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

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 $\alpha >0$, $\gamma >0$ and $\beta \in (-\sqrt{\alpha \gamma },0)\cup (0,\min \{\alpha ,\gamma \})\cup (\max \{\alpha ,\gamma \},+\mathrm{\infty })$, 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 $\frac{\alpha +\gamma -2\beta }{\alpha \gamma -{\beta }^2}$ tends to zero. Our results fill the blank in the system for high-dimensional concentrated normalized solutions.