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: $$-\Delta u=\lambda u+ |u|^{2^{\ast}-2}u, \quad u\in H_0^1 (\Omega),$$ where Ω is a smooth bounded domain in $${{\mathbb R}^N(N\ge3)}$$ and 2* is the critical Sobolev exponent. We show that, if N ≥ 5, this problem has at least $${\lceil\frac{N+1}{2}\rceil}$$ pairs of nontrivial solutions for each fixed λ ≥ λ1, where λ1 is the first eigenvalue of −Δ with the Dirichlet boundary condition. For N ≥ 3, we give energy estimates from below for ground state solutions.