Infinitely many nonradial positive solutions for multi-species nonlinear Schr\"odinger systems in ${\mathbb R}^N$

TL;DR

This paper proves the existence of infinitely many nonradial positive solutions for multi-species nonlinear Schrödinger systems in two and three dimensions, using Lyapunov-Schmidt reduction without symmetry assumptions.

Contribution

It constructs infinitely many solutions for coupled Schrödinger systems with minimal symmetry restrictions, extending prior results and confirming a conjecture by Pistoia and Vaira.

Findings

Existence of infinitely many nonradial solutions in bbr^N for N=2,3.

Solutions are positive and nonradial.

Results are almost optimal regarding coupling parameters.

Abstract

In this paper, we consider the multi-species nonlinear Schr\"odinger systems in $\bbr^N$: \begin{equation*} \left\{\aligned&-\Delta u_j+V_j(x)u_j=\mu_ju_j^3+\sum_{i=1;i\not=j}^d\beta_{i,j} u_i^2u_j\quad\text{in }\bbr^N, &u_j(x)>0\quad\text{in } {\mathbb R}^N, &u_j(x)\to0\quad\text{as }|x|\to+\infty,\quad j=1,2,\cdots,d,\endaligned\right. \end{equation*} where $N=2,3$, $\mu_j>0$ are constants, $\beta_{i,j}=\beta_{j,i}\not=0$ are coupling parameters, $d\geq2$ and $V_j(x)$ are potentials. By Ljapunov-Schmidt reduction arguments, we construct infinitely many nonradial positive solutions of the above system under some mild assumptions on potentials $V_j(x)$ and coupling parameters $\{\beta_{i,j}\}$, {\it without any symmetric assumptions on the limit case of the above system}. Our result, giving a positive answer to the conjecture in Pistoia and Vaira \cite{PV22} and extending the results in \cite{PW13,PV22}, reveals {\it new phenomenon} in the case of $N=2$ and $d=2$ and is {\it almost optimal} for the coupling parameters $\{\beta_{i,j}\}$.