Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case

Yongxiu Shi; Haitao Wan; Yongxiu Shi; Haitao Wan

doi:10.3934/era.2020119

Electronic Research Archive

2021, Volume 29, Issue 3: 2359-2373. doi: 10.3934/era.2020119

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Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case

Yongxiu Shi ,
Haitao Wan ^,

School of Mathematics and Information Science, Shandong Technology and Business University, Yantai 264005, China

Received: 01 August 2020 Revised: 01 October 2020 Published: 24 November 2020
Primary: 35J25, 35B40; Secondary: 35J62

This paper is considered with the quasilinear elliptic equation $ \Delta_{p}u = b(x)f(u),\,u(x)>0,\,x\in\Omega, $ where $ \Omega $ is an exterior domain with compact smooth boundary, $ b\in \rm C(\Omega) $ is non-negative in $ \Omega $ and may be singular or vanish on $ \partial\Omega $, $ f\in C[0, \infty) $ is positive and increasing on $ (0, \infty) $ which satisfies a generalized Keller-Osserman condition and is regularly varying at infinity with critical index $ p-1 $. By structuring a new comparison function, we establish the new asymptotic behavior of large solutions to the above equation in the exterior domain. We find that the lower term of $ f $ has an important influence on the asymptotic behavior of large solutions. And then we further establish the uniqueness of such solutions.
- p-Laplacian,
- large solutions,
- the exact asymptotic behavior,
- uniqueness,
- exterior domains
Citation: Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case[J]. Electronic Research Archive, 2021, 29(3): 2359-2373. doi: 10.3934/era.2020119

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Abstract

This paper is considered with the quasilinear elliptic equation $ \Delta_{p}u = b(x)f(u),\,u(x)>0,\,x\in\Omega, $ where $ \Omega $ is an exterior domain with compact smooth boundary, $ b\in \rm C(\Omega) $ is non-negative in $ \Omega $ and may be singular or vanish on $ \partial\Omega $, $ f\in C[0, \infty) $ is positive and increasing on $ (0, \infty) $ which satisfies a generalized Keller-Osserman condition and is regularly varying at infinity with critical index $ p-1 $. By structuring a new comparison function, we establish the new asymptotic behavior of large solutions to the above equation in the exterior domain. We find that the lower term of $ f $ has an important influence on the asymptotic behavior of large solutions. And then we further establish the uniqueness of such solutions.

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