\documentclass[reqno]{amsart} \usepackage{hyperref} \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 $1
0$.
\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
$-\Delta_{p}\underline{u}0$ 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}