Existence and uniqueness of entropy solution of a nonlinear elliptic equation in anisotropic Sobolev–Orlicz space

Our objective in this paper is to study a certain class of anisotropic elliptic equations with the second term, which is a low-order term and non-polynomial growth; described by an N-uplet of N-function satisfying the Δ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{2}$$\end{document}-condition in the framework of anisotropic Orlicz spaces. We prove the existence and uniqueness of entropic solution for a source in the dual or in L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{1}$$\end{document}, without assuming any condition on the behaviour of the solutions when x tends towards infinity. Moreover, we are giving an example of an anisotropic elliptic equation that verifies all our demonstrated results.