Variational analysis for an indefinite quasilinear problem with variable exponent

We study the nonlinear boundary value problem -div((vertical bar del u(x)vertical bar(p1(x)-2) + vertical bar del u(x)vertical bar(p2(x)-2))del u(x)) =lambda V-1(x)vertical bar u vertical bar(q(x)-2)u - mu V-2(x)vertical bar u vertical bar(alpha(x)-2)u in Omega, u = 0 on delta Omega, where Omega is a bounded domain in R-N with smooth boundary, lambda, mu are positive real numbers, p(1), p(2), q, alpha are continuous functions on Omega, V-1 and V-2 are weight functions in generalized Lebesgue spaces L-s1(x)(Omega) and L-s2(x)(Omega), respectively, such that V-1 > 0 in an open set Omega(0) subset of Omega with |Omega(0|) > 0 and V-2 >= 0 on Omega. We prove, under appropriate conditions that for any mu > 0 there exists lambda(*) sufficiently small such that for any lambda is an element of(0,lambda(*)) the above nonhomogeneous quasilinear problem has a nontrivial positive weak solution. The proof relies on some variational arguments based on Ekeland's variational principle.