On an elliptic problem with critical exponent and Hardy potential

In this paper, we study the following elliptic problem with critical exponent and a Hardy potential: -Delta u - mu/vertical bar x vertical bar(2) u = lambda u + vertical bar u vertical bar(2+) (- 2) u, u is an element of H-0(1) (Omega) where is Omega smooth open bounded domain in R-N (N >= 3) which contains the origin and 2* is the critical Sobolev exponent. We show that, if N >= 5 and is mu is an element of (0, (N-2/2)(2) - (N+2/N)(2)), this problem has a ground state solution for each fixed lambda > 0. Moreover, we give energy estimates from below and bounds on the number of nodal domains for these ground state solutions. If N >= 7 and mu is an element of (0, (N-2/2)(2) - 4), this problem has infinitely many sign-changing solutions for each fixed lambda > 0. (C) 2011 Elsevier Inc. All rights reserved.