that
\begin{equation} \label{ps7}
\int_\Omega\frac{1}{p(x)}\big( |\nabla u_n|^{p(x)}-\lambda |u_n|^{p(x)}\big)dx
=O(1)+o(1)\|u_n\|\,.
\end{equation}
Taking into account the definition of $\lambda^*$ and the fact that
$\lambda<\lambda^*$, relation \eqref{ps7} implies that $(u_n)$ is bounded in
$W^{1,p(x)}_0(\Omega)$, hence Claim \eqref{claim} is argued. So, up to a subsequence
\begin{gather} \label{cow}
u_n\rightharpoonup u\quad\text{in } W^{1,p(x)}_0(\Omega), \\
\label{col}
u_n\to u\quad\text{in}\ L^{p(x)}(\Omega).
\end{gather}
We prove in what follows that
\begin{equation} \label{cra0}(u_n)
\text{ contains a strongly convergent subsequence in $W^{1,p(x)}_0(\Omega)$}.
\end{equation}
We first observe that relation \eqref{ps4} yields
\begin{equation} \label{cra1}
\int_\Omega |\nabla u_n|^{p(x)-2}\nabla u_n\cdot\nabla v\,dx
=\int_\Omega \varrho(x,u_n)v\,dx
+ o(1)\|v\|\quad\text{as}\ n\to\infty\,,
\end{equation}
for all $v\in W^{1,p(x)}_0(\Omega)$ where $\varrho (x,w)=\lambda |w|^{p(x)-2}w+|w|^{q(x)-2}w$.
We assume in what follows that $p^+ 0\quad\text{and}\quad
B:=\int_{\omega} \phi^{q(x)}dx>0.
$$
We deduce that $\mathcal{J}(t\phi)<0$, provided that $t\in(0,1)$ is small enough.
\end{proof}
Fix $\mu^*>0$ as established in Lemma \ref{lema1} and let $\mu<\mu^*$.
Using Lemmata \ref{lema1} and \ref{lema2} we deduce that there exists $r>0$
such that
$$
\inf_{u\in\overline{B_r(0)}}\mathcal{J} (u)<0<\inf_{u\in\partial B_r(0)}\mathcal{J} (u),
$$
where $B_r(0):=\{u\in W^{1,p(x)}_0(\Omega); \|u\|\rho>0$
$$
\max\{\varphi(u_0),\varphi(u_1)\}<\inf[\varphi(u):\|u-u_0\|=\rho]=m_{\rho}
$$
and
$$c=\inf_{\gamma\in\Gamma}\max_{0\leq t\leq 1}\varphi(\gamma(t))\quad
\text{with}\quad \Gamma=\{\gamma\in C([0,1],X):\gamma(0)=u_0,\gamma(1)=u_1\}.
$$
Then $c\geq m_{\rho}$ and $c$ is a critical value of $\varphi$.
\end{theorem}
As pointed out by Brezis and Browder \cite{brebro}, the mountain pass theorem
``extends ideas already present in Poincar\'e and Birkhoff".
More generally, this result is in fact true in Banach-Finsler manifolds.
Assumption \eqref{pq} guarantees that the energy functional associated with
\eqref{prob} has a mountain pass geometry. We study in what follows a related
perturbed problem, provided that Theorem \ref{th1} cannot be applied.
Consider the problem
\begin{equation}\label{prob1}
\begin{gathered}
-\Delta_{p(x)} u=\lambda |u|^{p(x)-2}u+\mu |u|^{q(x)-2}u, \quad \text{in } \Omega\\
u=0, \quad \text{on } \partial\Omega.
\end{gathered}
\end{equation}
We say that $u$ is a \emph{weak solution} of problem \eqref{prob1} if
$u\in W^{1,p(x)}_0(\Omega)\setminus\{0\}$ and
$$
\int_\Omega \big( |\nabla u|^{p(x)-2}\nabla u\cdot\nabla v-\lambda|u|^{p(x)-2} u v \big)dx
-\mu\int_\Omega |u|^{q(x)-2}uv\,dx=0,
$$
for all $v\in W^{1,p(x)}_0(\Omega)$.
We assume that $p,q\in C_+(\overline\Omega)$ satisfy
\begin{equation} \label{pq1}
q^-