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 {a(u) + g(x, u) = f in ω u = 0 on ∂ω, (0.1) in the anisotropic orlicz-sobolev spaces, where a is a leray-lions operator a(u) = σ n i=1 − ∂/∂xi (ai(x, diu)), 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 ∆2-condition. within this framework, we prove the existence of a weak solution for our strongly nonlinear elliptic problem.