Nonlinear eigenvalue problems in Sobolev spaces with variable exponent

We study the boundary value problem -div⁡((|∇u|p1(x)-2+|∇u|p2(x)-2)∇u)=f(x,u) in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in ℝN. We focus on the cases when f±(x,  u)=±(-λ|u|m(x)-2u+|u|q(x)-2u), where m(x)≔max⁡⁡{p1(x),p2(x)}0. In the second case we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ℤ2-symmetric version for even functionals of the Mountain Pass Lemma and some adequate variational methods