Isolated singularities of solutions of certain quasi-linear elliptic inequalities

Abstract: We provide a complete classification of the asymptotic behavior of isolated singularities for solutions satisfying [ 0\le-\Delta_{p}u(x)\le \tau u^{\frac{n(p-1)}{n-p}} (x),\,\,u(x)\ge0,\,\,1<p<n,\,\,n\ge2, ]where $u(x)\in C^2(B(0,1)\backslash{0})$ and $B(0,1)\subset \mathbb R^n$.