On the classification of solutions to a weighted elliptic system involving the Grushin operator

TL;DR

This paper classifies positive solutions to a weighted elliptic system involving the Grushin operator, establishing new Liouville-type theorems that improve existing results and proving nonexistence of solutions for certain weighted equations.

Contribution

It introduces new Liouville-type theorems for stable solutions of a weighted Grushin system, extending and strengthening prior classifications and nonexistence results.

Findings

New Liouville-type theorems for stable solutions

Improved nonexistence results for weighted Grushin equations

Conditions under which solutions do not exist

Abstract

We investigate here the following weighted degenerate elliptic system \begin{align*} -\Delta_{s} u = \Big(1+\|\mathbf{x}\|^{2(s+1)}\Big)^{\frac{\alpha}{2(s+1)}} v^p, \quad -\Delta_{s} v = \Big(1+\|\mathbf{x}\|^{2(s+1)}\Big)^{\frac{\alpha}{2(s+1)}}u^\theta, \quad u,v>0\quad\mbox{in }\; \mathbb{R}^N:=\mathbb{R}^{N_1}\times \mathbb{R}^{N_2}. \end{align*} where $\Delta_{s}=\Delta_{x}+|x|^{2s}\Delta_{y},$ is the Grushin operator, $s \geq 0,$ $\alpha \geq 0$ and $1<p\leq\theta.$ Here $$\|\mathbf{x}\|=\Big(|x|^{2(s+1)}+|y|^2\Big)^{\frac{1}{2(s+1)}}, \;\mbox{and}\;\; \mathbf{x}:=(x, y)\in \mathbb{R}^N:=\mathbb{R}^{N_1}\times \mathbb{R}^{N_2}.$$ In particular, we establish some new Liouville-type theorems for stable solutions of the system, which recover and considerably improve upon the known results \cite{cow, Hfh, HU, Fa, DP}. As a consequence, we obtain a nonexistence result for the weighted Grushin equation \begin{align*} -\Delta_{s} u =\Big(1+\|\mathbf{x}\|^{2(s+1)}\Big)^{\frac{\alpha}{2(s+1)}} u^p,\;\; \quad u>0 \quad \mbox{in }\;\; \mathbb{R}^N. \end{align*}