Multiple solutions for the Brezis-Nirenberg problem

Abstract

We establish the existence of multiple solutions to the Dirichlet problem for the equation -δu = λu + |u|4/N-2 u on a bounded domain ω of ℝN, N ≥ 4. We show that, if λ > 0 is not a Dirichlet eigenvalue of -δ on ω, this problem has at least N+1/2 pairs of nontrivial solutions. If λ is an eigenvalue of multiplicity m then it has at least N+1-m/2 pairs of nontrivial solutions.