Ground state and multiple solutions for a critical exponent problem

We study the following Brezis-Nirenberg type critical exponent equation which is related to the Yamabe problem: -Delta u = lambda u + vertical bar u vertical bar(2*-2)u, u is an element of H-0(1)(Omega), where Omega is a smooth bounded domain in R-N (N >= 3) and 2* is the critical Sobolev exponent. We show that, if N >= 5, this problem has at least inverted right perpendicularN+1/2inverted left perpendicular pairs of nontrivial solutions for each fixed lambda >= lambda(1), where lambda(1) is the first eigenvalue of -Delta with the Dirichlet boundary condition. For N >= 3, we give energy estimates from below for ground state solutions.