NODAL Vector solutions with clustered peaks for a nonlinear elliptic equations in $\R^3$

TL;DR

This paper constructs multiple clustered spike solutions for a coupled nonlinear Schrödinger system in three-dimensional space, revealing how solutions behave as the semi-classical parameter approaches zero.

Contribution

It introduces a method to find nodal semi-classical bound states with clustered peaks for the system, extending understanding of solution structures in nonlinear Schrödinger equations.

Findings

Existence of nodal semi-classical bound states with clustered spikes.

Solutions' number tends to infinity as epsilon approaches zero.

Applicable under certain conditions on potentials and coupling constant.

Abstract

In this paper, we study the following coupled nonlinear Schr\"{o}dinger system in $\R^3$ $$ \left\{% \begin{array}{ll} -\epsilon^2\Delta u +P(x)u=\mu_1 u^3+\beta v^2u,~~&x\in \R^3,\vspace{0.15cm}\\ -\epsilon^2\Delta v +Q(x)v=\mu_2 v^3+\beta u^2v,~~&x\in \R^3,\\ \end{array}% \right. $$ where $\mu_1 >0,\mu_2>0$ and $\beta \in \R$ is a coupling constant. Whether the system is repulsive or attractive, we prove that it has nodal semi-classical segregated or synchronized bound states with clustered spikes for sufficiently small $\epsilon$ under some additional conditions on $P(x), Q(x)$ and $\beta$. Moreover, the number of this type of solutions will go to infinity as $\epsilon \to 0^+$.