高维临界椭圆系统的分离解
偏微分方程分析
2026-01-16 v2
摘要
我们研究临界耦合薛尔金子系统在高维情况下的多个分离解的存在性。该系统形式为：\n\n{ − Δ u 1 = K 1 ( ∣ y ∣ ) ∣ u 1 ∣ 2 ∗ − 2 u 1 + β ∣ u 2 ∣ 2 ∗ 2 ∣ u 1 ∣ 2 ∗ 2 − 2 u 1 , y ∈ R N , − Δ u 2 = K 2 ( ∣ y ∣ ) ∣ u 2 ∣ 2 ∗ − 2 u 2 + β ∣ u 1 ∣ 2 ∗ 2 ∣ u 2 ∣ 2 ∗ 2 − 2 u 2 , y ∈ R N , u 1 , u 2 ≥ 0 , u 1 , u 2 ∈ C 0 ( R N ) ∩ D 1 , 2 ( R N ) , \begin{cases} -\Delta u_{1} = K_1(| y|) | u_{1}|^{2^*-2}u_{1}+\beta | u_{2}|^{\frac{2^{*}}{2}}| u_{1}|^{\frac{2^{*}}{2}-2}u_{1}, & y\in \mathbb R^N,\\ -\Delta u_{2} = K_2(| y|) | u_{2}|^{2^*-2}u_{2}+\beta | u_{1}|^{\frac{2^{*}}{2}}| u_{2}|^{\frac{2^{*}}{2}-2}u_{2}, & y\in\mathbb R^N,\\ u_{1},u_{2}\geq0, u_{1},u_{2}\in C_0(\mathbb R^{N})\cap D^{1,2}(\mathbb R^N), \end{cases} ⎩ ⎨ ⎧ − Δ u 1 = K 1 ( ∣ y ∣ ) ∣ u 1 ∣ 2 ∗ − 2 u 1 + β ∣ u 2 ∣ 2 2 ∗ ∣ u 1 ∣ 2 2 ∗ − 2 u 1 , − Δ u 2 = K 2 ( ∣ y ∣ ) ∣ u 2 ∣ 2 ∗ − 2 u 2 + β ∣ u 1 ∣ 2 2 ∗ ∣ u 2 ∣ 2 2 ∗ − 2 u 2 , u 1 , u 2 ≥ 0 , u 1 , u 2 ∈ C 0 ( R N ) ∩ D 1 , 2 ( R N ) , y ∈ R N , y ∈ R N , N ≥ 5 N \geq 5 N ≥ 5 2 ∗ = 2 N N − 2 2^* = \frac{2N}{N-2} 2 ∗ = N − 2 2 N K 1 , K 2 > 0 K_1, K_2 > 0 K 1 , K 2 > 0 β < 0 \beta < 0 β < 0 K 1 K_1 K 1 K 2 K_2 K 2 r 0 ≠ ρ 0 r_0 \ne \rho_0 r 0 = ρ 0 k k k ( u 1 , k , u 2 , k ) (u_{1,k}, u_{2,k}) ( u 1 , k , u 2 , k ) r 0 r_0 r 0 ρ 0 \rho_0 ρ 0 2 ∗ / 2 − 1 < 1 2^*/2 -1 < 1 2 ∗ /2 − 1 < 1
引用
@article{arxiv.2511.14115,
title = {Segregated Solutions to Critical Elliptic Systems in High Dimensions ($N \geq 5$)},
author = {Zijuan Gao and Qing Guo and Chengxiang Zhang},
journal= {arXiv preprint arXiv:2511.14115},
year = {2026}
}