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

TL;DR

This work addresses the radial symmetry of positive finite-energy solutions to a quasilinear Hardy-Sobolev doubly critical system driven by the $p$-Laplacian. The authors develop a refined moving-plane method tailored to quasilinear equations with Hardy potentials, leveraging detailed decay estimates and weighted inequalities to control the nonlinear and coupling terms. They prove that any positive solution in $\mathcal{D}^{1,p}(\mathbb{R}^n)\times\mathcal{D}^{1,p}(\mathbb{R}^n)$ is radially symmetric and radially decreasing about the origin, including the case when the Hardy parameter $\gamma=0$. The results extend classical symmetry classifications from scalar and semilinear settings to this coupled, doubly critical quasilinear system, providing new qualitative insights with potential applications to models of coupled physical phenomena.

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} -Δ_p u\,=γ\frac{u^{p-1}}{|x|^p} + u^{p^*-1}+ ναu^{α-1} v^β& \text{in}\quad \mathbb{R}^n \\ -Δ_p v\,=γ\frac{v^{p-1}}{|x|^p} + v^{p^*-1}+ νβu^αv^{β-1} & \text{in}\quad\mathbb{R}^n, \end{cases} \end{equation} where $1<p<n$, $γ\in [0, Λ_{n,p})$ with $Λ_{n,p} = \left[(n-p)/p\right]^p$, $α, β> 1$ such that $α+ β= p^*=np/(n-p)$ and $ν>0$.

Theorems & Definitions (12)

• Theorem 1.1
• Remark 1.2
• Theorem 2.1: Proposition 1.1 of NS
• Remark 2.2
• Lemma 2.3
• Theorem 2.4
• Remark 2.5
• Theorem 2.6
• Theorem 2.7
• Lemma 2.8