On some anisotropic elliptic problem with measure data

Azraibia, Ouidad; Abdelkarim, Derham; El Haji, Badr

doi:10.22075/ijnaa.2024.33483.4992

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

10.22075/ijnaa.2024.33483.4992

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

References

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