ASYMPTOTIC BEHAVIOR OF POSITIVE SOLUTIONS TO SEMILINEAR ELLIPTIC EQUATIONS IN ℝn

Title & Authors
ASYMPTOTIC BEHAVIOR OF POSITIVE SOLUTIONS TO SEMILINEAR ELLIPTIC EQUATIONS IN ℝn
Lai, Baishun; Luo, Qing; Zhou, Shuqing;

Abstract
We investigate the asymptotic behavior of positive solutions to the elliptic equation (0.1) $\small{{\Delta}u+|x|^{l_1}u^p+|x|^{l_2}u^q=0}$ in $\small{\mathbb{R}^n}$. We obtain a conclusion that, for n $\small{\geq}$ 3, -2 < $\small{l_2}$ < $\small{l_1}$ $\small{\leq}$ 0 and q > p > 1, any positive radial solution to (0.1) has the following properties: $\small{lim_{r{\rightarrow}{\infty}}r^{\frac{2+l_1}{p-1}}\;u}$ and $\small{lim_{r{\rightarrow}0}r^{\frac{2+l_2}{q-1}}\;u}$ always exist if $\small{\frac{n+1_1}{n-2}}$ < p < q, $\small{p\;{\neq}\;\frac{n+2+2l_1}{n-2}}$, $\small{q\;{\neq}\;\frac{n+2+2l_2}{n-2}}$. 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;
Language
English
Cited by
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