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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.23127 |
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| _version_ | 1866911707476000768 |
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| author | Böer, Eduardo de Souza Santos, Ederson Moreira dos Ramos, Gustavo de Paula |
| author_facet | Böer, Eduardo de Souza Santos, Ederson Moreira dos Ramos, Gustavo de Paula |
| contents | This paper presents some qualitative properties of positive solutions to the strongly coupled system \[ \begin{cases} \displaystyle - Δu + τu =
\frac{2 p}{p + q} \left( I_α\ast |v|^q \right) |u|^{p - 2} u &\text{in} ~ \mathbb{R}^N, \\ \\ \displaystyle - Δv + ηv =
\frac{2 q}{p + q} \left( I_α\ast |u|^p \right) |v|^{q - 2} v &\text{in} ~ \mathbb{R}^N, \end{cases} \] with $τ, η> 0$, $N \in \mathbb{N}$, $0 < α< N$, \[ \max \left\{1, \frac{2 α}{N}\right\} < p, q < 2^* \quad \text{and} \quad \frac{2 (N + α)}{N} < p + q < 2_α^*, \] where $I_α$ denotes the Riesz potential, \[ 2^* := \begin{cases} \infty, &\text{if} ~ N \in \{1, 2\}, \\ \frac{2 N}{N - 2}, &\text{if} ~ N \geq 3, \end{cases} \quad \text{and} \quad 2_α^* := \begin{cases} \infty, &\text{if} ~ N \in \{1, 2\}, \\ \frac{2 (N + α)}{N - 2}, &\text{if} ~ N \geq 3. \end{cases} \] More precisely, by means of the moving planes method, we prove that positive solutions to this system are radially symmetric and strictly radially decreasing when $p, q \geq 2$, and we obtain a classification result for positive ground states in the case $p = q$ and $τ= η$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_23127 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Symmetry and classification of positive standing waves of nonlinear Hartree type equations Böer, Eduardo de Souza Santos, Ederson Moreira dos Ramos, Gustavo de Paula Analysis of PDEs This paper presents some qualitative properties of positive solutions to the strongly coupled system \[ \begin{cases} \displaystyle - Δu + τu = \frac{2 p}{p + q} \left( I_α\ast |v|^q \right) |u|^{p - 2} u &\text{in} ~ \mathbb{R}^N, \\ \\ \displaystyle - Δv + ηv = \frac{2 q}{p + q} \left( I_α\ast |u|^p \right) |v|^{q - 2} v &\text{in} ~ \mathbb{R}^N, \end{cases} \] with $τ, η> 0$, $N \in \mathbb{N}$, $0 < α< N$, \[ \max \left\{1, \frac{2 α}{N}\right\} < p, q < 2^* \quad \text{and} \quad \frac{2 (N + α)}{N} < p + q < 2_α^*, \] where $I_α$ denotes the Riesz potential, \[ 2^* := \begin{cases} \infty, &\text{if} ~ N \in \{1, 2\}, \\ \frac{2 N}{N - 2}, &\text{if} ~ N \geq 3, \end{cases} \quad \text{and} \quad 2_α^* := \begin{cases} \infty, &\text{if} ~ N \in \{1, 2\}, \\ \frac{2 (N + α)}{N - 2}, &\text{if} ~ N \geq 3. \end{cases} \] More precisely, by means of the moving planes method, we prove that positive solutions to this system are radially symmetric and strictly radially decreasing when $p, q \geq 2$, and we obtain a classification result for positive ground states in the case $p = q$ and $τ= η$. |
| title | Symmetry and classification of positive standing waves of nonlinear Hartree type equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.23127 |