Existence and multiplicity of eigenvalues for some double-phase problems involving an indefinite sign reaction term

Abstract

We study the following class of double-phase nonlinear eigenvalue problems − div ⁡ [ ϕ ( x , | ∇ u | ) ∇ u + ψ ( x , | ∇ u | ) ∇ u ] = λ f ( x , u ) in Ω , u = 0 on ∂ Ω , where Ω is a bounded domain from R N and the potential functions ϕ and ψ have ( p 1 ( x ) ; p 2 ( x ) ) variable growth. The primitive of the reaction term of the problem (the right-hand side) has indefinite sign in the variable u and allows us to study functions with slower growth near + ∞ , that is, it does not satisfy the Ambrosetti–Rabinowitz condition. Under these hypotheses we prove that for every parameter λ ∈ R + ∗ , the problem has an unbounded sequence of weak solutions. The proofs rely on variational arguments based on energy estimates and the use of Fountain Theorem.