Distribution of positive solutions to Schrödinger systems with linear and nonlinear couplings

Abstract

In this paper, we study the existence, nonexistence and uniqueness of positive solutions for the following two Schrödinger systems with linear and nonlinear couplings which arise from Bose–Einstein condensates:

$$\begin{aligned} \left\{ \begin{array}{l} -\triangle u+\mu u^{3}=\lambda u+\kappa v+\beta uv^{2} \ \ \text {in} \ \ \Omega ,\\ -\triangle v+\mu v^{3}=\lambda v+\kappa u+\beta vu^{2} \ \ \text {in} \ \ \Omega ,\\ u,v>0 \ \text {in} \ \Omega , \ u=v=0 \ \qquad \,\,\,\, \text {on} \ \partial \Omega , \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{array}{l} -\triangle u+\lambda u+\kappa v=\mu u^{3}+\beta uv^{2} \ \ \text {in} \ \ \Omega ,\\ -\triangle v+\lambda v+\kappa u=\mu v^{3}+\beta vu^{2} \ \ \text {in} \ \ \Omega ,\\ u,v>0 \ \text {in} \ \Omega , \ u=v=0 \ \qquad \quad \text {on} \ \partial \Omega \end{array}\right. \end{aligned}$$

on the range of $\lambda$ and the coupling constants $\kappa$, $\beta$, where $\Omega$ is a bounded smooth domain in ${\mathbb {R}}^{N}$, $N\ge 1$, $\lambda , \mu >0$. We obtain some interesting positive solutions distribution theorems in the $\kappa \lambda$-plane for fixed $\beta$ in different ranges. Especially we get some uniqueness results via synchronized solution techniques.