Ground state and multiple solutions for a critical exponent problem

We study the following Brezis?CNirenberg type critical exponent equation which is related to the Yamabe problem: $$-Delta u=lambda u+ |u|^{2^{ast}-2}u, quad uin H_0^1 (Omega),$$ where ?? is a smooth bounded domain in ${{mathbb R}^N(Nge3)}$ and 2^* is the critical Sobolev exponent. We show that, if N ?? 5, this problem has at least ${lceilfrac{N+1}{2}rceil}$ pairs of nontrivial solutions for each fixed ?? ?? ??₁, where ??₁ is the first eigenvalue of ??? with the Dirichlet boundary condition. For N ?? 3, we give energy estimates from below for ground state solutions.