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 $α>0$, $γ> 0$ and $\ β\in (-\sqrt{αγ},0)\cup(0,\min {α,γ})\cup \left(\max {α,γ} , + \infty\right)$, 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.