Fourth-order problems with Leray-Lions type operators in variable exponent spaces

Maria-Magdalena Boureanu

doi:10.3934/dcdss.2019016

Abstract

The Leray-Lions operators are versatile enough to be particularized to various elliptic operators, so they receive a lot of attention. This paper introduces to the mathematical literature Leray-Lions type operators that are appropriate for the study of the variable exponent problems of higher order. We establish some properties concerning these general operators and then we apply them to a fourth order problem with variable exponents.

Highlights

The variable exponent problems have numerous applications, and we can refer to those involving non-Newtonian fluids [30], elastic materials
Even if we do not take into account the fact that we work with variable exponents, our hypotheses on the Leray-Lions type operators are more general in the sense that in [31] the authors impose in addition the condition a(x, t)t ≤ pA(x, t) for a.e x ∈ Ω and all t ∈ R
To give an example of an elliptic problem in which the results proved in the previous section are valuable, we consider the following problem with Navier boundary condition:

Summary

Introduction

The variable exponent problems have numerous applications, and we can refer to those involving non-Newtonian fluids [30], elastic materials [33, 5], image restoration [8] etc. Until now there has not been considered a fourth-order elliptic problem with variable exponents and Leray-Lyons type operators, so we intend to fill this gap.

Objectives

Results

Conclusion