#### Vol. 309, No. 2, 2020

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Asymptotic behavior of solutions for some elliptic equations in exterior domains

### Zongming Guo and Zhongyuan Liu

Vol. 309 (2020), No. 2, 333–352
##### Abstract

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

 (1)

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

 (2)

where $B=\left\{x\in {ℝ}^{4}:\phantom{\rule{0.28em}{0ex}}|x|<1\right\}$ 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 $v\left(y\right)=u\left(x\right)$, $y=x∕|x{|}^{2}$):

 (3)

and

 (4)

respectively. We obtain the exact asymptotic behavior of solutions of (1) and (2) as $|x|\to \infty$. 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 $|y|\to 0$.

##### Keywords
asymptotic behavior of solutions, semilinear biharmonic problems, exterior domains, singular solutions
##### Mathematical Subject Classification
Primary: 35B45, 35J40