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

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.