An Existence Result for a Class of p(x)—Anisotropic Type Equations

Anass Ourraoui,Maria Alessandra Ragusa

doi:10.3390/sym13040633

Abstract

In this paper, we study a class of anisotropic variable exponent problems involving the p→(.)-Laplacian. By using the variational method as our main tool, we present a result regarding the existence of solutions without the so-called Ambrosetti–Rabinowitz-type conditions.

Highlights

In the early 1990s, the first anisotropic PDE model was proposed by the authors of [16], which was used for both image enhancement and denoising in terms of anisotropic PDEs as well as allowing the preservation of significant image features
We show that the mathematical model of homogeneous anisotropic elastic media movement can be introduced by dynamic system equations of elasticity; it is presented as a symmetrical hyperbolic system of the first order in term of velocity
We study the anisotropic nonlinear elliptic problem of the form

Summary

Introduction

The investigation of anisotropic problems has drawn the attention of many authors; for example, see the works presented in [1–15] and the references therein. | ∂ x i u | p i −2 ∂ x i u i =1 gives us another behavior for partial derivatives in several directions This differential operator involving a variable exponent can be regarded as an extension of the p( x )−Laplace operator for the anisotropic case; as far as we are aware, ∆ p ( x ) is not homogeneous, and so the p( x )−Laplacian has more complicated properties than the p−Laplacian. In [9], using an embedding theorem involving the critical exponent of anisotropic type, the authors presented some results regarding the existence and nonexistence of the following anisotropic quasilinear elliptic problem:.

Preliminaries

Main Results

Proofs

Conclusions

Methods