Infinitely many multi-peaks solutions for a nonlinear Hartree system
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914563399614464 |
|---|---|
| author | He, Qihan Wang, Qingfang |
| author_facet | He, Qihan Wang, Qingfang |
| contents | In this paper, we study the following nonlinear Hartree system: $-Δu_i + V_i(x)u_i = μ_i ϕ_{u_i}u_i + \sum_{j\neq i}β_{ij}ϕ_{u_j}u_i$ for $x\in\mathbb{R}^3$, with $u_i\in H^1(\mathbb{R}^3)$ ($i=1,2,3$), where $ϕ_u(x):=\int_{\mathbb{R}^3}\frac{u^2(y)}{|x-y|}dy$ for any $u\in H^1(\mathbb{R}^3)$, $V_i(x)$ ($i=1,2,3$) are continuous bounded radial functions, and $β_{ij}$ are coupling constants. We mainly investigate the effects of the potentials and the nonlinear coupling terms on the structure of solutions.
Applying the Lyapunov-Schmidt reduction method, we prove the existence of infinitely many solutions to the system. Specifically, the solutions we obtain satisfy that some components are synchronized with each other but segregated from the others, and that some components are positive while others are sign-changing. To the best of our knowledge, it is the first time that solutions possessing some positive components and some sign-changing ones have been constructed using Lyapunov-Schmidt reduction methods. Moreover, it is also the first attempt to investigate systems consisting of three Hartree equations with mixed couplings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13531 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Infinitely many multi-peaks solutions for a nonlinear Hartree system He, Qihan Wang, Qingfang Analysis of PDEs In this paper, we study the following nonlinear Hartree system: $-Δu_i + V_i(x)u_i = μ_i ϕ_{u_i}u_i + \sum_{j\neq i}β_{ij}ϕ_{u_j}u_i$ for $x\in\mathbb{R}^3$, with $u_i\in H^1(\mathbb{R}^3)$ ($i=1,2,3$), where $ϕ_u(x):=\int_{\mathbb{R}^3}\frac{u^2(y)}{|x-y|}dy$ for any $u\in H^1(\mathbb{R}^3)$, $V_i(x)$ ($i=1,2,3$) are continuous bounded radial functions, and $β_{ij}$ are coupling constants. We mainly investigate the effects of the potentials and the nonlinear coupling terms on the structure of solutions. Applying the Lyapunov-Schmidt reduction method, we prove the existence of infinitely many solutions to the system. Specifically, the solutions we obtain satisfy that some components are synchronized with each other but segregated from the others, and that some components are positive while others are sign-changing. To the best of our knowledge, it is the first time that solutions possessing some positive components and some sign-changing ones have been constructed using Lyapunov-Schmidt reduction methods. Moreover, it is also the first attempt to investigate systems consisting of three Hartree equations with mixed couplings. |
| title | Infinitely many multi-peaks solutions for a nonlinear Hartree system |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.13531 |