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

Abstract

In this paper, we study the following coupled nonlinear Schrödinger system in ${\mathbb{R}}^3$ $$ \left\{% \begin{array}{ll} -ε^2Δu +P(x)u=μ_1 u^3+βv^2u,~~&x\in \R^3,\vspace{0.15cm}\\ -ε^2Δv +Q(x)v=μ_2 v^3+βu^2v,~~&x\in \R^3,\\ \end{array}% \right. $$ where ${\mu }_1>0,{\mu }_2>0$ and $\beta \in \mathbb{R}$ 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 $\epsilon \to 0^{+}$.

arXiv comment36 pages

References in corpus (1)

• Radio-Frequency Spectroscopy of Ultracold Fermions