\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 97, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/97\hfil A subsolution-supersolution method]
{A subsolution-supersolution method for quasilinear systems}
\author[D. A. Kandilakis, M. Magiropoulos\hfil EJDE-2005/97\hfilneg]
{Dimitrios A. Kandilakis, Manolis Magiropoulos}

\address{Dimitrios A. Kandilakis \hfill\break Department of Sciences,
Technical University of Crete, 73100 Chania, Greece}
\email{dkan@science.tuc.gr}

\address{Manolis Magiropoulos \hfill\break 
Science Department, Technological
and Educational Institute of Crete, 71500 Heraklion, Greece}
\email{mageir@stef.teiher.gr}

\date{}
\thanks{Submitted April 18, 2005. Published September 4, 2005.}
\subjclass{35B38, 35D05, 35J50}
\keywords{Quasilinear System; supersolution; subsolution}

\begin{abstract}
 Assuming that a system of quasilinear equations of gradient type 
 admits a strict supersolution and a strict subsolution,
 we show that it also admits a positive solution.
\end{abstract}

\maketitle

\numberwithin{equation}{section} 
\newtheorem{theorem}{Theorem}[section] 
\newtheorem{remark}[theorem]{Remark} 
\newtheorem{corollary}[theorem]{Corollary}

\section{introduction}

Consider the quasilinear elliptic system 
\begin{equation}
\begin{gathered} -\Delta_{p}u=H_{u}(x,u,v)\quad \text{in }\Omega\\
-\Delta_{q}v=H_{v}(x,u,v)\quad \text{in }\Omega\\ u=v=0 \quad \text{on
}\partial\Omega, \end{gathered}  \label{sys}
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, $N\geq2$, with
boundary of class $C^{2}$, $\Delta_{p}$ and $\Delta_{q}$ are the $p-$ and $q-
$Laplace operators with $1<p,q<N$, and 
$H:\Omega\times\mathbb{R}\times\mathbb{R\to R}$ is a $C^{1}$ function.

The solvability of system \eqref{sys} has been extensively studied by
various methods, fibering \cite{boz-mit}, bifurcation \cite{dra-stav-zogr},
via the mountain pass theorem \cite{bo-fi} etc. We use the super- and sub-
solution method by assuming that \eqref{sys} admits a strict supersolution
and a strict subsolution and construct two sequences of approximate
solutions whose limit is shown to be a solution of the system. The same
approach can also be applied to nonvariational and Hamiltonian systems. It
is worth mentioning that, as far as \eqref{sys} is concerned, the classical
super- and sub- solution method is not directly applicable because the
``restriction" of the function $H(x,.,.)$ between the super- and sub-
solution is not necessarily differentiable.

We make the following assumptions:

\begin{itemize}
\item[(H1)]  $s\mapsto H_{u}(x,s,t)$ and $s\mapsto H_{v}(x,s,t)$ are
nondecreasing for a.e. $x\in\Omega$ and every $t>0$.

\item[(H2)] $t\mapsto H_{u}(x,s,t)$ and $t\mapsto H_{v}(x,s,t)$ are
nondecreasing for a.e. $x\in\Omega$ and every $s>0$.

\item[(H3)] $H_{u}(x,0,t)=H_{v}(x,s,0)=0$ for a.e. $x\in\Omega$ and every 
$s,t>0$.

\item[(H4)] There exists $C>0$ such that $|H_{u}(x,s,t)|\leq
C(1+|s|^{p^{\ast}-1}+|t|^{\frac{q^{\ast}(p^{\ast }-1)}{p^{\ast}}})$ and 
$|H_{v}(x,s,t)|\leq C(1+|s|^{\frac{p^{\ast}(q^{\ast}-1)}{q^{\ast}}
}+|t|^{q^{\ast}-1})$ a.e. in $\Omega$, where $p^{\ast}:=\frac{Np}{N-p}$ 
and $q^{\ast}:=\frac{Nq}{N-q}$ are the critical Sobolev exponents.
\end{itemize}

Note that if (H1)--(H4) are satisfied then 
\begin{equation*}
|H(x,s,t)|\leq c(1+|s|^{p^{\ast}}+|t|^{q^{\ast}})\text{ a.e.in }\Omega,
\end{equation*}
for some $c>0$.

Let $E=W_{0}^{1,p}(\Omega )\times W_{0}^{1,q}(\Omega )$. The energy
functional $\Phi :E\rightarrow \mathbb{R}$ associated to \eqref{sys} is
\begin{equation*}
\Phi (u,v)=\frac{1}{p}\int_{\Omega }\left\vert \nabla u\right\vert ^{p}+
\frac{1}{q}\int_{\Omega }\left\vert \nabla v\right\vert ^{q}-\int_{\Omega
}H(x,u(x),v(x))dx.
\end{equation*}
It is clear that if (H1)--(H4) are satisfied, then $\Phi $ is a 
$C^{1}$-functional whose critical points are solutions to \eqref{sys}.

\subsection*{Definition}

A pair of nonnegative functions $(\overline{u},\overline{v})\in C^{1} 
(\overline{\Omega})\times C^{1}(\overline{\Omega})$ is said to be a strict
supersolution for \eqref{sys} if 
$-\Delta_{p}\overline{u}>H_{u}(x,\overline {u},\overline{v})$ and 
$-\Delta_{q}\overline{v}>H_{v}(x,\overline{u} ,\overline{v})$ in $\Omega$. 
A pair of nonnegative functions $(\underline {u},
\underline{v})$ is said to be a strict subsolution if 
$-\Delta_{p}\underline{u}<H_{u}(x,\underline{u},\underline{v})$ and 
$-\Delta_{q}\underline{v}<H_{v}(x,\underline{u},\underline{v})$ a.e. in 
$\Omega$.

\begin{theorem} \label{T}
Assume that hypotheses (H1)--(H4) hold and \eqref{sys} admits a strict
supersolution $(\overline{u},\overline{v})$  and a strict subsolution
$(\underline{u},\underline{v})$ with $\underline{u}<\overline{u}$ and
$\underline{v}<\overline{v}$ in $\Omega$. Then \eqref{sys} has a
solution $(u_{0},v_{0})$ with $u_{0},v_{0}>0$ in $\Omega$.
\end{theorem}

\begin{proof}
For a function $F:\Omega\times\mathbb{R}\times\mathbb{R}\to \mathbb{R}$, we
define 
\begin{equation*}
\widehat{F}(x,s,t)= 
\begin{cases}
F(x,s,t) & \text{if }\underline{u}(x)\leq s\leq\overline{u}(x),\;
\underline {v}(x)\leq t\leq\overline{v}(x), \\ 
F(x,\underline{u}(x),t) & \text{if }s<\underline{u}(x),\; \underline{v}
(x)\leq t\leq\overline {v}(x), \\ 
F(x,s,\underline{v}(x)) & \text{if }\underline{u}(x)\leq s\leq\overline{u}
(x), \; t<\underline {v}(x), \\ 
F(x,\underline{u}(x),\underline{v}(x)) & \text{if }s<\underline{u}(x),\; 
t<\underline{v}(x), \\ 
F(x,\overline{u}(x),t) & \text{if }\overline{u}(x)<s,\; \underline{v}(x)\leq
t\leq\overline {v}(x), \\ 
F(x,\overline{u}(x),\overline{v}(x)) & \text{if }\overline{u}(x)<s,\; 
\overline{v}(x)<t, \\ 
F(x,\underline{u}(x),\overline{v}(x)) & \text{if }s<\underline{u}(x),\; 
\overline{v}(x)<t, \\ 
F(x,\overline{u}(x),\underline{v}(x)) & \text{if }\overline{u}(x)<s,\; t<
\underline{v}(x), \\ 
F(x,s,\overline{v}(x)) & \text{if }\underline{u}(x)\leq s\leq\overline{u}
(x),\; \overline{v}(x)<t.
\end{cases}
\end{equation*}
We will construct two sequences $u_{n}\in W_{0}^{1,p}(\Omega)$ and $v_{n}\in
W_{0}^{1,q}(\Omega)$, $n\in\mathbb{N}$, as follows: consider the problem 
\begin{equation}
\begin{gathered} -\Delta_{p}u=\widehat{H}_{u}(x,u,\overline{v})\quad
\text{in }\Omega\\ u=0 \quad \text{on }\partial\Omega. \end{gathered}
\label{eq1}
\end{equation}
The Euler-Lagrange functional associated with the above system is 
\begin{equation*}
\widehat{\Phi}(u)=\frac{1}{p}\int_{\Omega}\vert \nabla u\vert
^{p}-\int_{\Omega}\int_{0}^{u}\widehat{H}_{u}(x,s,\overline{v})ds\,dx 
\end{equation*}
which is bounded from below, weakly lower semicontinuous and coercive in 
$W_{0}^{1,p}(\Omega)$. Therefore, the infimum of $\widehat{\Phi}(.)$ is
achieved at some point 
$u_{1}\in W_{0}^{1,p}(\Omega)\cap C^{1}(\overline {\Omega})$ which is a 
solution of (\ref{eq1}). We claim that 
$\underline {u}(x)\leq u_{1}(x)\leq\overline{u}(x)$ for every $x\in\Omega$. 
Indeed, assume
that the set 
\begin{equation*}
\widetilde{\Omega}:=\{x\in\Omega:u_{1}(x)<\underline{u}(x)\} 
\end{equation*}
is nonempty. Since it is open, it must have positive measure and 
\begin{equation}
-\Delta_{p}u_{1}=H_{u}(x,\underline{u},\overline{v})\quad \text{in }
\widetilde {\Omega},  \label{a}
\end{equation}
while,
\begin{equation}
-\Delta_{p}\underline{u}<H_{u}(x,\underline{u},\overline{v})\quad \text{in }
\widetilde{\Omega}.  \label{b}
\end{equation}
Multiplying (\ref{a}) and (\ref{b}) with $\underline{u}-u_{1}$ and
integrating over $\widetilde{\Omega}$, we get 
\begin{equation*}
\int_{\Omega}\left\vert \nabla u_{0}\right\vert ^{p-2}\nabla u_{1}\nabla(
\underline{u}-u_{1})=\int_{\Omega}H_{u}(x,\underline{u},\overline {v})(
\underline{u}-u_{1}), 
\end{equation*}
and 
\begin{equation*}
\int_{\Omega}\left\vert \nabla\underline{u}\right\vert ^{p-2}\nabla 
\underline{u}\nabla(\underline{u}-u_{1})=\int_{\Omega}H_{u}(x,\underline {u},
\overline{v})(\underline{u}-u_{1}), 
\end{equation*}
which combined yield 
\begin{equation*}
\int_{\Omega}\left[ \left\vert \nabla\underline{u}\right\vert ^{p-2} \nabla
\underline{u}-\left\vert \nabla u_{1}\right\vert ^{p-2}\nabla u_{1}\right]
\nabla(\underline{u}-u_{1})<0, 
\end{equation*}
contradicting the strong monotonicity of the $-\Delta_{p}$ operator. Thus 
$\widetilde{\Omega}$ is empty. Similarly, $u_{1}(x)\leq\overline{u}(x)$ for
every $x\in\Omega$.

Consider the problem 
\begin{equation}
\begin{gathered} -\Delta_{q}v=\widehat{H}_{v}(x,u_{1},v)\quad \text{in
}\Omega\\ v=0 \quad \text{on }\partial\Omega. \end{gathered}   \label{eq2}
\end{equation}
Working as in (\ref{eq1}) we can show that it admits a solution $v_{1}\in
W_{0}^{1,p}(\Omega)\cap C^{1}(\overline{\Omega})$ with $\underline{v}(x)\leq
v_{1}(x)\leq\overline{v}(x)$. Assuming now that $u_{n}\in W_{0}^{1,p}(\Omega)
$ and $v_{n}\in W_{0}^{1,q}(\Omega)$, $n=1,\dots k-1$, have been defined, we
let $u_{k}\in W_{0}^{1,p}(\Omega)$ be a solution of (\ref{eq1}) with $v_{k-1}
$ in the place of $\overline{v}$ and $v_{k}\in W_{0}^{1,p}(\Omega)$ be a
solution of (\ref{eq2}) with $u_{k}$ in the place of $u_{1}$. Since 
$\widehat{H}_{u}(x,s,t)$ and $\widehat{H}_{v}(x,s,t)$ are bounded, the
sequences $u_{n}\in W_{0}^{1,p}(\Omega)$ and $v_{n}\in W_{0}^{1,q}(\Omega)$, 
$n\in\mathbb{N}$, are also bounded, so $u_{n}\to u_{0}$ weakly in 
$W_{0}^{1,p}(\Omega)$ and $v_{n}\to v_{0}$ weakly in $W_{0}^{1,q}(\Omega)$.
Exploiting the continuity of $H_{u}(x,.,.)$ and $H_{v}(x,.,.)$ and the
Sobolev embedding we easily deduce that $(u_{0},v_{0})$ is a solution of the
system 
\begin{equation*}
\begin{gathered}
-\Delta_{p}u=\widehat{H}_{u}(x,u,v)\quad \text{in }\Omega\\
-\Delta_{q}v=\widehat{H}_{v}(x,u,v)\quad \text{in }\Omega\\
u=v=0\quad \text{on }\partial\Omega,
\end{gathered}
\end{equation*}
while $\underline{u}(x)\leq u_{0}(x)\leq\overline{u}(x)$, $\underline {v}
(x)\leq v_{0}(x)\leq\overline{v}(x)$ for every $x\in\Omega$. Thus 
\begin{equation*}
\widehat{H}_{u}(x,u_{0},v_{0})=H_{u}(x,u_{0},v_{0}),\quad \widehat{H}
_{v}(x,u_{0},v_{0})=H_{v}(x,u_{0},v_{0}). 
\end{equation*}
Consequently, $(u_{0},v_{0})$ is a critical point of $\Phi(.,.)$ and
therefore a solution of \eqref{sys}. On account of (H1)(i), we have 
\begin{equation*}
-\Delta_{p}\underline{u}<H_{u}(x,\underline{u},\underline{v})\leq
H_{u}(x,u_{0},v_{0})=-\Delta_{p}u_{0}\quad \text{in }\Omega, 
\end{equation*}
and so, by the strong comparison principle in \cite[Proposition 2.2]{Gue-Ver}
, we deduce that 
\begin{equation*}
0\leq\underline{u}<u_{0}\text{ in }\Omega. 
\end{equation*}
Similarly, $v_{0}>0$ in $\Omega$.
\end{proof}

\begin{remark} \label{rmk4} \rm
In the case of a single equation, the existence of a solution is established
by minimizing (locally) the energy functional. By making use of the fact that
this solution is a minimizer, an application of the mountain pass principle
provides a second solution \cite[3]{A-B-C}. However, in our case it is not
clear that the solution $(u_{0},v_{0})$ provided by the previous Theorem is a
(local) minimizer of $\Phi(.,.)$.
\end{remark}

Let $\lambda_{1}$ denote the principal eigenvalue of the $p-$Laplace
operator and $\mu_{1}$ the principal eigenvalue of the $q-$Laplace operator
in $\Omega$.

\begin{corollary} \label{Cor}
Assume that hypotheses (H1)--(H4) hold. Then \eqref{sys} admits a
 strict supersolution $(\overline{u},\overline{v})$ and
\begin{equation}
\lim_{s\to 0^{+}}\frac{H_{u}(x,s,t)}{s^{p-1}}>\lambda
_{1}, \quad \lim_{t\to 0^{+}}\frac{H_{v}(x,s,t)}
{t^{q-1}}>\mu_{1} \label{sec}
\end{equation}
for a.e. $x\in\Omega$ and $s,t>0$. \textit{Then }\eqref{sys} has a solution
$(u_{0},v_{0})$ with $u_{0},v_{0}>0$ in $\Omega$.
\end{corollary}

\begin{proof}
Let $\varphi_{1}>0$ be an eigenfunction corresponding to $\lambda_{1}$ and 
$\psi_{1}>0$ an eigenfunction corresponding to $\mu_{1}$. In view of 
\eqref{sec} there exists $\varepsilon>0$ such that $(\varepsilon\varphi
_{1},\varepsilon\psi_{1})$ is a strict subsolution of \eqref{sys}.
Furthermore, as a consequence of the maximum principle \cite{Vas}, by taking 
$\varepsilon$ sufficiently small we have that 
$\varepsilon\varphi _{1}<\overline{u}$ and $\varepsilon\psi_{1}<\overline{v}$ 
in $\Omega$.
Theorem \ref{T} implies that \eqref{sys} has a solution $(u_{0},v_{0})$ with 
$u_{0},v_{0}>0$ in $\Omega$.
\end{proof}

We now present a simple (academic) example. Assume that $H(.,.,.)$ is a 
$C^{1}$ function satisfying (H1)--(H3) and 
\begin{equation*}
H_{u}(x,\xi s,\xi t)=\xi^{\alpha}H_{u}(x,s,t), \quad H_{v}(x,\xi s,\xi
t)=\xi^{\alpha}H_{v}(x,s,t) 
\end{equation*}
for some $\alpha \in[ 1,\min\{p-1,q-1\}] $ and every $s,t,\xi>0$. Then $H$
satisfies (H4) since 
\begin{align*}
H_{u}(s,t) &=H_{u}(\sqrt{s^{2}+t^{2}}\frac{s}{\sqrt{s^{2}+t^{2}}},\sqrt {
s^{2}+t^{2}}\frac{t}{\sqrt{s^{2}+t^{2}}}) \\
&=( s^{2}+t^{2}) ^{\frac{\alpha}{2}}H_{u}(\frac{s}{\sqrt {s^{2}+t^{2}}},
\frac{t}{\sqrt{s^{2}+t^{2}}})\leq M(s^{2}+t^{2}) ^{\frac{\alpha}{2}} \\
&\leq C_{1}(1+s^{\alpha}+t^{\alpha}),
\end{align*}
for some $C_{1}>0$, where $M=\sup\{H_{u}(s,t):s^{2}+t^{2}=1\}$. Similarly, 
$H_{u}(s,t)\leq C_{2}(1+s^{\alpha}+t^{\alpha})$ for some $C_{2}>0$.

If $\widehat{u}$, $\widehat{v}$ are the solutions of 
\begin{gather*}
-\Delta_{p}u=1 \quad \text{in }\Omega \\
-\Delta_{q}v=1 \quad \text{in }\Omega \\
u=v=0 \quad \text{on }\partial\Omega,
\end{gather*}
then there exists $\zeta>0$ such that $(\overline{u},\overline{v}):=(\zeta
\widehat{u},\zeta\widehat{v})$ is a strict supersolution of \eqref{sys}.
Indeed, if 
\begin{equation*}
M=\sup_{x\in\Omega} \big\{H_{u}(x,\widehat{u}(x),\widehat{v} (x)),H_{v}(x,
\widehat{u}(x),\widehat{v}(x))\big\}, 
\end{equation*}
then for $\zeta>\max\{M^{1/(1-p-\alpha)},M^{1/(1-q-\alpha)}\}$ we have 
\begin{equation*}
-\Delta_{p}(\zeta\widehat{u})=\zeta^{p-1}>M\zeta^{\alpha}\geq\zeta^{\alpha
}H_{u}(x,\widehat{u},\widehat{v})=H_{u}(x,\zeta\widehat{u},\zeta\widehat{v}
). 
\end{equation*}
Similarly, $-\Delta_{q}(\zeta\widehat{v})>H_{v}(x,\zeta\widehat{u} ,\zeta
\widehat{v})$. On the other hand, \eqref{sec} is satisfied because 
$\alpha<\min\{p-1,q-1\}$. By Corollary \ref{Cor}, \eqref{sys} admits a
positive solution.

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\end{document}