American Institute of Mathematical Sciences

doi: 10.3934/era.2020119

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

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

* Corresponding author: Haitao Wan

Received  August 2020 Revised  October 2020 Published  November 2020

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.

Citation: Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, doi: 10.3934/era.2020119
References:
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References:
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