Existence of solutions for $p(x)$-Laplacian problem on an unbounded domain

Abstract

In this paper we study the following $p(x)$-Laplacian
problem:
$$
\alignat 2
-\div(a(x)|\nabla u|^{p(x)-2}\nabla
u)+b(x)|u|^{p(x)-2}u&=f(x,u)
&\quad& x\in \Omega,\\
u&=0
&\quad&\text{on }\partial\Omega,
\endalignat
$$
where $1< p_{1}\le p(x)\le p_{2}< n$, $\Omega\subset {\mathbb R}^{n}$
is an exterior domain. Applying Mountain Pass Theorem we obtain
the existence of solutions in $W_{0}^{1,p(x)}(\Omega)$ for the
$p(x)$-Laplacian problem in the superlinear case.