  We consider nonlinear elliptic inclusion having a measure in the right-hand
side of the type $\beta(u)-div a(x,Du)\ni \mu$ in $\Omega$ 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.
