Ground state and multiple solutions for a critical exponent problem

Abstract

We study the following Brezis-Nirenberg type critical exponent equation which is related to the Yamabe problem, where Ω 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 ⌈N+1/2⌉ pairs of nontrivial solutions for each fixed λ ≥ λ 1, where λ 1 is the first eigenvalue of -Δ with the Dirichlet boundary condition. For N ≥ 3, we give energy estimates from below for ground state solutions. © 2011 Springer Basel AG.