On the elliptic equation Δu+K up = 0  in     $\mathbb{R}$n

Soohyun Bae

doi:10.3934/dcds.2013.33.555

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2013, Volume 33, Issue 2: 555-577. Doi: 10.3934/dcds.2013.33.555

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On the elliptic equation Δu+K u^p = 0 in $\mathbb{R}$ⁿ

Soohyun Bae^1,

1.
Faculty of Liberal Arts and Sciences, Hanbat National University, Daejeon, 305-719

Received: June 30, 2011

Revised: March 31, 2012

Published: January 2013

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Abstract

This paper deals with the elliptic equation Δu+K(|x|) u^p = 0 in $\mathbb{R}$ⁿ\{0} when $r^{-l}K(r)$ for $l>-2$ behaves monotonically near $r=0$ or $\infty$ with $r=|x|$. By the method of phase plane, we present a new proof for the structure of positive radial solutions, and analyze the asymptotic behavior at $\infty$. We also employ the approach to classify singular solutions in terms of the asymptotic behavior at $0$. In particular, when $p=\frac{n+2+2l}{n-2}$, we establish the uniqueness of solutions with asymptotic self-similarity at $0$ and at $\infty$, and the existence of multiple solutions of Delaunay-Fowler type at $0$ and $\infty$.

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Mathematics Subject Classification: Primary: 35J61, 35B40, 35C06; Secondary: 70K05.

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