S^N 球面冠上的 Emden-Fowler 方程
偏微分方程分析
2019-12-25 v1
摘要
设 S N ⊂ R N + 1 \mathbb{S}^N\subset\mathbb{R}^{N+1} S N ⊂ R N + 1 N ≥ 3 N\ge 3 N ≥ 3 S Θ ⊂ S N S_{\Theta}\subset\mathbb{S}^N S Θ ⊂ S N Θ ∈ ( 0 , π ) \Theta\in(0,\pi) Θ ∈ ( 0 , π ) S Θ S_{\Theta} S Θ { ( Θ , ∥ U ∥ ∞ ) } ⊂ R 2 \{(\Theta,\left\|U\right\|_{\infty})\}\subset\mathbb{R}^2 {( Θ , ∥ U ∥ ∞ )} ⊂ R 2 Δ S N U + U p = 0 \Delta_{\mathbb{S}^N}U+U^p=0 Δ S N U + U p = 0 S Θ S_{\Theta} S Θ U = 0 U=0 U = 0 ∂ S Θ \partial S_{\Theta} ∂ S Θ U > 0 U>0 U > 0 S Θ S_{\Theta} S Θ p > 1 p>1 p > 1 p > p S : = ( N − 2 ) / ( N + 2 ) p>p_{\rm S}:=(N-2)/(N+2) p > p S := ( N − 2 ) / ( N + 2 ) Θ ‾ ∈ ( 0 , π ) \underline{\Theta}\in(0,\pi) Θ ∈ ( 0 , π ) Θ ∈ ( Θ ‾ , π ) \Theta\in(\underline{\Theta},\pi) Θ ∈ ( Θ , π ) Θ ∈ ( 0 , Θ ‾ ) \Theta\in(0,\underline{\Theta}) Θ ∈ ( 0 , Θ ) Θ \Theta Θ π \pi π p S < p < p J L p_{\rm S}<p<p_{\rm JL} p S < p < p JL Θ ∗ ∈ ( Θ ‾ , π ) \Theta^*\in(\underline{\Theta},\pi) Θ ∗ ∈ ( Θ , π ) Θ = Θ ∗ \Theta=\Theta^* Θ = Θ ∗ N ≥ 11 N\ge 11 N ≥ 11 p J L = 1 + 4 N − 4 − 2 N − 1 p_{\rm JL}= 1+\frac{4}{N-4-2\sqrt{N-1}} p JL = 1 + N − 4 − 2 N − 1 4 2 ≤ N ≤ 10 2\le N\le 10 2 ≤ N ≤ 10 p J L = ∞ p_{\rm JL}=\infty p JL = ∞ p → ∞ p\to\infty p → ∞ p ↓ 1 p\downarrow 1 p ↓ 1
引用
@article{arxiv.1912.11239,
title = {The Emden-Fowler equation on a spherical cap of $\mathbb{S}^N$},
author = {Atsushi Kosaka and Yasuhito Miyamoto},
journal= {arXiv preprint arXiv:1912.11239},
year = {2019}
}