Double phase problems with variable growth and convection for the Baouendi\u2013Grushin operator

Anouar Bahrouni,Vicenţiu D Rădulescu,Patrick Winkert

doi:10.1007/s00033-020-01412-7

Abstract

In this paper we study a class of quasilinear elliptic equations with double phase energy and reaction term depending on the gradient. The main feature is that the associated functional is driven by the Baouendi–Grushin operator with variable coefficient. This partial differential equation is of mixed type and possesses both elliptic and hyperbolic regions. We first establish some new qualitative properties of a differential operator introduced recently by Bahrouni et al. (Nonlinearity 32(7):2481–2495, 2019). Next, under quite general assumptions on the convection term, we prove the existence of stationary waves by applying the theory of pseudomonotone operators. The analysis carried out in this paper is motivated by patterns arising in the theory of transonic flows.

Highlights

Let Ω ⊂ RN, N > 1, be a bounded domain with smooth boundary ∂Ω and let n, m be nonnegative integers such that N = n + m
The novelty of our paper is the fact that we combine a double phase operator driven by the Baouendi–Grushin operator with variable growth and a right-hand side which depends on the gradient of the solution
Bahrouni et al [5] proved a new version of a Caffarelli–Kohn–Nirenberg inequality with variable exponent for the Baouendi–Grushin operator ΔG

Summary

Introduction

The novelty of our paper is the fact that we combine a double phase operator driven by the Baouendi–Grushin operator with variable growth and a right-hand side which depends on the gradient of the solution. Bahrouni et al [5] proved a new version of a Caffarelli–Kohn–Nirenberg inequality with variable exponent for the Baouendi–Grushin operator ΔG. New properties concerning the Baouendi–Grushin operator will be discussed, and in the last section we state and prove our main result concerning the existence of a weak solution to problem (1.1) New properties concerning the Baouendi–Grushin operator will be discussed in Sect. 3, and in the last section we state and prove our main result concerning the existence of a weak solution to problem (1.1)

Terminology and the abstract setting

Properties of the double phase operator and the corresponding function space

Existence of a solution