Existence of solutions for $p(x)$-Laplacian problem on an unbounded domain - Yongqiang, Fu (Topological Methods in Nonlinear Analysis, 2007)
In this paper we study the following $p(x)$-Laplacianproblem:$$\alignat 2-\div(a(x)| abla u|^{p(x)-2} ablau)+b(x)|u|^{p(x)-2}u&=f(x,u) &\quad& x\in \Omega,\\ u&=0&\quad&\text{on }\partial\Omega,\endalignat$$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 obtainthe existence of solutions in $W_{0}^{1,p(x)}(\Omega)$ for the$p(x)$-Laplacian problem in the superlinear case.