Existence results for singular double phase problem with variable exponents

Abstract

The goal of this paper is to prove the existence of two weak solutions for the following problem: − div(|∇u|p(x)−2∇u+μ(x)|∇u|q(x)−2∇u) =λh(x)u −η(x)+ξ(x)|u|s(x)−2u in Ω, u = 0 on ∂Ω, where Ω ⊂ RN, N ≥ 2 is a bounded domain with smooth boundary ∂Ω and λ > 0 is a real parameter. The functions h(x), ξ(x) ∈ C(Ω) are positive with compact support in Ω.We assume some suitable conditions on functions p, q, η and s. We use the Nehari manifold method based on fibering maps to establish our results. Mathematics Subject Classification. 35J30, 35J75, 35D30. Keywords. Nehari manifold,Musielak–Sobolev spaces, fibering map, double phase operator with variable exponents