非线性Hartree型方程正驻波的对称性与分类
偏微分方程分析
2026-05-25 v1
摘要
本文研究了强耦合系统正解的一些定性性质：{ − Δ u + τ u = 2 p p + q ( I α ∗ ∣ v ∣ q ) ∣ u ∣ p − 2 u in R N , − Δ v + η v = 2 q p + q ( I α ∗ ∣ u ∣ p ) ∣ v ∣ q − 2 v in R N , \begin{cases} \displaystyle - \Delta u + \tau u = \frac{2 p}{p + q} \left( I_\alpha \ast |v|^q \right) |u|^{p - 2} u &\text{in} ~ \mathbb{R}^N, \\ \\ \displaystyle - \Delta v + \eta v = \frac{2 q}{p + q} \left( I_\alpha \ast |u|^p \right) |v|^{q - 2} v &\text{in} ~ \mathbb{R}^N, \end{cases} ⎩ ⎨ ⎧ − Δ u + τ u = p + q 2 p ( I α ∗ ∣ v ∣ q ) ∣ u ∣ p − 2 u − Δ v + η v = p + q 2 q ( I α ∗ ∣ u ∣ p ) ∣ v ∣ q − 2 v in R N , in R N , τ , η > 0 \tau, \eta > 0 τ , η > 0 N ∈ N N \in \mathbb{N} N ∈ N 0 < α < N 0 < \alpha < N 0 < α < N max { 1 , 2 α N } < p , q < 2 ∗ and 2 ( N + α ) N < p + q < 2 α ∗ , \max \left\{1, \frac{2 \alpha}{N}\right\} < p, q < 2^* \quad \text{and} \quad \frac{2 (N + \alpha)}{N} < p + q < 2_\alpha^*, max { 1 , N 2 α } < p , q < 2 ∗ and N 2 ( N + α ) < p + q < 2 α ∗ , I α I_\alpha I α 2 ∗ : = { ∞ , if N ∈ { 1 , 2 } , 2 N N − 2 , if N ≥ 3 , and 2 α ∗ : = { ∞ , if N ∈ { 1 , 2 } , 2 ( N + α ) N − 2 , if N ≥ 3. 2^* := \begin{cases} \infty, &\text{if} ~ N \in \{1, 2\}, \\ \frac{2 N}{N - 2}, &\text{if} ~ N \geq 3, \end{cases} \quad \text{and} \quad 2_\alpha^* := \begin{cases} \infty, &\text{if} ~ N \in \{1, 2\}, \\ \frac{2 (N + \alpha)}{N - 2}, &\text{if} ~ N \geq 3. \end{cases} 2 ∗ := { ∞ , N − 2 2 N , if N ∈ { 1 , 2 } , if N ≥ 3 , and 2 α ∗ := { ∞ , N − 2 2 ( N + α ) , if N ∈ { 1 , 2 } , if N ≥ 3. p , q ≥ 2 p, q \geq 2 p , q ≥ 2 p = q p = q p = q τ = η \tau = \eta τ = η
引用
@article{arxiv.2605.23127,
title = {Symmetry and classification of positive standing waves of nonlinear Hartree type equations},
author = {Eduardo de Souza Böer and Ederson Moreira dos Santos and Gustavo de Paula Ramos},
journal= {arXiv preprint arXiv:2605.23127},
year = {2026}
}
备注
18 pages