Asymptotic Behaviors of Solutions to quasilinear elliptic Equations with critical Sobolev growth and Hardy potential

TL;DR

This paper investigates the asymptotic behavior of solutions to a class of quasilinear elliptic equations with critical Sobolev growth and Hardy potential, providing optimal estimates at both the origin and infinity.

Contribution

It derives precise asymptotic estimates for weak solutions of quasilinear elliptic equations involving Hardy potential and critical Sobolev growth, extending understanding of their behavior.

Findings

Established optimal asymptotic estimates at the origin and infinity.

Analyzed solutions under conditions on the Hardy potential parameter.

Extended previous results to more general quasilinear equations.

Abstract

Optimal estimates on the asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to 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 $Q\in L^{\infty}(\mathbb{R}^{N})$.