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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.