Liouville type theorems for stable solutions of the weighted system involving the Grushin operator with negative exponents
Mtiri Foued
TL;DR
The paper addresses Liouville-type nonexistence of stable solutions for a weighted Grushin system with negative exponents $\Delta_s u = \rho(\mathbf{x}) v^{-p}$ and $\Delta_s v = \rho(\mathbf{x}) u^{-\theta}$ in $\mathbb{R}^N$, under the condition $p\geq \theta>1$ and a lower bound on the weight $\rho$ with parameter $\alpha$. It develops a polynomial-based framework, introduces a detailed Grushin-geometry analysis, and proves a stability-based a priori inequality, culminating in a bootstrap argument that yields nonexistence whenever the homogeneous dimension $N_s$ satisfies $N_s<2\left[1+(2+\alpha)x_0\right]$, with $x_0$ the largest root of a specific polynomial $H$. The method hinges on a comparison principle between $u$ and $v$, integral estimates derived from stability, and a careful spectral-parameter analysis of $H$ (via a related $L(z)$ and $z_0$). This work extends known results for the Grushin setting and negatively exponent weighted equations, both strengthening nonexistence ranges and broadening applicability to the weighted equation $\Delta_s u = \rho(\mathbf{x}) u^{-p}$ with $p>1$.
Abstract
The aim of this paper is to study the stability of solutions to the general weighted system with negative exponents: \( Δ_s u = ρ(\mathbf{x}) v^{-p}, \quad Δ_s v = ρ(\mathbf{x}) u^{-θ}, \quad u,v > 0 \) in \( \mathbb{R}^N \), where \( p \geq θ> 1 \) and \( s \geq 0 \). Here, \( Δ_s u = Δ_x u + |x|^{2s} Δ_y u \) is the Grushin operator, and \( ρ\) is a nonnegative continuous function satisfying certain conditions. We show that the system has no stable solution if \( p \geq θ> 1 \) and \( N_s < 2 \left[ 1 + (2 + α)x_0 \right] \), where \( x_0 \) is the largest root of the equation \( x^4 - \frac{16pθ(p-1)}{θ-1} \left( \frac{1}{p+θ+2} \right)^2 \left[ x^2 + \frac{p+θ-2}{(p+θ+2)(θ-1)} x + \frac{p-1}{(θ-1)(p+θ+2)^2} \right] = 0 \). Our result improves previous work and also applies to the weighted equation \( Δ_s u = ρ(\mathbf{x}) u^{-p} \) in \( \mathbb{R}^N \), where \( p > 1 \).