Liouville type theorems for stable solutions of the weighted elliptic system

2015/02/14 by Liang-Gen Hu, Jing Zeng, Hu, Liang-Gen +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.1502.04157

openalex publication_date 2015/02/14 · arxiv created 2015/02/28 · arxiv updated 2015/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We examine the weighted elliptic system \begincases -Δu=(1+|x|²)^\fracα2 v,
-Δv=(1+|x|²)^\fracα2 u^p, \endcases in ℝ^N,where N ≥ 5, p>1 and α>0. We prove Liouville type results for the classical positive (nonnegative) stable solutions in dimension N<ℓ+\dfracα(ℓ-2)2 (N <ℓ+\dfracα(ℓ-2)(p+3)4(p+1)) and ℓ ≥ 5, p ∈ (1,p_*(ℓ)). In particular, for any p>1 and α> 0, we obtain the nonexistence of classical positive (nonnegative) stable solutions for any N ≤ 12+5 α (N≤ 12+\dfrac5α(p+3)2(p+1)).

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