Standing waves for nonlinear Hartree type equations: existence and qualitative properties
Abstract
We consider systems of the form \[ \left\{ \begin{array}{l} -\Delta u + u = \frac{2p}{p+q}(I_\alpha \ast |v|^{q})|u|^{p-2}u \ \ \textrm{ in } \mathbb{R}^N, \\ -\Delta v + v = \frac{2q}{p+q}(I_\alpha \ast |u|^{p})|v|^{q-2}v \ \ \textrm{ in } \mathbb{R}^N, \end{array} \right. \] for $\alpha\in (0, N)$, $\max\left\{\frac{2\alpha}{N}, 1\right\} < p, q < 2^*$ and $\frac{2(N+\alpha)}{N} < p+ q < 2^{*}_{\alpha}$, where $I_\alpha$ denotes the Riesz potential, \[ 2^* = \left\{ \begin{array}{l}\frac{2N}{N-2} \ \ \text{for} \ \ N\geq 3,\\ +\infty \ \ \text{for} \ \ N =1,2, \end{array}\right. \quad \text{and} \quad 2^*_{\alpha} = \left\{ \begin{array}{l}\frac{2(N+\alpha)}{N-2} \ \ \text{for} \ \ N\geq 3,\\ +\infty \ \ \text{for} \ \ N =1,2. \end{array} \right. \] 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 $p+q=\frac{2(N+\alpha)}{N}$ and $ p+ q = 2^{*}_{\alpha}$ are critical for the existence of solutions.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2024
- DOI:
- 10.48550/arXiv.2409.19885
- arXiv:
- arXiv:2409.19885
- Bibcode:
- 2024arXiv240919885D
- Keywords:
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- Mathematics - Analysis of PDEs;
- 35B06;
- 35B40;
- 35J47;
- 35J50;
- 35J60;
- 35Q40;
- 35Q92
- E-Print:
- 35 pages, 7 figures, few typos were fixed