Gradient Estimates for Solutions To Quasilinear Elliptic Equations with Critical Sobolev Growth and Hardy Potential

This note is a continuation of the work \cite{CaoXiangYan2014}. We study the following quasilinear elliptic equations $$-{\mathrm{\Delta }}_{p}u-\frac{\mu }{\mathrm{\mid }x{\mathrm{\mid }}^p}\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u=Q(x)\mathrm{\mid }u{\mathrm{\mid }}^{\frac{Np}{N-p}-2}u,x\in {\mathbb{R}}^N,$$ where $1<p<N,0\le \mu <{((N-p)\mathrm{/}p)}^p$ and $Q\in L^{\mathrm{\infty }}({\mathbb{R}}^N)$. Optimal asymptotic estimates on the gradient of solutions are obtained both at the origin and at the infinity.