Summability of the Solutions of Nonlinear Elliptic Equations with Right Hand Side Measures

We prove a regularity result for weak solutions of nonlinear elliptic problems of the type $$-\text{div}(a(x,\nabla u)) = \mu,\ \ u \in W^{1,1}_0(\Omega)$$ using an approximation technique. Here $\mu$ is a bounded Radon measure and the operator $A(u) = -\text{div}(a(x,\nabla u))$ is assumed to be coercive and monotone, acting between $W^{1,p}_0(\Omega)$ and $W^{-1,p'}_0(\Omega)$.