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

Vasile Uța

doi:10.14232/ejqtde.2022.1.5

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.

Highlights

The study of variational problems with nonstandard growth conditions has been developed extensively over the last years
The primitive of the reaction term of the problem 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
As new types of materials arose in the domains that we mentioned before, new problems arose in the field of variable exponent analysis and partial differential equations which

Summary

Introduction

The study of variational problems with nonstandard growth conditions has been developed extensively over the last years. High energy solutions for similar problems were studied under more restrictive hypotheses in the following works: [19, 22], where the reaction function is supposed to satisfy the so called (AR)-condition, or in [23] where the differential operator enables us to study some simple case, where in order to make connections to our problem the potential function φ is supposed to verify just the case (C2) and the potential function ψ ≡ 0, but the nonlinearity in the right-hand side of the problem is more general than the one used in [19] and [22] This generality comes at a cost, that is, the parameter λ is allowed to take values just in a bounded interval near the origin. We draw a parallel between previous results and the new results presented in this paper as well as some future perspectives of research in this direction

The functional framework

Basic hypotheses and auxiliary results

Main results

Findings

Some examples and final remarks