Multiple solutions for the Brezis-Nirenberg problem

Abstract

We establish the existence of multiple solutions to the Dirichlet problem for the equation \[ -\Delta u=\lambda u+|u|^{\frac{4}{N-2}}u \] on a bounded domain $\Omega$ of $\mathbb{R}^{N},$ $N\geq4.$ We show that, if $\lambda>0$ is not a Dirichlet eigenvalue of $-\Delta$ on $\Omega,$ this problem has at least $\frac{N+1}{2}$ pairs of nontrivial solutions. If $\lambda$ is an eigenvalue of multiplicity $m$ then it has at least $\frac{N+1-m}{2}$ pairs of nontrivial solutions.