Standing waves for nonlinear Hartree type equations: existence and qualitative properties
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| Format: | Preprint |
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2024
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| _version_ | 1866910970092191744 |
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| author | Böer, Eduardo de Souza Santos, Ederson Moreira dos |
| author_facet | Böer, Eduardo de Souza Santos, Ederson Moreira dos |
| contents | We consider systems of the form \[ \left\{ \begin{array}{l} -Δu + u = \frac{2p}{p+q}(I_α\ast |v|^{q})|u|^{p-2}u \ \ \textrm{ in } \mathbb{R}^N, \\ -Δv + v = \frac{2q}{p+q}(I_α\ast |u|^{p})|v|^{q-2}v \ \ \textrm{ in } \mathbb{R}^N, \end{array} \right. \] for $α\in (0, N)$, $\max\left\{\frac{2α}{N}, 1\right\} < p, q < 2^*$ and $\frac{2(N+α)}{N} < p+ q < 2^{*}_α$, where $I_α$ denotes the Riesz potential, \[ 2^* = \left\{ \begin{array}{l}\frac{2N}{N-2} \ \ \text{for} \ \ N\geq 3,\\ +\infty \ \ \text{for} \ \ N =1,2, \end{array}\right. \quad \text{and} \quad 2^*_α = \left\{ \begin{array}{l}\frac{2(N+α)}{N-2} \ \ \text{for} \ \ N\geq 3,\\ +\infty \ \ \text{for} \ \ N =1,2. \end{array} \right. \] This type of systems arises in the study of standing wave solutions for a certain approximation of the Hartree theory for a two-component attractive interaction. We prove existence and some qualitative properties for ground state solutions, such as definite sign for each component, radial symmetry and sharp asymptotic decay at infinity, and a regularity/integrability result for the (weak) solutions. Moreover, we show that the straight lines $p+q=\frac{2(N+α)}{N}$ and $ p+ q = 2^{*}_α$ are critical for the existence of solutions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_19885 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Standing waves for nonlinear Hartree type equations: existence and qualitative properties Böer, Eduardo de Souza Santos, Ederson Moreira dos Analysis of PDEs 35B06, 35B40, 35J47, 35J50, 35J60, 35Q40, 35Q92 We consider systems of the form \[ \left\{ \begin{array}{l} -Δu + u = \frac{2p}{p+q}(I_α\ast |v|^{q})|u|^{p-2}u \ \ \textrm{ in } \mathbb{R}^N, \\ -Δv + v = \frac{2q}{p+q}(I_α\ast |u|^{p})|v|^{q-2}v \ \ \textrm{ in } \mathbb{R}^N, \end{array} \right. \] for $α\in (0, N)$, $\max\left\{\frac{2α}{N}, 1\right\} < p, q < 2^*$ and $\frac{2(N+α)}{N} < p+ q < 2^{*}_α$, where $I_α$ denotes the Riesz potential, \[ 2^* = \left\{ \begin{array}{l}\frac{2N}{N-2} \ \ \text{for} \ \ N\geq 3,\\ +\infty \ \ \text{for} \ \ N =1,2, \end{array}\right. \quad \text{and} \quad 2^*_α = \left\{ \begin{array}{l}\frac{2(N+α)}{N-2} \ \ \text{for} \ \ N\geq 3,\\ +\infty \ \ \text{for} \ \ N =1,2. \end{array} \right. \] This type of systems arises in the study of standing wave solutions for a certain approximation of the Hartree theory for a two-component attractive interaction. We prove existence and some qualitative properties for ground state solutions, such as definite sign for each component, radial symmetry and sharp asymptotic decay at infinity, and a regularity/integrability result for the (weak) solutions. Moreover, we show that the straight lines $p+q=\frac{2(N+α)}{N}$ and $ p+ q = 2^{*}_α$ are critical for the existence of solutions. |
| title | Standing waves for nonlinear Hartree type equations: existence and qualitative properties |
| topic | Analysis of PDEs 35B06, 35B40, 35J47, 35J50, 35J60, 35Q40, 35Q92 |
| url | https://arxiv.org/abs/2409.19885 |