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

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