Isolated singularities of certain quasi-linear elliptic inequalities II

In this paper, we study the asymptotic behavior of isolated singularities for quasilinear elliptic equations of $p$-Laplacian type. We focus on the $p$-Laplace equation with the critical Serrin exponent $n(p-1)\mathrm{/}(n-p)$ and the $n$-Laplace equation with critical exponential growth. We prove that any positive solution either admits a $C^{1,\alpha }$ extension to the origin or exhibits a precise singular asymptotic behavior. These results provide a complement to our previous study on $p$-Laplacian type inequalities.