Multiple solutions to the pure critical exponent problem in domains with a hole of arbitrary size

We consider the problem -Delta u = vertical bar u vertical bar(2*-2) u in Omega, u = 0 on partial derivative Omega, where Omega is a bounded smooth domain in R-N, N >= 3, and 2* = 2N/N- 2 is the critical Sobolev exponent. We assume that Omega is annular shaped, i.e. there are constants R-2 > R-1 > 0 such that {x epsilon R-N : R-1 < vertical bar x vertical bar < R-2} subset of Omega and 0 is not an element of Omega. We also assume that Omega is invariant under a group Gamma of orthogonal transformations of R-N without fixed points. We establish the existence of multiple sign changing solutions if, either Gamma is arbitrary and R-1/R-2 is small enough, or R-1/R-2 is arbitrary and the minimal Gamma-orbit of Omega is large enough. We believe this is the first existence result for sign changing solutions in domains with holes of arbitrary size. The proof takes advantage of the invariance of this problem under the group of Mobius transformations.