Positive solutions to a semilinear elliptic equation with a Sobolev-Hardy term

In this paper, we study the existence of positive solutions to the following semilinear elliptic equation with a Sobolev-Hardy term {- Delta u - lambda u = u(2#-1)/vertical bar y vertical bar x is an element of Omega, u > 0, x is an element of Omega, (0.1) u is an element of H-0(1) (Omega), where Omega is a bounded domain with smooth boundary in R-N (N >= 3), x = (y, z) is an element of Omega subset of R-k x RN-k = R-N, 2 <= k < N, 2(#) := 2(N-1)/N-2 is the corresponding critical exponent and 0 < lambda < lambda(1) where lambda(1) is the first eigenvalue of -Delta in H-0(1) (Omega). When N >= 4, we prove that problem (0.1) has at least one positive solution by using the mountain-pass lemma and a global compactness result. The case N = 3 is quite different and we deal with this case by using the method in Jannelli (1999) [20] to prove the existence result. Moreover, we obtain the nonexistence result of (0.1) in a star shaped domain. Our main results extend a recent result of Castorina et al. (2009) [10] where lambda = 0 and Omega = R-N. Crown Copyright (C) 2011 Published by Elsevier Ltd. All rights reserved.