Measure Data for a General Class of Nonlinear Elliptic Problems

We consider nonlinear elliptic inclusion having a measure in the right-hand side of the type $\beta (u)-diva(x,Du)\ni \mu$ in $\mathrm{\Omega }$ a bounded domain in ${\mathbb{R}}^N,$ with $\beta$ is a maximal monotone graph in ${\mathbb{R}}^2$ and $a(x,Du)$ is a Leray-Lions type operator. We study a suitable notion of solution for this kind of problem. The functional setting involves anisotropic Sobolev spaces.