Multiple nodal solutions having shared componentwise nodal numbers for coupled Schr\"{o}dinger equations

TL;DR

This paper proves the existence of infinitely many shared-component nodal solutions for coupled nonlinear Schrödinger equations with repulsive interactions, revealing complex solution structures in radial domains.

Contribution

It introduces a novel approach combining heat flow deformation and symmetric mountain pass techniques to establish multiple solutions with prescribed nodal properties.

Findings

Existence of infinitely many solutions with shared componentwise nodal numbers.

Solutions exhibit componentwise sign-changing behavior with prescribed zeros.

Method applies to systems with non-symmetric coupling, broadening applicability.

Abstract

We investigate the structure of nodal solutions for coupled nonlinear Schr\"{o}dinger equations in the repulsive coupling regime. Among other results, for the following coupled system of $N$ equations, we prove the existence of infinitely many nodal solutions which share the same componentwise-prescribed nodal numbers \begin{equation}\label{ab} \left\{ \begin{array}{lr} -{\Delta}u_{j}+\lambda u_{j}=\mu u^{3}_{j}+\sum_{i\neq j}\beta u_{j}u_{i}^{2} \,\,\,\,\,\,\, in\ \W , u_{j}\in H_{0,r}^{1}(\W), \,\,\,\,\,\,\,\,j=1,\dots,N, \end{array} \right. \end{equation} where $\W$ is a radial domain in $\mathbb R^n$ for $n\leq 3$, $\lambda>0$, $\mu>0$, and $\beta <0$. More precisely, let $p$ be a prime factor of $N$ and write $N=pB$. Suppose $\beta\leq-\frac{\mu}{p-1}$. Then for any given non-negative integers $P_{1},P_{2},\dots,P_{B}$, (\ref{ab}) has infinitely many solutions $(u_{1},\dots,u_{N})$ such that each of these solutions satisfies the same property: for $b=1,...,B$, $u_{pb-p+i}$ changes sign precisely $P_b$ times for $i=1,...,p$. The result reveals the complex nature of the solution structure in the repulsive coupling regime due to componentwise segregation of solutions. Our method is to combine a heat flow approach as deformation with a minimax construction of the symmetric mountain pass theorem using a $\mathbb Z_p$ group action index. Our method is robust, also allowing to give the existence of one solution without assuming any symmetry of the coupling.