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

TL;DR

This paper derives optimal gradient estimates for solutions to a class of quasilinear elliptic equations with critical Sobolev growth and Hardy potential, extending previous work and providing precise asymptotic behavior at zero and infinity.

Contribution

It provides the first optimal asymptotic gradient estimates for solutions to these equations with Hardy potential and critical Sobolev growth.

Findings

Established gradient bounds at the origin and infinity.

Extended previous results to include Hardy potential effects.

Provided precise asymptotic behavior of solutions' gradients.

Abstract

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