Structure of positive solutions to a Schrödinger system

Abstract

The paper is concerned with the existence and uniqueness of positive solutions of the coupled nonlinear Schrödinger system arising in nonlinear optics and in Hartree-Fock theory for a double condensate

$$\begin{aligned}\left\{ \begin{array}{lll} -\Delta u+\mu _1u^3&{}=\lambda u+\beta uv^2&{}\quad \text { in }\Omega ,\\ -\Delta v+\mu _2v^3&{}=\lambda v+\beta u^2v&{}\quad \text { in }\Omega ,\\ u,\,v>0\text { in }&{}\Omega ,\quad u,\,v=0&{}\quad \text { on }\partial \Omega , \end{array}\right. \end{aligned}$$

on the range of $\lambda$ and the coupling constant $\beta$, where $\Omega \subset \mathbb R^N(N\ge 1)$ is a bounded smooth domain, $\lambda >0$ and $0<\mu _1\le \mu _2$. Under some conditions, we obtain some interesting solution structure theorems in the $\beta \lambda$-plane. Especially we obtain interesting uniqueness results via synchronized solution techniques.