Existence and Uniqueness of Renormalized Solution to Nonlinear Anisotropic Elliptic Problems with Variable Exponent and L 1 -Data

Abstract

Nonlinear partial differential equations are considered as an essential tool for describing the behavior of many natural phenomena. The modeling of some phenomena requires to work in Sobolev spaces with constant exponent. But for others, such as electrorheological fluids, the properties of classical spaces are not sufficient to have precision. To overcome this difficulty, we work in the appropriate spaces called Lebesgue and Sobolev spaces with variable exponent. In recent works, researchers are attracted by the study of mathematical problems in the context of variable exponent. This great interest is motivated by their applications in many fields such as elastic mechanics, fluid dynamics, and image restoration. In this paper, we combine the technic of monotone operators in Banach spaces and approximation methods to prove the existence of renormalized solutions of a class of nonlinear anisotropic problem involving $\stackrel{⟶}{p}\left(.\right)-$ Leray–Lions operator, a graph, and $L^1$ data. In particular, we establish the uniqueness of the solution when the graph data are considered a strictly increasing function.