On a nonlinear eigenvalue problem in Sobolev spaces with variable   exponent
  We consider a class of nonlinear Dirichlet problems involving the $p(x)$--Laplace operator. Our framework is based on the theory of Sobolev spaces with variable exponent and we establish the existence of a weak solution in such a space. The proof relies on the Mountain Pass Theorem. 
Teodora Liliana Dinu