Multiple positive and 2-nodal symmetric solutions of elliptic problems with critical nonlinearity

Abstract

We consider the problem − Δ u + a ( x ) u = f ( x ) | u | 2 * − 2 u in Ω, u = 0 on ∂ Ω, where Ω is a bounded smooth domain in R N , N ⩾ 4 , 2 * : = 2 N N − 2 is the critical Sobolev exponent, and a , f are continuous functions. We assume that Ω, a and f are invariant under the action of a group of orthogonal transformations. We obtain multiplicity results which contain information about the symmetry and symmetry-breaking properties of the solutions, and about their nodal domains. Our results include new multiplicity results for the Brezis–Nirenberg problem − Δ u + λ u = | u | 2 * − 2 u in Ω, u = 0 on ∂ Ω.