Nonlinear eigenvalue problems in Sobolev spaces with variable exponent - Teodora-Liliana Dinu (Journal of Function Spaces and Applications)
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)}<q(x)<N⋅m(x)N-m(x) for any x∈Ω̅. In the first case we show the existence of infinitely many weak solutions for any λ>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.