Infinitely many multi-peaks solutions for a nonlinear Hartree system

TL;DR

This paper proves the existence of infinitely many solutions for a three-component nonlinear Hartree system, revealing complex solution structures influenced by potentials and couplings, using Lyapunov-Schmidt reduction.

Contribution

First construction of solutions with positive and sign-changing components in a three-component Hartree system using Lyapunov-Schmidt reduction.

Findings

Existence of infinitely many solutions with mixed component signs.

Solutions exhibit synchronization and segregation among components.

First analysis of three Hartree equations with mixed couplings.

Abstract

In this paper, we study the following nonlinear Hartree system: $-\Delta u_i + V_i(x)u_i = \mu_i \phi_{u_i}u_i + \sum_{j\neq i}\beta_{ij}\phi_{u_j}u_i$ for $x\in\mathbb{R}^3$, with $u_i\in H^1(\mathbb{R}^3)$ ($i=1,2,3$), where $\phi_u(x):=\int_{\mathbb{R}^3}\frac{u^2(y)}{|x-y|}dy$ for any $u\in H^1(\mathbb{R}^3)$, $V_i(x)$ ($i=1,2,3$) are continuous bounded radial functions, and $\beta_{ij}$ 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.