Liouville type theorems for stable solutions of the weighted system involving the Grushin operator with negative exponents (2505.05637v1)

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 ).