Abstract

This paper is concerned with the asymptotic behavior of solutions of the problems

where $B=\left\{x\in {ℝ}^2:\left|x\right|<1\right\}$ is the unit ball of ${ℝ}^2$, and

where $B=\left\{x\in {ℝ}^4:\left|x\right|<1\right\}$ is the unit ball of ${ℝ}^4$. It is seen that the asymptotic behavior of solutions for (1) and (2) is equivalent to the asymptotic behavior of singular solutions of the related problems (via the transformation $v\left(y\right)=u\left(x\right)$, $y=x∕|x{|}^2$):

and

respectively. We obtain the exact asymptotic behavior of solutions of (1) and (2) as $\left|x\right|\to \infty$. Meanwhile, we find that the singular solutions of the related problems (3) and (4) in $B\setminus \left\{0\right\}$ are asymptotic radial solutions and obtain the corresponding asymptotic behavior as $\left|y\right|\to 0$.

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Mathematical Subject Classification

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