We study the existence of nonnegative solutions to the Dirichlet problem L M p,q u := −∆u+u p −M |∇u| q = µ in a domain Ω ⊂ R N where µ is a nonnegative Radon measure, when p > 1, q > 1 and M ≥ 0. We also give conditions under which nonnegative solutions of L M p,q u = 0 in Ω \ K where K is a compact subset of Ω can be extended as a solution of the same equation in Ω.