SUPERCRITICAL ELLIPTIC PROBLEMS FROM A PERTURBATION VIEWPOINT

Abstract

Abstract. We review some recent existence results for the elliptic problem u+up = 0, u > 0 in an exterior domain, = RN \D under zero Dirichlet and vanishing conditions, where D is smooth and bounded, and p > N+2 N−2 . We prove that the associated Dirichlet problem has infinitely many positive solutions. We establish analogous results for the standing-wave supercritical nonlinear Schr¨odinger equation u− V (x)u + up = 0 where V 0 and V (x) = o(|x|−2) at infinity. In addition we present existence results for the Dirichlet problem in bounded domains with a sufficiently small spherical hole if p differs from certain sequence of resonant values which tends to infinity.