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

Δu = eu  in 2B,2Beu(x)dx < , (1)

where B = {x 2 : |x| < 1} is the unit ball of 2, and

Δ2u = eu  in 4B,4Beu(x)dx < , (2)

where B = {x 4 : |x| < 1} is the unit ball of 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(y) = u(x), y = x|x|2):

Δyv = |y|4ev  in B{0},B{0}|y|4ev(y)dy < (3)

and

Δy2v = |y|8ev  in B{0},B{0}|y|8ev(y)dy < , (4)

respectively. We obtain the exact asymptotic behavior of solutions of (1) and (2) as |x|. Meanwhile, we find that the singular solutions of the related problems (3) and (4) in B{0} are asymptotic radial solutions and obtain the corresponding asymptotic behavior as |y| 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
Authors
Zongming Guo
Department of Mathematics
Henan Normal University
Xinxiang
China
Zhongyuan Liu
School of Mathematics and Statistics
Henan University
Kaifeng
China