Existence of a renormalized solution for some nonlinear anisotropic elliptic problems

Abstract

In this paper, we study a general class of nonlinear anisotrpic elliptic problems associated with the differential inclusion β(u) - div(a(x, D(u) + F(u)) ∋ f in Ω, where f ∈ L∞(Ω). A vector field a(. , .) is a Caratheodory function. Using trunction techniques and the generalized monotonicity method in the functional spaces we prove the existence of renormalized solutions for L∞-data.

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

“…Reasoning as in [25] (see also [21,32]), we can prove that the operator A ϵ is pseudomonotone, coercive, and bounded. Ten, we deduce from [38] (Teorem 2.7) that A ϵ is surjective.…”

“…Taking into account the monotonicity of a i and β ϵ and following the same lines as in [21,32], we establish the following comparison principle which will be essential in the proof of uniqueness of the solution.…”

“…In this paper, we extend the recent works [21,24,25,32] by using the ideas developed in [21,32]. More precisely, we used the technic of monotone operators in Banach spaces and approximation methods to prove the existence and uniqueness of a renormalized solution of problem (E, f) in the context of anisotropic space involving variable exponents W 1, p → (.)…”

“…. , p N ) are constants, the authors in [32] studied the problem (E, f) and established the existence and uniqueness of renormalized solution in the anisotropic Sobolev space W (E, f) is more general than the one which appears in [32].…”

“…in Ω. Indeed, the proof relies on Lemma 4.6 and subdiferential argument (see [21,32,39] for details).…”