Nonvariational and singular double phase problems for the Baouendi-Grushin operator

Anouar Bahrouni,Vicenţiu D Rădulescu,Dušan D Repovš

doi:10.1016/j.jde.2021.09.033

Abstract

In this paper we introduce a new double phase Baouendi-Grushin type operator with variable coefficients. We give basic properties of the corresponding functions space and prove a compactness result. In the second part, using topological argument, we prove the existence of weak solutions of some nonvariational problems in which this new operator is present. The present paper extends and complements some of our previous contributions related to double phase anisotropic variational integrals.

Highlights

The present paper is motivated by recent fundamental enrichment to the mathematical analysis of nonlinear models with unbalanced growth
We mainly refer to the pioneering contributions of Marcellini [19,20] who studied lower semicontinuity and regularity properties of minimizers of certain quasiconvex integrals
Related problems are inspired by models arising in nonlinear elasticity and they describe the deformation of an elastic body, see Ball [1,2]

Summary

Introduction

The present paper is motivated by recent fundamental enrichment to the mathematical analysis of nonlinear models with unbalanced growth. The main aim of our work is to introduce a new double phase Baouendi-Grushin type operator with variable exponents and its suitable functions space. Our abstract results related to the new function space are motivated by the existence of solutions for nonvariational problems of type (1.1). We are able to introduce the new Baouendi-Grushin type operator with variable coefficients, which is defined by. The main goal of our recent paper [6] was to study a singular system in the whole space RN in which the Baouendi-Grushin operator (−G(x,y) ) is present. We are not able to study a large number of equations driven by −G(x,y) in the whole space RN For this reason and in order to get a better compactness result, we introduced the new operator. We refer to the monograph by Papageorgiou, Rădulescu & Repovš [28] as a general reference for the abstract methods used in this paper

Terminology and the abstract setting

Double phase Baouendi-Grushin operators

Nonvariational case

Singular problem