Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 33, pp. 1-11.
Title: Existence of multiple solutions for a p(x)-Laplace equation
Author: Duchao Liu (Lanzhou Univ., Lanzhou, China)
Abstract:
This article shows the existence of at least three
nontrivial solutions to the quasilinear elliptic equation
$$
-\Delta_{p(x)}u+|u|^{p(x)-2}u=f(x,u)
$$
in a smooth bounded domain $\Omega\subset\mathbb{R}^{n}$,
with the nonlinear boundary condition
$|\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}=g(x,u)$
or the Dirichlet boundary condition $u=0$ on $\partial\Omega$.
In addition, this paper proves that one solution is positive,
one is negative, and the last one is a sign-changing solution.
The method used here is based on Nehari results, on three
sub-manifolds of the space $W^{1,p(x)}(\Omega)$.
Submitted September 26, 2008. Published March 03, 2010.
Math Subject Classifications: 35B38, 35D05, 35J20.
Key Words: Critical points; p(x)-Laplacian; integral functionals;
generalized Lebesgue-Sobolev spaces.