Circle-like concentrated solutions for two-component Bose-Einstein condensates

Abstract

We investigate the normalized solutions of the following two-component Bose-Einstein condensates (BEC) system \begin{equation}\left\{ \begin{split} -Δu + (λ+P(x))u &= αu^3 +βuv^2, && \text{in } \mathbb{R}^2,\\-Δv + (λ+Q(x))v &= γv^3 +βu^2 v, && \text{in } \mathbb{R}^2, \end{split} \right.\end{equation} with $L^2$-constraint $${\int }_{{\mathbb{R}}^2}(u^2+v^2)dx=1.$$ For any $\hat{I}\pm >0$, ${\hat{I}}^3>0$ and ${\hat{I}}^2\in (-\sqrt{\hat{I}\pm {\hat{I}}^3},0)\cup (0,\min \{\hat{I}\pm ,{\hat{I}}^3\})\cup \left(\max \{\hat{I}\pm ,{\hat{I}}^3\},+\infty \right)$, we establish the existence of synchronized solutions concentrating on high-dimensional subsets of ${\mathbb{R}}^2$ by employing a finite-dimensional reduction method combined with some local Pohozaev identities. More precisely, we construct vector radial solutions that concentrate on circles when $ \frac{α+ γ- 2β}{αγ- β^2}$ tends to zero. Our results fill the blank in the system for high-dimensional concentrated normalized solutions.