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

We consider the problem −Δu+a(x)u=f(x)|u|2*−2u in Ω, u=0 on ∂Ω, where Ω is a bounded smooth domain in RN, N⩾4, 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 −Δu+λu=|u|2*−2u in Ω, u=0 on ∂Ω.