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

We discuss the problem - div(a(x, del u)) = m(x)vertical bar u vertical bar(r(x)-2) u + n(x)vertical bar u vertical bar(s(x)-2)u in Omega, where Omega is a bounded domain with smooth boundary in R-N(N >= 2), and div(a(x, del 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 W-0(1,p(x))(Omega). 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.