Liouville type theorems for stable solutions of the weighted system involving the Grushin operator with negative exponents

TL;DR

This paper establishes Liouville type theorems for stable solutions of a weighted system involving the Grushin operator with negative exponents, identifying conditions under which no stable solutions exist.

Contribution

It extends Liouville theorems to a weighted Grushin system with negative exponents, improving previous results and covering more general weighted equations.

Findings

No stable solutions exist if certain dimensional and parameter conditions are met.

The results generalize to weighted equations with negative exponents.

The work refines previous stability and nonexistence criteria.

Abstract

The aim of this paper is to study the stability of solutions to the general weighted system with negative exponents: \( \Delta_s u = \rho(\mathbf{x}) v^{-p}, \quad \Delta_s v = \rho(\mathbf{x}) u^{-\theta}, \quad u,v > 0 \) in \( \mathbb{R}^N \), where \( p \geq \theta > 1 \) and \( s \geq 0 \). Here, \( \Delta_s u = \Delta_x u + |x|^{2s} \Delta_y u \) is the Grushin operator, and \( \rho \) is a nonnegative continuous function satisfying certain conditions. We show that the system has no stable solution if \( p \geq \theta > 1 \) and \( N_s < 2 \left[ 1 + (2 + \alpha)x_0 \right] \), where \( x_0 \) is the largest root of the equation \( x^4 - \frac{16p\theta(p-1)}{\theta-1} \left( \frac{1}{p+\theta+2} \right)^2 \left[ x^2 + \frac{p+\theta-2}{(p+\theta+2)(\theta-1)} x + \frac{p-1}{(\theta-1)(p+\theta+2)^2} \right] = 0 \). Our result improves previous work and also applies to the weighted equation \( \Delta_s u = \rho(\mathbf{x}) u^{-p} \) in \( \mathbb{R}^N \), where \( p > 1 \).