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

Abstract. We consider the problem u + a(x)u = f(x) juj22 u in u = 0 on @; where a bounded smooth domain in RN, N 4, 2 := 2N N2 is the critical Sobolev exponent, and a; f are continuous functions. We assume that; 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 fo