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

Supercritical elliptic problems from a perturbation viewpoint

Manuel del Pino 1,

1. 

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

Received  March 2007 Published  February 2008

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