On On the Existence of Entropy Solutions for Some Nonlinear Elliptic Problem in the Setting of Generalized Sobolev Spaces

Abstract

The novelty of this note is to establish  existence result for the following  anisotropic  elliptic equation $$ -\operatorname{div} B(x, \vartheta, \nabla \vartheta)=f $$ where the datum  $f \in L^1(\Omega)$. Furthermore only the large monotonicity conditions will be assumed on $B(x, s, \xi)$. To overcome this difficulty we will  use the approach of Minty's lemma in the anisotropic weighted Sobolev spaces.