Existence Results for Nonlinear Anisotropic Elliptic Equation - Youssef Akdim; Mostafa El moumni; Abdelhafid Salmani (Advances in Science, Technology and Engineering Systems Journal, 2017)
In this work, we shall be concerned with the existence of weak solutions of anisotropic elliptic operators [latex]Au +\sum_{i=1}^{N}g_{i}(x, u, \nabla u)+\sum_{i=1}^{N}H_{i}(x, \nabla u)=f-\sum_{i=1}^{N} \frac{\partial }{\partial x_{i}}k_{i}[/latex] where the right hand side [latex]f[/latex] belongs to [latex]L^{p^{'}_{∞}}(Ω)[/latex] and [latex]k_{i}[/latex] belongs to [latex]L^{p_{i}^{'}}(Ω)[/latex] for [latex]i=1,...,N[/latex] and [latex]A[/latex] is a Leray-Lions operator. The critical growth condition on [latex]g_{i}[/latex] is the respect to [latex]\nabla u[/latex] and no growth condition with respect to u, while the function [latex]H_{i}[/latex] grows as [latex]|\nabla u|^{p_{i}-1}[/latex].
Keywords: Anisotropic elliptic equations; Weak solutions; Nonlinear operators