Entropy solutions of anisotropic elliptic nonlinear obstacle problem with measure data

Abstract

We prove the existence of an entropy solution for a class of nonlinear anisotropic elliptic unilateral problem associated to the following equation

$$\begin{aligned} -\sum _{i=1}^{N} \partial _i a_i(x,u, \nabla u) -\sum _{i=1}^{N}\partial _{i}\phi _{i}( u)=\mu , \end{aligned}$$

where the right hand side $\mu$ belongs to $L^{1}(\Omega )+ W^{-1, \vec {p'}}(\Omega )$. The operator $-\sum _{i=1}^{N} \partial _i a_i(x,u, \nabla u)$ is a Leray–Lions anisotropic operator and $\phi _{i} \in C^{0}({\mathbb {R}}, {\mathbb {R}})$.