\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 02, pp. 1--11.\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 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/02\hfil Nonexistence results for semilinear systems]
{Nonexistence results for semilinear systems in unbounded domains}

\author[B. Khodja, A. Moussaoui \hfil EJDE-2009/02\hfilneg]
{Brahim Khodja, Abdelkrim Moussaoui}  % in alphabetical order

\address{Brahim Khodja \newline
Department of mathematics, Badji Mokhtar University,
B.P. 12 Annaba, Algeria}
\email{bmkhodja@yahoo.fr}

\address{Abdelkrim Moussaoui \newline
Department of mathematics, Bejaia University, Targa
Ouzemour Bejaia, Algeria}
 \email{remdz@yahoo.fr}

\thanks{Submitted April 10, 2008. Published January 2, 2009.}
\subjclass[2000]{35J45, 35J55}
\keywords{Semi linear systems; Pohozaev identity; trivial solution;
\hfill\break\indent Robin boundary condition}

\begin{abstract}
 This paper concerns the non-existence of nontrivial solutions for the
 semi-linear system of gradient type
\begin{equation*}
\lambda \frac{\partial ^{2}u_{k}}{\partial t^{2}}
-\sum_{i=1}^n \frac{\partial }{\partial x_{i}}(p_{i}(x)\frac{
\partial u_{k}}{\partial x_{i}})+f_{k}(x,u_{1},\dots ,u_{m})
=0\quad \text{in }\Omega ,\; k=1,\dots ,m
\end{equation*}
with Dirichlet, Neumann or Robin boundary conditions. The functions $f_{k}:
\mathcal{D}\times
\mathbb{R}^{m}\to \mathbb{R}$
$(k=1,\dots ,m)$ are locally Lipschitz continuous and satisfy
\begin{equation*}
2H(x,u_{1},\dots ,u_{m})-\sum_{k=1}^m
u_{k}f_{k}(x,u_{1},\dots ,u_{m})\geq 0\quad (\text{resp.}\leq 0)
\end{equation*}
for $\lambda >0$ (resp. $\lambda <0$). We establish the non-existence of
nontrivial solutions using Pohozaev-type identities.
Here $u_{1},\dots ,u_{m}$ are in $H^{2}(\Omega )\cap L^{\infty }(\Omega )$,
$\Omega =\mathbb{R}\times \mathcal{D}$ with
$\mathcal{D}=\prod_{i=1}^n  (\alpha _{i},\beta _{i})$ and
$H\in \mathcal{C}^{1}(
\overline{\mathcal{D}}\times \mathbb{R}^{m})$ such that
$\frac{\partial H}{\partial u_{k}}=f_{k}$,
$k=1,\dots ,m $.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

In this paper we study the  semi-linear system
\begin{equation}
\begin{gathered}
\lambda \frac{\partial ^{2}u_{1}}{\partial t^{2}}-\underset{i=1}{\overset{n}{
\sum }}\frac{\partial }{\partial x_{i}}(p_{i}(x)\frac{
\partial u_{1}}{\partial x_{i}})+f_{1}(x,u_{1},\dots ,u_{m})
=0\quad\text{in }\Omega , \\
\lambda \frac{\partial ^{2}u_{2}}{\partial t^{2}}-\underset{i=1}{\overset{n}{
\sum }}\frac{\partial }{\partial x_{i}}(p_{i}(x)\frac{
\partial u_{2}}{\partial x_{i}})+f_{2}(x,u_{1},\dots ,u_{m})
=0\quad\text{in }\Omega , \\
\dots  \\
\lambda \frac{\partial ^{2}u_{m}}{\partial t^{2}}-\underset{i=1}{\overset{n}{
\sum }}\frac{\partial }{\partial x_{i}}(p_{i}(x)\frac{
\partial u_{m}}{\partial x_{i}})+f_{m}(x,u_{1},\dots ,u_{m})
=0\quad\text{in }\Omega ,
\end{gathered}  \label{ep1}
\end{equation}
under Dirichlet, Neumann or Robin boundary conditions. Here
$\Omega =\mathbb{R}\times \mathcal{D}$ where
$\mathcal{D}=\prod_{i=1}^n (\alpha _{i},\beta _{i})$,
$\lambda $ is a real parameter,
$f_{k}:\mathcal{D}\times \mathbb{R}^{m}\to \mathbb{R}$
$(k=1,\dots ,m)$ are locally Lipschitz continuous functions such
that
\begin{equation*}
f_{k}(x,0,\dots ,0)=0\quad \text{in }\mathcal{D},
\end{equation*}
so that $(u_{1},\dots ,u_{m})=0$ is a solution of
 \eqref{ep1} and $p_{i}:\overline{\mathcal{D}}\to \mathbb{R}$
 $(i=1,\dots ,n)$ are continuous functions satisfying
\begin{equation*}
p_{i}(x)>0\text{ or }p_{i}(x)<0\quad \text{in }\mathcal{D}.
\end{equation*}

We assume that  system \eqref{ep1} is of the gradient type;
that is, there is a real-valued differentiable function
$H(x,u_{1},\dots ,u_{m})$ such that
\begin{equation*}
\frac{\partial H}{\partial u_{k}}=f_{k},\quad
H(x,0,\dots ,0) =0\quad \text{for }x\in \mathcal{D}.
\end{equation*}
For $k=1,\dots ,m$, $u_{k}$ are in
$H^{2}(\Omega )\cap L^{\infty}(\Omega )$ and satisfy
\begin{equation} \label{ed}
u_{k}(t,s)=0,\quad (t,s)\in \mathbb{R} \times \partial \mathcal{D}
\end{equation}
(Dirichlet boundary condition), or
\begin{equation} \label{en}
\frac{\partial u_{k}(t,s)}{\partial n}=0,\quad (t,s)\in \mathbb{R}
\times \partial \mathcal{D}
\end{equation}
(Neumann boundary condition), or
\begin{equation} \label{er}
(u_{k}+\varepsilon \frac{\partial u_{k}}{\partial n})(
t,s)=0,\quad (t,s)\in \mathbb{R} \times \partial \mathcal{D}
\end{equation}
(Robin boundary condition),
where $\varepsilon $ is a positive real number. Throughout this paper we
denote the boundary of $\Omega $ by
\begin{equation*}
\partial \Omega =\mathbb{R}\times \partial \mathcal{D}
=\Gamma _{\alpha _{1}}\cup \Gamma _{\beta_{1}}\cup \Gamma _{\alpha _{2}}
\cup \Gamma _{\beta _{2}}\dots \cup \Gamma _{\alpha _{n}}\cup
\Gamma _{\beta _{n}},
\end{equation*}
where
\[
\Gamma _{\mu _{s}}=\left\{ (t,x_{1},\dots ,x_{s-1},\mu
_{s},x_{s+1},\dots  x_{n}),\; t\in
\mathbb{R},\quad 1\leq s\leq n\right\},
\]
$(t,x)=(t,x_{1},\dots ,x_{n})$, and
$$
n(t,s)=(0,n_{1}(t,s),n_{2}(t,s) ,\dots ,n_{n}(t,s))
$$
is the outward normal to $\partial \Omega $ at the point $(t,s)$.
If $x\in \prod_{i=1}^n (\alpha _{i},\beta
_{i})$, $l=1,2,\dots ,n$ and $\tau \in \left\{ \alpha _{1},\beta
_{1},\alpha _{2},\beta _{2},\dots ,\alpha _{n},\beta _{n}\right\} $
one writes
\begin{equation*}
x_{l}^{\tau }=(x_{1},\dots ,x_{l-1},\tau ,x_{l+1},\dots ,x_{n}),
\quad
dx_{l}^{\ast }=dx_{1}\dots dx_{l-1}dx_{l+1}\dots dx_{n}
\end{equation*}
and
\begin{align*}
&\int_{\alpha _{1}}^{\beta _{1}}\dots \int_{\alpha _{i-1}}^{\beta
_{i-1}}\int_{\alpha _{i+1}}^{\beta _{i+1}}\dots \int_{\alpha
_{n}}^{\beta _{n}}f_{k}(x,r_{1},\dots ,r_{m})dx_{1}\dots dx_{i-1}
dx_{i+1}\dots dx_{n}
\\
&=\int_{\mathcal{D}_{i}^{\ast }}f_{k}(x,r_{1},\dots ,r_{m})dx_{i}^{\ast }
\quad \text{for all }k=1,\dots ,m.
\end{align*}

The question of non-existence of nontrivial solutions for elliptic problems
has been studied extensively in both bounded and unbounded domain
(see \cite{3},\cite{4},\cite{7}-\cite{8} and their references).
In particular, Amster et al. in \cite{1} showed the non-solvability
of the gradient elliptic system
\begin{gather*}
-\Delta u_{i}=g_{i}(u)\quad\text{in }\Omega , \\
u_{i}=0\quad \text{on }\partial \Omega ,\; i=1,\dots ,n,
\end{gather*}
where $\Omega $ is a starshaped domain. A similar result was given for
Hamiltonian systems by N. M. Chuong and T. D. Ke \cite{2} in $k$-starshaped
domain and by Khodja \cite{6} in unbounded domain
$\mathbb{R}^{+}\times\mathbb{R}$.

In the scalar case, when $\Omega $ is an unbounded domain, Haraux and Khodja
\cite{4} established that under assumptions
\begin{gather*}
f(0)=0, \\
2F(u)-uf(u)\leq 0, \quad u\neq 0
\end{gather*}
$(F(u)=\int_{0}^{u}f(s)ds)$, the problem
\begin{gather*}
-\Delta u+f(u)=0\quad \text{in }\Omega , \\
(u\text{ or }\frac{\partial u}{\partial n})=0\quad \text{on }
\partial \Omega ,
\end{gather*}
has only a trivial solution in
$H^{2}(\Omega )\cap L^{\infty}(\Omega )$, where
$\Omega =J\times \omega $, $J\subset\mathbb{R}$ is an unbounded
interval and $\omega $ a domain in $\mathbb{R}^{N}$. The case of
Robin boundary conditions was treated by Khodja \cite{5}
and it was shown nonexistence results for the equation
\begin{equation*}
\lambda \frac{\partial ^{2}u}{\partial t^{2}}
-\frac{\partial }{\partial x}\big(p(x,y)\frac{\partial u}{\partial x}\big)
-\frac{ \partial }{\partial y}\big(q(x,y)\frac{\partial u}{\partial y}
\big)+f(x,y,u)=0\quad\text{in }\Omega ,
\end{equation*}
where $\Omega =\mathbb{R}\times ] \alpha _{1},\beta _{1}
[ \times ] \alpha _{2},\beta_{2}[ $.
In the above works, the integral identity of Pohozaev was
adapted for each problem treated and applied to obtain the non-existence
results. The present study extends and complements these works. We shall
prove the non-solvability results to the class of semi-linear system of
gradient type \eqref{ep1} under Dirichlet, Neumann or Robin
boundary conditions. By using a Pohozaev-type identity, our demonstration
strategy will be to show that the  function
\begin{equation*}
\mathcal{E}(t)=\int_{\mathcal{D}}
\big(\sum_{k=1}^m \vert u_{k}(t,x)\vert ^{2}\big)dx
\end{equation*}
is convex in $\mathbb{R}$, and then, from the Maximum Principle,
we obtain that any solution $(u_{1},\dots ,u_{m})$ to the problems
\eqref{ep1}-\eqref{ed}, \eqref{ep1}-\eqref{en} and
\eqref{ep1}-\eqref{er} is trivial. We draw the attention
of the reader to the use of the Pohozaev-type identity which, to the best of
our knowledge, was not explored before in connection with gradient systems
in an unbounded cylindrical-type domain.

This paper is organized as follows. In the next section, we give a
Pohozaev-type identity adapted to the systems with Dirichlet, Neumann and
Robin boundary conditions; section $3$ gives our main results and some
examples will be illustrated in section 4.

\section{Integral identities}

The proof of our main results which will appear in the next section
use the following type of Pohozaev identity, adapted for systems.

\begin{theorem} \label{thm1}
Let $u_{1},\dots ,u_{m}$ in $H^{2}(\Omega )\cap L^{\infty }(\Omega )$
be a solution of problem \eqref{ep1}--\eqref{er}. Then for each
$t\in\mathbb{R}$ and $\varepsilon >0$, we have
\begin{equation}
\begin{aligned}
&\int_{\mathcal{D}}\Big[ \frac{\lambda }{2}\sum_{k=1}^m
\vert \frac{\partial u_{k}}{\partial t}\vert ^{2}
+\sum_{i=1}^n \frac{p_{i}(x)}{2}\Big(\sum_{k=1}^m
\vert \frac{\partial u_{k}}{\partial x_{i}}
\vert ^{2}\Big)+H(x,u_{1},\dots ,u_{m})\Big] dx
\\
& +\frac{1}{2\varepsilon }\sum_{i=1}^n \int_{\mathcal{D}_{i}^{\ast }}
\Big[ p_{i}(x_{i}^{\beta _{i}})\big(\sum_{k=1}^m | u_{k}| ^{2}\big)
(t,x_{i}^{\beta _{i}})
  +p_{i}(x_{i}^{\alpha _{i}})\big(\sum_{k=1}^m
  | u_{k}| ^{2}\big)(t,x_{i}^{\alpha _{i}})\Big] dx_{i}^{\ast }=0.
\end{aligned}\label{e22}
\end{equation}
\end{theorem}

\begin{proof}
For $t\in\mathbb{R}$ we consider a function
\begin{equation*}
\mathcal{K}(t)=\int_{\mathcal{D}}\big[ \frac{\lambda }{2}
\sum_{k=1}^m | \frac{\partial u_{k}}{\partial t
}| ^{2}+\sum_{i=1}^n \frac{p_{i}(x)}{2}(\sum_{k=1}^m |
\frac{\partial u_{k}}{\partial x_{i}}| ^{2})+H(x,u_{1},\dots ,u_{m})\big] dx.
\end{equation*}
The hypothesis on $u_{k}$, $f_{k}$ ($k=1,\dots ,m$) and $p_{i}$
($i=1,\dots ,n$) implies that $\mathcal{K}$ is absolutely continuous
 and thus differentiable almost everywhere on $\mathbb{R}$; we have
\begin{equation}
\begin{aligned}
\frac{d\mathcal{K}(t)}{dt}
&=\int_{\mathcal{D}}\big[ \lambda
\sum_{k=1}^m \frac{\partial u_{k}}{\partial t}\frac{
\partial ^{2}u_{k}}{\partial t^{2}}+\sum_{i=1}^n
p_{i}(x)\Big(\sum_{k=1}^m \frac{\partial
u_{k}}{\partial x_{i}}\frac{\partial ^{2}u_{k}}{\partial t\partial x_{i}}\Big)\\
&\quad +\sum_{k=1}^m \frac{\partial u_{k}}{
\partial t}f_{k}(x,u_{1},\dots ,u_{m})\big] dx.
\end{aligned}\label{e5}
\end{equation}
Fubini's theorem and an integration by part give
\begin{align*}
&\int_{\mathcal{D}}\sum_{i=1}^n p_{i}(x)
\Big(\sum_{k=1}^m \frac{\partial u_{k}}{\partial x_{i}
}\frac{\partial ^{2}u_{k}}{\partial t\partial x_{i}}\Big)(t,x)dx
\\
&=-\int_{\mathcal{D}}\sum_{i=1}^n \big[ \underset{k=1}{
\overset{m}{\sum }}\frac{\partial }{\partial x_{i}}(p_{i}(
x)\frac{\partial u_{k}}{\partial x_{i}})\frac{\partial u_{k}}{
\partial t}\big] (t,x)dx
\\
&\quad +\sum_{i=1}^n \int_{\mathcal{D}_{i}^{\ast }}
\big[ p_{i}(x_{i}^{\beta _{i}})(\underset{k=1}{\overset{m
}{\sum }}\frac{\partial u_{k}}{\partial x_{i}}\frac{\partial u_{k}}{\partial
t})(t,x_{i}^{\beta _{i}})
  -p_{i}(x_{i}^{\alpha _{i}})\Big(
\sum_{k=1}^m \frac{\partial u_{k}}{\partial x_{i}}\frac{
\partial u_{k}}{\partial t}\Big)(t,x_{i}^{\alpha _{i}})
\big] dx_{i}^{\ast }.
\end{align*}
Replacing in \eqref{e5} we find
\begin{align*}
&\frac{d\mathcal{K}(t)}{dt}\\
&=\sum_{k=1}^m \int_{\mathcal{D}}
 \Big[ \lambda \frac{\partial ^{2}u_{k}}{\partial t^{2}}-
\sum_{i=1}^n \frac{\partial }{\partial x_{i}}(
p_{i}(x)\frac{\partial u_{k}}{\partial x_{i}})
+f_{k}(x,u_{1},\dots ,u_{m})\Big] (t,x)\frac{
\partial u_{k}}{\partial t}dx
\\
&\quad +\sum_{i=1}^n \int_{\mathcal{D}_{i}^{\ast }}
\Big[ p_{i}(x_{i}^{\beta _{i}})(\underset{k=1}{\overset{m
}{\sum }}\frac{\partial u_{k}}{\partial x_{i}}\frac{\partial u_{k}}{\partial
t})(t,x_{i}^{\beta _{i}})
  -p_{i}(x_{i}^{\alpha _{i}})
\Big(\sum_{k=1}^m \frac{\partial u_{k}}{\partial x_{i}
}\frac{\partial u_{k}}{\partial t}\Big)(t,x_{i}^{\alpha
_{i}})\Big] dx_{i}^{\ast }.
\end{align*}
Let us consider on $\partial \Omega $ the expression
$u_{k}+\varepsilon \frac{\partial u_{k}}{\partial n}=0$.
For $k=1,\dots ,m$
\begin{equation*}
u_{k}+\varepsilon \frac{\partial u_{k}}{\partial n}=0\Longleftrightarrow
\begin{cases}
(u_{k}-\varepsilon \frac{\partial u_{k}}{\partial x})
(t,x_{i}^{\alpha _{i}})=0, \\
(u_{k}+\varepsilon \frac{\partial u_{k}}{\partial x})
(t,x_{i}^{\beta _{i}})=0, \\
t\in \mathbb{R},\alpha _{i}<x_{i}<\beta _{i},i=1,\dots n.
\end{cases}
\end{equation*}
Then for $\varepsilon >0$, one can write
\begin{align*}
&\sum_{i=1}^n \int_{\mathcal{D}_{i}^{\ast }}\Big[
 p_{i}(x_{i}^{\beta _{i}})\Big(\sum_{k=1}^m \frac{\partial u_{k}}{\partial x_{i}}\frac{\partial u_{k}}{\partial t}
 \Big)(t,x_{i}^{\beta _{i}})
-p_{i}(x_{i}^{\alpha _{i}})\big(
 \sum_{k=1}^m \frac{\partial u_{k}}{\partial x_{i}}\frac{
 \partial u_{k}}{\partial t}\Big)(t,x_{i}^{\alpha _{i}})
 \Big] dx_{i}^{\ast }
 \\
&=\frac{-1}{\varepsilon }\sum_{i=1}^n \int_{\mathcal{D}
 _{i}^{\ast }}\Big[ p_{i}(x_{i}^{\beta _{i}})(\underset{
 k=1}{\overset{m}{\sum }}u_{k}\frac{\partial u_{k}}{\partial t})(
 t,x_{i}^{\beta _{i}})
+p_{i}(x_{i}^{\alpha _{i}})\Big(
 \sum_{k=1}^m u_{k}\frac{\partial u_{k}}{\partial t}
 \Big)(t,x_{i}^{\alpha _{i}})\Big] dx_{i}^{\ast }
\\
&=\frac{-1}{2\varepsilon }\frac{d}{dt}(\sum_{i=1}^n
 \int_{\mathcal{D}_{i}^{\ast }}\Big[ p_{i}(x_{i}^{\beta _{i}})
 (\sum_{k=1}^m | u_{k}| ^{2})(t,x_{i}^{\beta _{i}})
 +p_{i}(x_{i}^{\alpha _{i}})
 \big(\sum_{k=1}^m | u_{k}|^{2}\big)(t,x_{i}^{\alpha _{i}})\Big]
 dx_{i}^{\ast }).
\end{align*}
Therefore,
\begin{align*}
&\frac{d}{dt}\Big(\mathcal{K}(t)+\frac{1}{2\varepsilon }
\sum_{i=1}^n \int_{\mathcal{D}_{i}^{\ast }}\Big[
p_{i}(x_{i}^{\beta _{i}})(\sum_{k=1}^m | u_{k}| ^{2})(t,x_{i}^{\beta _{i}})
\\
& +p_{i}(x_{i}^{\alpha _{i}})(\sum_{k=1}^m
| u_{k}| ^{2})(t,x_{i}^{\alpha _{i}})\Big] dx_{i}^{\ast }\Big)=0.
\end{align*}
Integrating with respect to $t$, we obtain
\begin{align*}
&\mathcal{K}(t)+\frac{1}{2\varepsilon }\underset{i=1}{\overset{n}
{\sum }}\int_{\mathcal{D}_{i}^{\ast }}\Big[ p_{i}(x_{i}^{\beta
_{i}})\big(\sum_{k=1}^m | u_{k}| ^{2}\big)(t,x_{i}^{\beta _{i}})\\
& +p_{i}(x_{i}^{\alpha _{i}})\big(\sum_{k=1}^m | u_{k}| ^{2}\big)(t,x_{i}^{\alpha _{i}})\Big] dx_{i}^{\ast }=\text{const}
\end{align*}
and since $(u_{1}(t,x),\dots ,u_{m}(t,x))\in (H^{2}(\Omega )
\cap L^{\infty }(\Omega ))^{m}$, one must get
\begin{align*}
&\int_{\mathbb{R}}(\mathcal{K}(t)+\frac{1}{2\varepsilon }
\sum_{i=1}^n \int_{\mathcal{D}_{i}^{\ast }}\Big[ p_{i}(
x_{i}^{\beta _{i}})(\sum_{k=1}^m
| u_{k}| ^{2})(t,x_{i}^{\beta _{i}}) \\
& +p_{i}(x_{i}^{\alpha _{i}})\big(\sum_{k=1}^m | u_{k}| ^{2}\big)
(t,x_{i}^{\alpha _{i}})\Big] dx_{i}^{\ast })dt<\infty .
\end{align*}
It follows that the constant must be $0$, which is the desired result.
\end{proof}

For the Dirichlet or Neumann boundary conditions, we have the
integral identity given in the following theorem.

\begin{theorem} \label{thm2}
Let $u_{1},\dots ,u_{m}$ in $H^{2}(\Omega )\cap L^{\infty }(\Omega )$
be a solution of problems \eqref{ep1}-\eqref{ed} or
\eqref{ep1}-\eqref{en}. Then for each $t\in\mathbb{R}$, we have
\begin{equation}
\int_{\mathcal{D}}\Big[ \frac{\lambda }{2}\sum_{k=1}^m
| \frac{\partial u_{k}}{\partial t}| ^{2}
+\sum_{i=1}^n \frac{p_{i}(x)}{2}
\Big(\sum_{k=1}^m | \frac{\partial u_{k}}{\partial x_{i}}
| ^{2}\Big)+H(x,u_{1},\dots ,u_{m})\Big] dx=0.
\label{e100}
\end{equation}
\end{theorem}

\begin{proof}
To prove \eqref{e100} it suffices to check that the expression
\begin{equation*}
\sum_{i=1}^n \int_{\mathcal{D}_{i}^{\ast }}
\Big[p_{i}(x_{i}^{\beta _{i}})\big(\sum_{k=1}^m | u_{k}| ^{2}\big)
(t,x_{i}^{\beta _{i}}) +p_{i}(x_{i}^{\alpha _{i}})
\big(\sum_{k=1}^m | u_{k}| ^{2}\big)(t,x_{i}^{\alpha _{i}})\Big] dx_{i}^{\ast } \\
\end{equation*}
vanishes if
\begin{equation} \label{e1.2}
u_{1}(t,s)=u_{2}(t,s)=\dots =u_{m}(t,s)
=0,\text{ }(t,s)\in \mathbb{R}\times \partial \mathcal{D}
\end{equation}
or
\begin{equation} \label{e1.3}
\frac{\partial u_{1}(t,s)}{\partial n}=\frac{\partial
u_{2}(t,s)}{\partial n}=\dots =\frac{\partial u_{m}(
t,s)}{\partial n}=0,\text{ }(t,s)\in \mathbb{R}
\times \partial \mathcal{D}.
\end{equation}
Indeed, suppose that \eqref{e1.2} holds then it is known that
\[
\nabla u_{k}=\frac{\partial u_{k}}{\partial n}\cdot n,\quad
k=1,\dots ,m;
\]
i.e.,
\[
\begin{bmatrix}
\frac{\partial u_{k}}{\partial t}(t,s)\\
\frac{\partial u_{k}}{\partial x_{1}}(t,s)\\
\dots  \\
\frac{\partial u_{k}}{\partial x_{n}}(t,s)
\end{bmatrix}
=\begin{bmatrix}
0 \\
n_{1}\frac{\partial u_{k}}{\partial n}(t,s)\\
\dots  \\
n_{n}\frac{\partial u_{k}}{\partial n}(t,s)
\end{bmatrix}
,\quad (t,s)\in \mathbb{R}\times \partial \mathcal{D},\quad
k=1,\dots ,m.
\]
Consequently, for $k=1,\dots ,m$,
\begin{equation*}
\frac{\partial u_{k}}{\partial t}(t,x_{i}^{\alpha _{i}})
=\frac{ \partial u_{k}}{\partial t}(t,x_{i}^{\beta _{i}})=0,\quad
i=1,\dots ,n.
\end{equation*}
Now if the boundary condition is \eqref{e1.3}, then for $k=1,\dots ,m$,
one can write
\begin{equation*}
0=\frac{\partial u_{k}}{\partial n}(t,s)=\left\langle \nabla
u_{k},n\right\rangle \text{ on }\Gamma _{\alpha _{1}}\cup \Gamma _{\beta
_{1}}\cup \Gamma _{\alpha _{2}}\cup \Gamma _{\beta _{2}}\dots \cup \Gamma
_{\alpha _{n}}\cup \Gamma _{\beta _{n}};
\end{equation*}
i.e.,
\begin{equation*}
\frac{\partial u_{k}}{\partial x_{i}}(t,x_{i}^{\alpha _{i}})=
\frac{\partial u_{k}}{\partial x_{i}}(t,x_{i}^{\beta _{i}})=0,
\quad \text{for all }t\in \mathbb{R},
\quad i=1,\dots ,n,\; k=1,\dots ,m.
\end{equation*}
In both cases
$\frac{d\mathcal{K}(t)}{dt}=0\text{ for all }t\in\mathbb{R}$
which completes the proof.
\end{proof}

\section{Main results}

Before giving our main results, we note that the parameter $\lambda $ plays,
in fact, an important part as it allows \eqref{ep1} to be
dealt with in two manners based on whether its value is positive or
negative. Indeed, if $\lambda $ is positive (resp. negative), the system
$\eqref{ep1}$ is a hyperbolic (resp. elliptic) problem.

\subsection{Semi-linear hyperbolic problems}

Using identity \eqref{e22} we obtain the following first
result.

\begin{theorem} \label{thm3}
Let $\lambda >0$ and
$u_{1},\dots ,u_{m}\in H^{2}(\Omega )\cap L^{\infty }(\Omega )$.
Assume $p_{i}(x)>0$ in $\mathcal{D}$ $(i=1,\dots ,n)$ and
$f_{k}$ $(k=1,\dots ,m)$ satisfying
\begin{equation*}
H(x,u_{1},\dots ,u_{m})\geq 0.
\end{equation*}
Then problems \eqref{ep1}-\eqref{ed}, \eqref{ep1})-\eqref{en} and
\eqref{ep1}-\eqref{er} have no nontrivial solutions.
\end{theorem}

\begin{proof}
Applying formula \eqref{e22} (resp. \eqref{e100}) we immediately obtain
\begin{equation*}
\frac{\partial u_{k}}{\partial t}(t,x)=\frac{\partial u_{k}}{
\partial x_{i}}(t,x)=0\quad \text{in }\Omega ,\;
i=1,\dots ,n, \; k=1,\dots ,m.
\end{equation*}
Thus $u_{1},\dots ,u_{m}$ are constant and since for $k=1,\dots ,m$,
\begin{equation*}
\int_{\Omega }| u_{k}(t,x)| ^{2}dxdt\leq 0,
\end{equation*}
these constants are necessarily zero.
\end{proof}

The next theorem gives a non-existence result if the functions $f_{k}$
($k=1,\dots ,m$) satisfy another type of non-linearity.

\begin{theorem} \label{thm4}
Let $\lambda >0$ and $u_{1},\dots ,u_{m}:\Omega \to\mathbb{R}$ be a
solution of problem \eqref{ep1}-\eqref{er}.
Suppose that
$u_{1},\dots ,u_{m}\in H^{2}(\Omega )\cap L^{\infty}(\Omega )$
and $f_{k}$ $(k=1,\dots ,m)$ verify the
following condition
\begin{equation}
2H(x,u_{1},\dots ,u_{m})-\sum_{k=1}^m
u_{k}f_{k}(x,u_{1},\dots ,u_{m})\geq 0.
\label{e50}
\end{equation}
Then  problem \eqref{ep1}-\eqref{er} has no
nontrivial solutions.
\end{theorem}

\begin{remark} \label{rmk1} \rm
Since $u_{1},\dots ,u_{m}$ are bounded in $\Omega $, from the Maximum
Principle, the function $\mathcal{E}(t)$ is convex in
$\mathbb{R}$ which implies that the solution to the problem
\eqref{ep1}-\eqref{er} is identically equal to zero.
\end{remark}

\begin{proof}[Proof of Theorem \ref{thm4}]
It is easy to see that almost everywhere in $\Omega $
\begin{equation*}
(u_{k}\frac{\partial ^{2}u_{k}}{\partial t^{2}})(
t,x)=\Big(\frac{1}{2}\frac{\partial ^{2}(u_{k}^{2})}{
\partial t^{2}}-| \frac{\partial u_{k}}{\partial t}|
^{2}\Big)(t,x),\quad k=1,\dots ,m.
\end{equation*}
Let us multiply the $k$-th equation of \eqref{ep1} by
$u_{k}/2$ and integrate over $\mathcal{D}$ we obtain
\begin{equation}
\begin{aligned}
&\int_{\mathcal{D}}\Big[ \lambda \frac{\partial ^{2}u_{k}}{\partial t^{2}}
\frac{u_{k}}{2}-\sum_{i=1}^n \frac{\partial }{\partial
x_{i}}(p_{i}(x)\frac{\partial u_{k}}{\partial x_{i}}
)\frac{u_{k}}{2}+f_{k}(x,u_{1},\dots ,u_{m})\frac{u_{k}}{2}
\Big] (t,x)dx
\\
& =\int_{\mathcal{D}}\big[ \frac{\lambda }{4}\frac{\partial
^{2}(u_{k}^{2})}{\partial t^{2}}-\frac{\lambda }{2}|
\frac{\partial u_{k}}{\partial t}| ^{2}\big] (t,x)dx
\\
&\quad +\int_{\mathcal{D}}\Big[ -\sum_{i=1}^n \frac{
\partial }{\partial x_{i}}(p_{i}(x)\frac{\partial u_{k}}{
\partial x_{i}})\frac{u_{k}}{2}+f(x,u_{1},\dots ,u_{m})
\frac{u_{k}}{2}\Big] (t,x)dx.
\end{aligned}
\label{e12}
\end{equation}
Let us transform
\begin{align*}
&\int_{\mathcal{D}}\Big(-\sum_{i=1}^n \frac{\partial }{
\partial x_{i}}(p_{i}(x)\frac{\partial u_{k}}{\partial
x_{i}})\frac{u_{k}}{2}\Big)(t,x)dx\\
&=\int_{\mathcal{D}} \sum_{i=1}^n \frac{p_{i}(x)}{2}|
\frac{\partial u_{k}(t,x)}{\partial x_{i}}| ^{2}dx
\\
&\quad -\frac{1}{2}\sum_{i=1}^n \int_{\mathcal{D}_{i}^{\ast }}
\Big[ p_{i}(x_{i}^{\beta _{i}})(u_{k}\frac{\partial u_{k}
}{\partial x_{i}})(t,x_{i}^{\beta _{i}})-p_{i}(
x_{i}^{\alpha _{i}})(u_{k}\frac{\partial u_{k}}{\partial x_{i}}
)(t,x_{i}^{\alpha _{i}})\Big] dx_{i}^{\ast }.
\end{align*}
The substitution of this formula in \eqref{e12} gives
\begin{equation}
\begin{aligned}
&\int_{\mathcal{D}}\Big[ \lambda \frac{\partial ^{2}u_{k}}{\partial t^{2}}
\frac{u_{k}}{2}-\sum_{i=1}^n \frac{\partial }{\partial
x_{i}}(p_{i}(x)\frac{\partial u_{k}}{\partial x_{i}}
)\frac{u_{k}}{2}+f(x,u_{1},\dots ,u_{m})\frac{u_{k}}{2}
\Big] (t,x)dx
\\
&=\int_{\mathcal{D}}\Big(\frac{\lambda }{4}\frac{\partial ^{2}(
u_{k}^{2})}{\partial t^{2}}-\frac{\lambda }{2}| \frac{
\partial u_{k}}{\partial t}| ^{2}\Big)(t,x)dx
\\
&\quad +\int_{\mathcal{D}} \sum_{i}^n
\frac{p_{i}(x)}{2}|
\frac{\partial u_{k}(t,x)}{\partial x_{i}}| ^{2}dx
+\int_{\mathcal{D}}(\frac{u_{k}}{2}f(x,u_{1},\dots ,u_{m}))(t,x)dx
\\
&\quad -\frac{1}{2}\sum_{i=1}^n \int_{\mathcal{D}_{i}^{\ast }}
\Big[ p_{i}(x_{i}^{\beta _{i}})(u_{k}\frac{\partial u_{k}
}{\partial x_{i}})(t,x_{i}^{\beta _{i}})-p_{i}(
x_{i}^{\alpha _{i}})(u_{k}\frac{\partial u_{k}}{\partial x_{i}}
)(t,x_{i}^{\alpha _{i}})\Big] dx_{i}^{\ast }
\\
&=\int_{\mathcal{D}}(\frac{\lambda }{4}\frac{\partial ^{2}(
u_{k}^{2})}{\partial t^{2}}-\frac{\lambda }{2}| \frac{
\partial u_{k}}{\partial t}| ^{2})(t,x)dx
\\
&\quad +\int_{\mathcal{D}}\sum_{i=1}^n \frac{p_{i}(
x)}{2}| \frac{\partial u_{k}(t,x)}{\partial
x_{i}}| ^{2}dx+\int_{\mathcal{D}}(\frac{u_{k}}{2}f(
x,u_{1},\dots ,u_{m}))(t,x)dx
\\
&\quad +\frac{1}{2\varepsilon }\sum_{i=1}^n \int_{\mathcal{D}
_{i}^{\ast }}\left[ p_{i}(x_{i}^{\beta _{i}})|
u_{k}(t,x_{i}^{\beta _{i}})| ^{2}+p_{i}(
x_{i}^{\alpha _{i}})| u_{k}(t,x_{i}^{\alpha
_{i}})| ^{2}\right] dx_{i}^{\ast }.
\end{aligned}
\label{e15}
\end{equation}
Adding these identities for $k=1,\dots ,k_{0}$, we get
\begin{equation*}
\begin{aligned}
&\frac{\lambda }{4}\int_{\mathcal{D}}\Big(\sum_{k=1}^m
\frac{\partial ^{2}(u_{k}^{2})}{\partial t^{2}}\Big)(t,x)dx
-\frac{\lambda }{2}\int_{\mathcal{D}}
\Big(\sum_{k=1}^m | \frac{\partial u_{k}}{\partial t}|
^{2}\Big)(t,x)dx
\\
&\quad +\int_{\mathcal{D}}\sum_{i=1}^n \frac{p_{i}(
x)}{2}(\sum_{k=1}^m | \frac{
\partial u_{k}}{\partial x_{i}}| ^{2})(t,x)dx+
\frac{1}{2}\int_{\mathcal{D}}(\sum_{k=1}^m
u_{k}f_{k}(x,u_{1},\dots ,u_{m}))(t,x)dx \\
\\
& +\frac{1}{2\varepsilon }\sum_{i=1}^n \int_{\mathcal{D}
_{i}^{\ast }}\Big[ p_{i}(x_{i}^{\beta _{i}})
\big(\sum_{k=1}^m | u_{k}| ^{2}\big)(t,x_{i}^{\beta _{i}})
 +p_{i}(x_{i}^{\alpha _{i}})
 \big(\sum_{k=1}^m | u_{k}| ^{2}\big)(t,x_{i}^{\alpha _{i}})\Big]
 dx_{i}^{\ast }=0,
\end{aligned}
\end{equation*}
which combined with \eqref{e22} yields
\begin{equation}
\begin{aligned}
\frac{\lambda }{4}\frac{d^{2}}{dt^{2}}\Big(\int_{\mathcal{D}}\big(
\sum_{k=1}^m u_{k}^{2}\big)(t,x)dx\Big)
&=\lambda \int_{\mathcal{D}}(\sum_{k=1}^m |
\frac{\partial u_{k}}{\partial t}| ^{2})(t,x)dx
 +\int_{\mathcal{D}}\Big[ H(x,u_{1},\dots ,u_{m})\\
 &\quad -\frac{1}{2}(\sum_{k=1}^m u_{k}f_{k}(x,u_{1},
  \dots ,u_{m}))(t,x,y)\Big] dx.
\end{aligned}
\label{e52}
\end{equation}
The assumptions \eqref{e50} and $\lambda >0$ enable us to
assert that
\[
\frac{\lambda }{4}\frac{d^{2}}{dt^{2}}
\Big(\int_{\mathcal{D}}
\big(\sum_{k=1}^m u_{k}^{2}\big)(t,x)dx\Big)
\geq \lambda \int_{\mathcal{D}}\Big(
\sum_{k=1}^m | \frac{\partial u_{k}}{\partial t
}| ^{2}\Big)(t,x)dx\geq 0,
\]
 for all $t\in \mathbb{R}$.
This completes the proof.
\end{proof}

\begin{theorem} \label{thm5}
Let $\lambda >0$ and $f_{k}$ be as described in Theorem \ref{thm4}. Assume that
$u_{1},\dots ,u_{m}\in H^{2}(\Omega )\cap L^{\infty }(\Omega)$
is a solution of \eqref{ep1}-\eqref{ed}
or \eqref{ep1}-\eqref{en}. Then problems
\eqref{ep1})-\eqref{ed} and \eqref{ep1}-\eqref{en} have no nontrivial solutions.
\end{theorem}

\begin{proof}
By a similar arguments as in the proof of Theorem \ref{thm4} we obtain
\begin{align*}
&\frac{\lambda }{4}\int_{\mathcal{D}}\Big(\sum_{k=1}^m
\frac{\partial ^{2}(u_{k}^{2})}{\partial t^{2}}\Big)(t,x)dx
-\frac{\lambda }{2}\int_{\mathcal{D}}
\Big(\sum_{k=1}^m | \frac{\partial u_{k}}{\partial t}|^{2}\Big)(t,x)dx
\\
& +\int_{\mathcal{D}}\sum_{i=1}^n \frac{p_{i}(x)}{2}
\Big(\sum_{k=1}^m | \frac{\partial u_{k}}{\partial x_{i}}| ^{2}\Big)(t,x)dx
\\
& +\frac{1}{2}\int_{\mathcal{D}}\Big(\sum_{k=1}^m
u_{k}f_{k}(x,u_{1},\dots ,u_{m})\Big)(t,x)dx
\\
& +\frac{1}{2\varepsilon }\sum_{i=1}^n \int_{\mathcal{D}
_{i}^{\ast }}\Big[ p_{i}(x_{i}^{\beta _{i}})
\big(\sum_{k=1}^m | u_{k}| ^{2}\big)(t,x_{i}^{\beta _{i}})
 +p_{i}(x_{i}^{\alpha _{i}})
 \big(\sum_{k=1}^m | u_{k}| ^{2}\big)(t,x_{i}^{\alpha _{i}})\Big]
dx_{i}^{\ast }=0.
\end{align*}
If
\begin{equation*}
u_{1}(t,s)=\dots =u_{m}(t,s)=0,\quad (t,s)\in\mathbb{R}\times
\partial \mathcal{D}
\end{equation*}
or
\begin{equation*}
\frac{\partial u_{1}(t,s)}{\partial n}=\dots =\frac{\partial
u_{m}(t,s)}{\partial n}=0,\text{ }(t,s)\in
\mathbb{R}\times \partial \mathcal{D},
\end{equation*}
this formula reduces to
\begin{align*}
&\frac{\lambda }{4}\int_{\mathcal{D}}\Big(\sum_{k=1}^m
\frac{\partial ^{2}(u_{k}^{2})}{\partial t^{2}}\Big)(t,x)dx
-\frac{\lambda }{2}\int_{\mathcal{D}}
\Big(\sum_{k=1}^m | \frac{\partial u_{k}}{\partial t}|^{2}\Big)(t,x)dx\\
&+\int_{\mathcal{D}}\sum_{i=1}^n \frac{p_{i}(x)}{2}
\Big(\sum_{k=1}^m | \frac{\partial u_{k}}{\partial x_{i}}| ^{2}\Big)(t,x)dx
\\
&+\frac{1}{2}\int_{\mathcal{D}}\Big(
\sum_{k=1}^m u_{k}f_{k}(x,u_{1},\dots ,u_{m})
\Big)(t,x)dx=0.
\end{align*}
We can now employ \eqref{e100} to transform this identity into
the  form
\begin{equation}
\begin{aligned}
&\frac{\lambda }{4}\int_{\mathcal{D}}\Big(\sum_{k=1}^m
\frac{\partial ^{2}(u_{k}^{2})}{\partial t^{2}}\Big)(t,x)dx\\
&=\lambda \int_{\mathcal{D}}
 \Big(\sum_{k=1}^m | \frac{\partial u_{k}}{\partial t}| ^{2}\Big)(t,x)dx
\\
&\quad +\int_{\mathcal{D}}\Big[ H(x,u_{1},\dots ,u_{m})-
\frac{1}{2}\Big(\sum_{k=1}^m u_{k}f_{k}(
x,u_{1},\dots ,u_{m})\Big)(t,x,y)\Big] dx\,.
\end{aligned} \label{e53}
\end{equation}
This completes the proof.
\end{proof}

\subsection{Semi-linear elliptic problems}

We shall prove that a dual result holds for $\lambda <0$.

\begin{theorem} \label{thm6}
Let $(u_{1},\dots ,u_{m})\in (H^{2}(\Omega )
\cap L^{\infty }(\Omega ))^{m}$ be a solution of
\eqref{ep1})-\eqref{er}, $\lambda <0$ and $f_{k}$ $(k=1,\dots ,m)$ satisfying
\begin{equation}
2H(x,u_{1},\dots ,u_{m})-\sum_{k=1}^m
u_{k}f_{k}(x,u_{1},\dots ,u_{m})\leq 0.
\label{e51}
\end{equation}
Then problem \eqref{ep1}-\eqref{er} has no
nontrivial solutions.
\end{theorem}

\begin{proof}
Formula \eqref{e52} combined with the assumption \eqref{e51} yields
\begin{equation*}
\frac{\lambda }{4}\frac{d^{2}}{dt^{2}}
\Big(\int_{\mathcal{D}}\big(\sum_{k=1}^m u_{k}^{2}\big)(t,x)dx\Big)
\leq \lambda \int_{\mathcal{D}}
\Big(\sum_{k=1}^m | \frac{\partial u_{k}}{\partial t}| ^{2}\Big)
(t,x)dx,\quad \text{for all }t\in\mathbb{R}
\end{equation*}
and $\lambda <0$ gives the desired result.
\end{proof}

\begin{theorem} \label{thm7}
Let $\lambda <0$ and $f_{k}$ $(k=1,\dots ,m)$ be as described in
Theorem \ref{thm6}. We assume that
\begin{equation*}
u_{1},\dots ,u_{m}\in H^{2}(\Omega )\cap L^{\infty }(\Omega)
\end{equation*}
is a solution of \eqref{ep1}-\eqref{ed} or \ref{ep1}-\eqref{en}.
Then problems \eqref{ep1}-\eqref{ed} and \eqref{ep1}-\eqref{en}
 have no nontrivial solutions.
\end{theorem}

This theorem follows from \eqref{e53} and \eqref{e51}
 with $\lambda <0$.

\section{Examples}

In this section, we illustrate our theoretical results by giving some
 examples.

\subsection*{Example 1}
Let $\theta :\mathcal{D}\to\mathbb{R}$ be a continuous function,
 the exponents $\alpha _{s}>0$, $s=1,\dots ,m$ and
\begin{equation*}
p_{i}(x)>0\quad \text{or}\quad p_{i}(x)<0\text{ in }\mathcal{D},\quad
i=1,\dots ,n.
\end{equation*}
Then  system \eqref{ep1} with
\begin{equation*}
f_{k}(x,u_{1},\dots ,u_{m})=\theta (x)
\Big[ \prod_{s=1,\,s\neq k}^m \frac{1}{\alpha _{s}+1}
| u_{s}| ^{\alpha _{s}+1}\Big] |
u_{k}| ^{\alpha _{k}-1}u_{k},\quad k=1,\dots ,m
\end{equation*}
subject to Dirichlet, Neumann or Robin boundary conditions, does not have
nontrivial solutions.
Indeed,  when
$\lambda >0$  and $p_{i},\theta >0$  in $\mathcal{D}$, ($i=1,\dots ,n$),
we have
\begin{equation*}
H(x,u_{1},\dots ,u_{m})=\theta (x)\Big[ \prod_{s=1}^m
\frac{1}{\alpha _{s}+1}| u_{s}|^{\alpha _{s}+1}\Big]
\end{equation*}
and Theorem \ref{thm3} gives the desired result.

When $\lambda >0$ (resp. $\lambda <0$),
$\theta (x)\leq 0$ (resp. $\theta (x)\geq 0$) in $\mathcal{D}$ and
$p_{i}(x)>0$ or $p_{i}(x)<0$  in $\mathcal{D}$, $i=1,\dots ,n$,
we have
\begin{align*}
&2H(x,u_{1},\dots ,u_{k_{0}})-\sum_{k=1}^m
u_{k}f_{k}(x,u_{1},\dots ,u_{m})\\
&=\theta (x)\frac{2-\sum_{k=1}^m (\alpha _{k}+1)}
{\prod_{k=1}^m (\alpha _{k}+1)}
\prod_{k=1}^m | u_{k}| ^{\alpha _{k}+1}\leq 0 \quad
(\text{resp. }\geq 0)\,.
\end{align*}
We conclude by using Theorem \ref{thm4} or Theorem \ref{thm5}
 (resp. Theorem \ref{thm6} or
Theorem \ref{thm7}) as the system is subject to Robin, Neumann or Dirichlet
boundary conditions.


\subsection*{Example 2}
Let us consider the system \eqref{ep1} with $m=2$ and
\begin{gather*}
f_{1}(x,u_{1},u_{2})=\rho (x)u_{2}(
| u_{1}| ^{\alpha -1}u_{1}+\frac{1}{\beta +1}|
u_{2}| ^{\beta -1}u_{2}),
\\
f_{2}(x,u_{1},u_{2})=\rho (x)u_{1}(\frac{1}{
\alpha +1}| u_{1}| ^{\alpha -1}u_{1}+|
u_{2}| ^{\beta -1}u_{2}),
\end{gather*}
where the continuous function $\rho (x)$ is positive (resp.
negative) and $\alpha ,\beta $ are positive real number. Then this problem
does not have nontrivial solutions.\newline
It suffices to remark that
\begin{equation*}
H(x,u_{1},u_{2})=\rho (x)(u_{2}\frac{| u_{1}| ^{\alpha +1}}{\alpha +1}
+u_{1}\frac{|u_{2}| ^{\beta +1}}{\beta +1})\\
\end{equation*}
and a simple computation gives
\begin{align*}
& 2H(x,u_{1},u_{2})-u_{1}f_{1}(x,u_{1},u_{2})
-u_{2}f_{2}(x,u_{1},u_{2})
\\
&=\rho (x)\Big[ (\frac{1}{\alpha +1}-1)
| u_{1}| ^{\alpha +1}u_{2}+(\frac{1}{\beta +1}
-1)| u_{2}| ^{\beta +1}u_{1}\Big] \leq 0\quad
(\text{resp. }\geq 0).
\end{align*}
The conclusion is the same as in the previous example.


\subsection{Example3}
For the  scalar case ($m=1$), let
$\theta _{1},\theta _{2}:\overline{\mathcal{D}}\to \mathbb{\mathbb{R}}$
be two nonnegative continuous functions, $p,q\geq 1$ and
\begin{equation*}
f(x,u)=\delta u+\theta _{1}(x)| u|
^{p-1}u+\theta _{2}(x)| u| ^{q-1}u,
\end{equation*}
where $\delta $ is a real constant. Then the problem
\begin{gather*}
-\frac{\partial ^{2}u}{\partial t^{2}}-\sum_{i=1}^{n}\frac{\partial }{
\partial x_{i}}(p_{i}(x)\frac{\partial u}{\partial y_{i}})+f(x,u)=0\quad
\text{in }\Omega , \\
u+\varepsilon \frac{\partial u}{\partial n}=0\quad \text{on }\partial \Omega ,
\end{gather*}   %\label{e4.2}
does not have nontrivial solutions.
A simple computation gives
\begin{equation*}
2H(x,u)-uf(x,u)
 =\theta _{1}(x)(\frac{2}{p+1}-1)|
u| ^{p+1}+\theta _{2}(x)(\frac{2}{q+1}-1)|
u| ^{q+1}\leq 0,
\end{equation*}
and an application of Theorem \ref{thm6} gives the desired result.


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\end{document}