Radial symmetry of positive solutions to quasilinear Hardy-Sobolev doubly critical 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 $\nu>0$.