Existence and multiplicity of weak solutions for nonuniformly elliptic equations with nonstandard growth condition

We discuss the problem - div(a(x, ∇u)) = m(x){divides}u{divides}^r(x)-2u + n(x){divides}u{divides}^s(x)-2u in Ω, where Ω is a bounded domain with smooth boundary in ℝ^N(N ≥ 2), and div(a(x, ∇u)) is a p(x)-Laplace type operator with 1 & r(x) & p(x) & s(x). We show the existence of infinitely many nontrivial weak solutions in. Our approach relies on the theory of the variable exponent Lebesgue and Sobolev spaces combined with adequate variational methods and a variation of the Mountain Pass lemma and critical point theory.