Existence of solutions for p(x)-Laplacian equations

We discuss the problem {-div (vertical bar Delta(u)vertical bar(p(x)-2)del(u))=lambda(a(x)vertical bar u vertical bar(q(x)-2) u + b(x)vertical bar u vertical bar(h(x)-2)u), for x is an element of Omega, u=0, for x is an element of partial derivative Omega. where Omega is a bounded domain with smooth boundary in R-N (N >= 2) and p is Lipschitz continuous, q and h are continuous functions on (Omega) over bar such that 1 < q(x) < p(x) < h(x) < p*(x) and p(x) < N. We show the existence of at least one nontrivial weak solution. Our approach relies on the variable exponent theory of Lebesgue and Sobolev spaces combined with adequate variational methods and the Mountain Pass Theorem.