Infinitely many segregated vector solutions of Schrodinger system

TL;DR

This paper constructs infinitely many segregated vector solutions for a coupled Schrödinger system, where one component peaks at the origin and the other exhibits multiple peaks arranged on a circle, extending previous work with nonlinear coupling terms.

Contribution

It introduces a new class of solutions with complex peak arrangements in coupled Schrödinger systems, advancing the understanding of multi-peak solutions with nonlinear interactions.

Findings

Existence of infinitely many segregated solutions with prescribed peak structures.

Extension of previous results to systems with nonlinear coupling terms.

Construction of solutions with one peak at the origin and multiple peaks on a circle.

Abstract

We consider the following system of Schr\"odinger equations \begin{equation*}\left.\begin{cases} -\Delta U + \lambda U = \alpha_0 U^3+ \beta UV^2 -\Delta V + \mu(y) V = \alpha_1 V^3+\beta U^2V \end{cases}\right. \text{in} \quad \mathbb{R}^N, \ N=2, 3,\end{equation*} where $\lambda$, $\alpha_0$, $\alpha_1>0$ are positive constants, $\beta \in \mathbb{R}$ is the coupling constant, and $\mu: \mathbb{R}^N \rightarrow \mathbb{R}$ is a potential function. Continuing the work of Lin and Peng \cite{lin_peng_2014}, we present a solution of the type where one species has a peak at the origin and the other species has many peaks over a circle, but as seen in the above, coupling terms are nonlinear.