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

Abstract

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 $$\Delta _{2}$$ -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 $$L^{1}$$ , 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.