International Journal of Nonlinear Analysis and Applications

Existence of a solution for a strongly nonlinear elliptic perturbed problem in anisotropic Orlicz-Sobolev space

Document Type : Research Paper

Authors

Equipe EDP et calcul scientifique, laboratoire de mathematiques et leurs interactions, Faculte des Sciences, Moulay Ismail University, Meknes, Morocco

Abstract

This paper is devoted to studying the existence of a solution to the  Dirichlet problem for a specific class of elliptical anisotropic equations  of the type
\begin{eqnarray}\label{P.1}
\left \{\begin{array}{rl}
&A(u)+g(x,u)= f  \ \  in\
\Omega
\\
& u=0\ \ on\ \partial \Omega,
\end{array}
\right.
\end{eqnarray}
in the anisotropic Orlicz-Sobolev spaces, where A is a Leray-Lions operator  $A(u)=\displaystyle\sum_{i=1}^{N}-\frac{\partial}{\partial x_{i}} (a_{i}(x,D^{i} u)),$   the Carathéodory function $g(x, s )$ is a non-linear lower order term that verify some natural growth and sign conditions, where the data $f$ is framed in anisotropic Orlicz-Sobolev spaces, and it is described by an Orlicz function that does not meet the $\Delta_2$-condition. Within this framework, we prove the existence of a weak solution for our strongly nonlinear elliptic problem.

Keywords

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International Journal of Nonlinear Analysis and Applications
Volume 16, Issue 4
April 2025
Pages 161-168
  • Receive Date: 28 January 2024
  • Accept Date: 19 March 2024