Radial symmetry of positive solutions to quasilinear Hardy-Sobolev doubly critical systems

TL;DR

This paper proves that positive finite energy solutions to a certain class of quasilinear Hardy-Sobolev systems are radially symmetric, extending symmetry results to doubly critical systems with Hardy potential and nonlinear coupling.

Contribution

It establishes radial symmetry for positive solutions of a doubly critical quasilinear system involving Hardy potential, a significant extension of symmetry results in nonlinear PDEs.

Findings

Positive solutions are radially symmetric.

Solutions have finite energy.

Results apply to a broad class of quasilinear systems.

Abstract

The aim of this paper is to prove radial symmetry results for positive weak solutions with finite energy to the following quasilinear doubly critical system \begin{equation} \begin{cases} -\Delta_p u\,=\gamma \frac{u^{p-1}}{|x|^p} + u^{p^*-1}+ \nu \alpha u^{\alpha-1} v^\beta & \text{in}\quad \mathbb{R}^n \\ -\Delta_p v\,=\gamma \frac{v^{p-1}}{|x|^p} + v^{p^*-1}+ \nu \beta u^\alpha v^{\beta-1} & \text{in}\quad\mathbb{R}^n, \end{cases} \end{equation} where $1<p<n$, $\gamma \in [0, \Lambda_{n,p})$ with $\Lambda_{n,p} = \left[(n-p)/p\right]^p$, $\alpha, \beta > 1$ such that $\alpha + \beta = p^*=np/(n-p)$ and $\nu>0$.