Electron. J. Diff. Equ., Vol. 2010(2010), No. 33, pp. 1-11.

Existence of multiple solutions for a p(x)-Laplace equation

Duchao Liu

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 3, 2010.
Math Subject Classifications: 35B38, 35D05, 35J20.
Key Words: Critical points; p(x)-Laplacian; integral functionals; generalized Lebesgue-Sobolev spaces.

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Duchao Liu
School of Mathematics and Statistics
Lanzhou University, Lanzhou 730000, China
email: liudch06@lzu.cn

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