On the existence and multiplicity of eigenvalues for a class of double-phase non-autonomous problems with variable exponent growth

Vasile Uta

doi:10.14232/ejqtde.2020.1.28

Abstract

On the existence and multiplicity of eigenvalues for a class of double-phase non-autonomous problems with variable exponent growth

Highlights

In Ω, u = 0 on ∂Ω, where Ω is a bounded domain from RN with smooth boundary and the potential functions φ and ψ have (p1(x); p2(x)) variable growth
The proofs rely on variational arguments based on the Ekeland’s variational principle, the mountain pass theorem, the fountain theorem and energy estimates
The recent study of various mathematical models described by variational problems with nonstandard variable growth conditions is motivated by many phenomena that arise in applied sciences

Summary

Introduction

The recent study of various mathematical models described by variational problems with nonstandard variable growth conditions is motivated by many phenomena that arise in applied sciences. (Pλ) where Ω is a bounded domain in RN with Lipschitz boundary and λ ∈ R is a real parameter The study of these types of problems was motivated by the fact that we may need to model a composite that changes its hardening exponent according to the point. The first solution is obtained as a local minimum near the origin To this end we refer to [9,17] and [24, Chapter 2] for more details about the method used to point out this type of solutions. More details about existence and nonexistence results related to variable exponent equations can be found in the following works [4, 11], while more critical point techniques and qualitative analysis for double-phase operators can be found in [1, 5, 20]. In the final part of this paper are given some examples and remarks in order to illustrate the validity of the general results obtained throughout this work

The functional framework

Basic hypotheses and auxiliary results

Findings

Multiple types of solutions