A Note on Asymptotic Behaviors Of Solutions To Quasilinear Elliptic Equations with Hardy Potential

TL;DR

This paper derives optimal asymptotic estimates for weak solutions of a class of quasilinear elliptic equations with Hardy potential, at both the origin and infinity, enhancing understanding of their solution behaviors.

Contribution

It provides new optimal estimates on the asymptotic behavior of solutions to quasilinear elliptic equations involving Hardy potential, at both zero and infinity.

Findings

Established optimal decay and growth rates of solutions near the origin.

Determined asymptotic profiles of solutions at infinity.

Extended previous results to a broader class of equations with Hardy potential.

Abstract

Optimal estimates on asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations \begin{eqnarray*} -\Delta_{p}u-\frac{\mu}{|x|^{p}}|u|^{p-2}u+m|u|^{p-2}u=f(u), & & x\in\R^{N}, \end{eqnarray*} where $1<p<N$, $0\leq\mu<\left((N-p)/p\right)^{p}$, $m>0$ and $f$ is a continuous function.