Uniqueness of Positive Radial Solutions To Singular Critical Growth Quasilinear Elliptic Equations

TL;DR

This paper proves the uniqueness of positive radial solutions for a class of singular quasilinear elliptic equations with critical growth, extending understanding of solution behavior in such nonlinear PDEs.

Contribution

It establishes the first uniqueness result for positive radial solutions to a singular critical growth quasilinear elliptic equation with specific parameter conditions.

Findings

Uniqueness of positive radial solutions under given conditions

Analysis of a related limiting problem

Extension of solution theory for singular critical growth equations

Abstract

In this paper, we prove that there exists at most one positive radial weak solution to the following quasilinear elliptic equation with singular critical growth \[ \begin{cases} -\Delta_{p}u-{\displaystyle \frac{\mu}{|x|^{p}}|u|^{p-2}u}{\displaystyle =\frac{|u|^{\frac{(N-s)p}{N-p}-2}u}{|x|^{s}}}+\lambda|u|^{p-2}u & \text{in }B,\\ u=0 & \text{on }\partial B, \end{cases} \] where $B$ is an open finite ball in $\mathbb{R}^{N}$ centered at the origin, $1<p<N$, $-\infty<\mu<((N-p)/p)^{p}$, $0\le s<p$ and $\lambda\in\mathbb{R}$. A related limiting problem is also considered.