On (p(x), q(x))‐Laplace equations in ℝN without Ambrosetti‐Rabinowitz condition

Abstract

In the present work, we consider a (p(x), q(x))‐elliptic equation describing the behavior of a double‐phase anisotropic problem which has relevance in electrorheological fluid applications. The analysis leads to the existence of weak (nonnegative) solutions in the special case of potential terms with critical frequency and a superlinear reaction term. In order to prove the existence result, we combine critical point theory of mountain pass type with related topological and variational methods. Basically, the approach is variational, but we do not impose any Ambrosetti‐Rabinowitz type condition for the superlinearity of the reaction. More specifically, we apply the Euler‐Lagrange functional approach to the variational formulation of the above‐mentioned model problem. We note that we work in the whole space and so we have to consider non‐compact embeddings. This aspect constitutes an additional difficulty in our study.