Multiplicity results of solutions to the double phase anisotropic variational problems involving variable exponent

Abstract

Aim of this paper is to discuss the existence of multiple solutions to double phase anisotropic variational problems for the case of a combined effect of concave-convex nonlinearities. Especially the superlinear (convex) term to the given problem substantially fulfills a weaker condition as well as Ambrosetti-Rabinowitz condition. To achieve these results, we apply the variational methods such as the famous mountain pass theorem and Ekeland's type variational principle when an energy functional corresponding to our problem satisfies the compactness condition of the Palais-Smale type. In particular, we establish several existence results of a sequence of infinitely many solutions by employing the Cerami compactness condition. The key tools for obtaining these results are the fountain theorem and the dual fountain theorem.