Supercritical elliptic problems from a perturbation viewpoint

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January 2008, 21(1): 69-89. doi: 10.3934/dcds.2008.21.69

Supercritical elliptic problems from a perturbation viewpoint

1.	Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

Received March 2007 Published February 2008

Abstract
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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.

Keywords: supercritical elliptic problems, exterior domains., Critical Sobolev exponent.

Mathematics Subject Classification: Primary: 35J25, 35J2.

Citation: Manuel del Pino. Supercritical elliptic problems from a perturbation viewpoint. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 69-89. doi: 10.3934/dcds.2008.21.69

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