Regularity of Entropy Solutions of Quasilinear Elliptic Problems Related to Hardy-Sobolev Inequalities

Abstract

Abstract This article is concerned with the regularity of the entropy solution of where Ω is a smooth bounded domain Ω of ℝN such that 0 ∈ Ω, 1 < p < N. and γ < (N-p)/p. Assuming f ∈ Lq(Ω, |x|α(q-1)dx) for some q ≥ 1 and Nγp/(N-p) ≤ α ≤ (γ+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.