Regularity of entropy solutions of quasilinear elliptic problems related to Hardy-Sobolev inequalities

This article is concerned with the regularity of the entropy solution of -div (\x\(-gamma p)\del u\(p-2)del u) = f(x) in Omega, u = 0 on partial derivative Omega, where Omega is a smooth bounded domain Omega of R-N such that 0 is an element of Omega, 1 < p < N, and gamma < (N - p) /p. Assuming f is an element of L-q (Omega, \x\(alpha(q-1)) dx) for some q >= 1 and N gamma p/ (N - p) <= alpha <= (gamma + 1)p, we obtain estimates for the entropy solution u and its weak gradient in Lebesgue spaces with weights. Moreover, we introduce some explicit examples showing the optimality of our results and a relation between our problem and a Hardy-Sobolev type inequality.