A Nonlinear Elliptic PDE with Two Sobolev-Hardy Critical Exponents

In this paper, we consider the following PDE involving two Sobolev-Hardy critical exponents, Delta u + lambda u(2)*((s1)-1)/vertical bar X vertical bar(s1) + u(2)*((s2)-1)/vertical bar X vertical bar(s2) = 0 in Omega, u = 0 on Omega, where 0 <= s(2) < s1 <= 2, 0 not equal lambda is an element of R and 0 is an element of partial derivative Omega. The existence (or nonexistence) for least-energy solutions has been extensively studied when s(1) = 0 or s(2) = 0. In this paper, we prove that if 0 < s(2) < s(1) < 2 and the mean curvature of partial derivative Omega at 0 H(0) < 0, then (0.1) has a least-energy solution. Therefore, this paper has completed the study of (0.1) for the least-energy solutions. We also prove existence or nonexistence of positive entire solutions of (0.1) with Omega = R-+(N) under different situations of s(1), s(2) and lambda.