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

Document Type : Research Paper


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



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
\left \{\begin{array}{rl}
&A(u)+g(x,u)= f  \ \  in\
& u=0\ \ on\ \partial \Omega,
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.


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Articles in Press, Corrected Proof
Available Online from 16 May 2024
  • Receive Date: 28 January 2024
  • Accept Date: 19 March 2024