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Fast and slow decay solutions for supercritical elliptic problems in exterior domains

Authors:	Juan Dávila Manuel del Pino Monica Musso Juncheng Wei

Institution:	1.Departamento de Ingeniería Matemática and CMM,Universidad de Chile,Santiago,Chile;2.Departamento de Matemática,Pontificia Universidad Católica de Chile,Macul,Chile;3.Dipartimento di Matematica,Politecnico di Torino,Torino,Italy;4.Department of Mathematics,Chinese University of Hong Kong,Shatin,Hong Kong

Abstract:	We consider the elliptic problem Δu + u ^p = 0, u > 0 in an exterior domain, ${\Omega = \mathbb{R}^N{\setminus}\mathcal{D}}$ under zero Dirichlet and vanishing conditions, where ${\mathcal{D}}$ is smooth and bounded in ${\mathbb{R}^N}$ , N ≥ 3, and p is supercritical, namely ${p > \frac{N+2}{N-2}}$ . We prove that this problem has infinitely many solutions with slow decay ${O(\|x\|^{-\frac2{p-1}})}$ at infinity. In addition, a solution with fast decay O(\|x\|^2-N) exists if p is close enough from above to the critical exponent.

Keywords:	Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 35J60 35B20 35B33 35J20
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