Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 72, pp. 1-12.

Title: Existence of solutions for a $p(x)$-Laplacian  non-homogeneous
       equations

Author: Ionica Andrei (High School of Cujmir, Romania)

Abstract:
 We study the boundary value problem
 
 $$\displaylines{
 -\hbox{\rm div}(|\nabla u|^{p(x)-2}\nabla  u)=f(x,u)\quad
 \hbox{in }\Omega, \cr
 u=0\quad \hbox{on }\partial \Omega,
 }$$
 where $\Omega$  is a smooth bounded domain  in $\mathbb{R}^N$. 
 Our attention is focused on the cases when
 $$
 f(x,u)=\pm (-\lambda |u|^{p(x)-2}u+|u|^{q(x)-2}u),
 $$
 where $ p(x)<q(x)<N\cdot p(x)/(N-p(x))$ for $x$ in $\Omega$.
 
Submitted March 9, 2009. Published June 02, 2009.
Math Subject Classifications: 35D05, 35J60, 58E05
Key Words: p(x)-Laplace operator; generalized
           Lebesgue-Sobolev space; critical point; weak solution.