Supercritical elliptic problems from a perturbation viewpoint

We review some recent existence results for the elliptic problem $\Delta u + u^p =0$, $u>0$ in an exterior domain, $\Omega = \R^N\setminus \D$ under zero Dirichlet and vanishing conditions, where $\D$ is smooth and bounded, and $p>\frac{N+2}{N-2}$. We prove that the associated Dirichlet problem has infinitely many positive solutions. We establish analogous results for the standing-wave supercritical nonlinear Schrödinger equation $\Delta u - V(x)u + u^p = 0 $ where $V\ge 0$ 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.