Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2011; 48(2): 431-447

Printed March 1, 2011

https://doi.org/10.4134/JKMS.2011.48.2.431

Copyright © The Korean Mathematical Society.

Asymptotic behavior of positive solutions to semilinear elliptic equations in $\mathbb R^n$

Baishun Lai, Qing Luo, and Shuqing Zhou

Henan University, Henan University, Hunan Normal University

Abstract

We investigate the asymptotic behavior of positive solutions to the elliptic equation \begin{equation} \Delta u+|x|^{l_{1}}u^{p}+|x|^{l_{2}}u^{q}=0\ \mbox{in}\ \ \mathbb R^n.\ \end{equation} We obtain a conclusion that, for $n\geq 3, -2 < l_{2} < l_{1}\leq 0$ and $q > p >1$, any positive radial solution to (0.1) has the following properties: $\lim_{r\to\infty}r^{\frac{2+l_{1}}{p-1}}u$ and $\lim_{r\to0}r^{\frac{2+l_{2}}{q-1}}u$ always exist if $\frac{n+l_{1}}{n-2} < p < q, \ \ p\neq\frac{n+2+2l_{1}}{n-2},\ \ q \neq\frac{n+2+2l_{2}}{n-2}.$ In addition, we prove that the singular positive solution of (0.1) is unique under some conditions.

Keywords: semilinear elliptic equation, positive solutions, asymptotic behavior, singular solutions

MSC numbers: Primary 35J60; Secondary 35B05, 35B40