International Journal of Nonlinear Analysis and Applications

On some anisotropic elliptic problem with measure data

Document Type : Research Paper

Authors

1 Laboratory LAMA, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, Sidi Mohammed Ben Abdallah University, PB 1796 Fez-Atlas, Fez, Morocco

2 Laboratory LaR2A, Departement of Mathematics, Faculty of Sciences Tetouan, Abdelmalek Essaadi University, BP 2121, Tetouan, Morocco

Abstract

We prove optimal existence results for entropy solutions to some anisotropic boundary value problems like
\begin{equation}\label{pro}
\left\{\begin{array}{lll}
-\sum_{i=1}^N D^i A_i(x, w, \nabla w)= f-\operatorname{div} F(w) \textrm{ in }\Omega, & \textrm{in }&\Omega, \\
v=0 & \textrm{on } &\partial \Omega,
\end{array}\right.
\end{equation}
where $ f \in L^{1}(\Omega) $, $ F = (F_{1}, . . . , F_{N}) $ satisfies $ F \in (C^{0}(\mathbb{R}))^{N}. $and $\Omega $ is a bounded, open subset of ${\mathbb{R}^{N}}$, $ N\geq 2$, and the function $A_{i}(x, s, \xi)$ verify the large monotonicity condition. The construction of the proof of our theorem is done by using Minty's Lemma in its modified version.

Keywords

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International Journal of Nonlinear Analysis and Applications
Volume 16, Issue 5
May 2025
Pages 1-11
  • Receive Date: 09 February 2024
  • Revise Date: 18 March 2024
  • Accept Date: 19 March 2024