### 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.

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.

#### Keywords

Existence; p(x)-Laplacian problem; unbounded domain

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