A nonlinear degenerate elliptic problem with W^{1,1}_0 solutions

We study a nonlinear equation with an elliptic operator having degenerate coercivity. We prove the existence of a unique W^{1,1}_0 distributional solution under suitable summability assumptions on the source in Lebesgue spaces. Moreover, we prove that our problem has no solution if the source is a Radon measure concentrated on a set of zero harmonic capacity.