INFINITELY MANY SIGN-CHANGING SOLUTIONS FOR THE BREZIS-NIRENBERG PROBLEM INVOLVING HARDY POTENTIAL

In this article, we give a new proof on the existence of infinitely many sign changing solutions for the following Brezis-Nirenberg problem with critical exponent and a Hardy potential -Delta u - mu u/vertical bar x vertical bar(2) = lambda u + vertical bar u vertical bar(2*-2)u in Omega, u = 0 on partial derivative Omega, where Omega is a smooth open bounded domain of R-N which contains the origin, 2* =2N/N-2 is the critical Sobolev exponent. More precisely, under the assumptions that N >= 7, mu [0, (mu) over bar - 4), and (mu) over bar = (N-2)(2)/4, we show that the problem admits infinitely many sign-changing solutions for each fixed lambda > 0. Our proof is based on a combination of invariant sets method and Ljusternik-Schnirelman theory.