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

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 \( p > 1 \).