高阶 Lane-Emden 型方程组解的存在性
偏微分方程分析
2017-12-20 v2
摘要
我们证明如下 Lane-Emden 型方程组的存在性结果 { ( − Δ ) α u = ∣ v ∣ q ( − Δ ) β v = ∣ u ∣ p in B 1 ⊂ R N ∂ r u ∂ ν r = 0 , r = 0 , … , α − 1 , on ∂ B 1 ∂ r v ∂ ν r = 0 , r = 0 , … , β − 1 , on ∂ B 1 . \begin{cases} \begin{aligned} (-\Delta)^{\alpha} u=\left| v \right|^q \\ (-\Delta)^{\beta} v= \left| u \right|^p \end{aligned} \text{ in } B_1 \subset \mathbb{R}^N \\ \frac{\partial^{r} u}{\partial \nu^{r}}=0, \, r=0, \dots, \alpha-1, \text{ on } \partial B_1 \\ \frac{\partial^{r} v}{\partial \nu^{r}}=0, \, r=0, \dots, \beta-1, \text{ on } \partial B_1. \end{cases} ⎩ ⎨ ⎧ ( − Δ ) α u = ∣ v ∣ q ( − Δ ) β v = ∣ u ∣ p in B 1 ⊂ R N ∂ ν r ∂ r u = 0 , r = 0 , … , α − 1 , on ∂ B 1 ∂ ν r ∂ r v = 0 , r = 0 , … , β − 1 , on ∂ B 1 . B 1 B_1 B 1 R N \mathbb{R}^N R N N > max { 2 α , 2 β } N >\max \{2\alpha, 2\beta \} N > max { 2 α , 2 β } ν ν ν α , β ∈ N \alpha, \beta \in \mathbb{N} α , β ∈ N α , β ≥ 1 \alpha, \beta \ge 1 α , β ≥ 1 ( − Δ ) α = − Δ ( ( − Δ ) α − 1 ) (-\Delta)^{\alpha}= -\Delta((-\Delta)^{\alpha-1}) ( − Δ ) α = − Δ (( − Δ ) α − 1 ) α = 2 \alpha=2 α = 2 β = 1 \beta=1 β = 1 p , q > 1 p, q>1 p , q > 1
引用
@article{arxiv.1711.06887,
title = {Existence of solutions to higher order Lane-Emden type systems},
author = {Delia Schiera},
journal= {arXiv preprint arXiv:1711.06887},
year = {2017}
}
备注
minor corrections