Entropy solutions of anisotropic elliptic nonlinear obstacle problem with measure data

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 $$phi _{i} in C^{0}({mathbb {R}}, {mathbb {R}})$$.