### Electronic Research Archive

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

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

• 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.

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

### Related Papers:

• 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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