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ASYMPTOTIC BEHAVIOR OF POSITIVE SOLUTIONS TO SEMILINEAR ELLIPTIC EQUATIONS IN ℝn

  • Lai, Baishun (INSTITUTE OF CONTEMPORARY MATHEMATICS HENAN UNIVERSITY, SCHOOL OF MATHEMATICS HENAN UNIVERSITY) ;
  • Luo, Qing (SCHOOL OF MATHEMATICS HENAN UNIVERSITY) ;
  • Zhou, Shuqing (SCHOOL OF MATHEMATICS AND COMPUTER SCIENCE HUNAN NORMAL UNIVERSITY)
  • Received : 2009.11.11
  • Published : 2011.03.01

Abstract

We investigate the asymptotic behavior of positive solutions to the elliptic equation (0.1) ${\Delta}u+|x|^{l_1}u^p+|x|^{l_2}u^q=0$ in $\mathbb{R}^n$. 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: $lim_{r{\rightarrow}{\infty}}r^{\frac{2+l_1}{p-1}}\;u$ and $lim_{r{\rightarrow}0}r^{\frac{2+l_2}{q-1}}\;u$ always exist if $\frac{n+1_1}{n-2}$ < 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.

Acknowledgement

Supported by : Natural Science Foundation of China, Hunan Normal University

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