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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.05637 |
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Table of Contents:
- 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 \( p > 1 \).