Revisiting an idea of Brezis and Nirenberg

Let n >= 3 and Omega be a C-1 bounded domain in R-n with 0 is an element of partial derivative Omega. Suppose partial derivative Omega is C-2 at 0 and the mean curvature of partial derivative Omega at 0 is negative, we prove the existence of positive solutions for the equation: {Delta u + lambda u(n+2/n-2) + u(2*(s)-1)/vertical bar x vertical bar(s) = 0 in Omega, u = 0 on partial derivative Omega, where lambda > 0, 0 < s < 2, 2*(s) = (2(n-s)n-2) and n >= 4. For n = 3, the existence result holds for 0 < s < 1. Under the same assumption of the domain Omega, for p <= 2* (s) - 1, we also prove the existence of a positive solution for the following equation: {Delta u + lambda u(p) + u(2*(s)-1)/vertical bar x vertical bar(s) = 0 in Omega, u = 0 on partial derivative Omega, where lambda > 0 and 1 <= p N n/n-2 (C) 2010 Elsevier Inc. All rights reserved.