Asymptotic behavior of solutions for some elliptic equations in exterior domains

Zongming Guo and Zhongyuan Liu

Vol. 309 (2020), No. 2, 333–352

DOI: 10.2140/pjm.2020.309.333

Abstract

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

$$-\Delta u=e^u\text{ in }{ℝ}^2\setminus B,{\int \; <!--nolimits-->}_{{ℝ}^2\setminus B}{e}^{u\left(x\right)}dx<\infty ,$$

(1)

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

$${\Delta }^{2}u=e^u\text{ in }{ℝ}^4\setminus B,{\int \; <!--nolimits-->}_{{ℝ}^4\setminus B}{e}^{u\left(x\right)}dx<\infty ,$$

(2)

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 $v\left(y\right)=u\left(x\right)$, $y=x∕|x{|}^2$):

$$-{\Delta }_{y}v=|y{|}^{-4}{e}^v\text{ in }B\setminus \left\{0\right\},{\int \; <!--nolimits-->}_{B\setminus \left\{0\right\}}|y{|}^{-4}{e}^{v\left(y\right)}dy<\infty$$

(3)

and

$${\Delta }_y^{2}v=|y{|}^{-8}{e}^v\text{ in }B\setminus \left\{0\right\},{\int \; <!--nolimits-->}_{B\setminus \left\{0\right\}}|y{|}^{-8}{e}^{v\left(y\right)}dy<\infty ,$$

(4)

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 $\left|y\right|\to 0$.

Keywords

asymptotic behavior of solutions, semilinear biharmonic problems, exterior domains, singular solutions

Mathematical Subject Classification

Primary: 35B45, 35J40

Milestones

Received: 30 April 2020

Revised: 26 August 2020

Accepted: 29 August 2020

Published: 14 January 2021