Chongyang Tian | Searcharxiv

arXiv2026

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} -\Delta u + (\lambda+P(x))u &= \alpha u^3 +\beta uv^2, && \text{in } \mathbb{R}^2,\\-\Delta v + (\lambda+Q(x))v &= \gamma v^3 +\beta u^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 $\ \beta \in (-\sqrt{\alpha\gamma},0)\cup(0,\min \{\alpha,\gamma\})\cup \left(\max \{\alpha,\gamma\} , + \infty\right)$, 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{\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.

math.AP↗

arXiv2025

Asymptotic Sphere Concentration at Infinity for NLS with L^2 Constraint

We consider the nonlinear Schr\"odinger equation$$-\Delta u + V(x)\,u = a\,u^p + \mu u \quad \text{in }\mathbb{R}^n,\qquad \int_{\mathbb{R}^n} u^2 = 1,$$modeling attractive Bose--Einstein condensates. For all dimensions $n\ge 2$ and all exponents $p>1$, we prove the existence of normalized solutions whose $L^2$-mass concentrates on spheres with radii diverging to infinity. In particular, the concentration set escapes to infinity rather than remaining on a fixed compact hypersurface, which makes our regime qualitatively different both from classical point-concentration phenomena and from concentrating profiles in unconstrained problems. Our approach combines a tailored finite-dimensional reduction with a blow-up analysis based on Pohozaev identities and, in this way, extends the two-dimensional mass-critical result for $(n,p)=(2,3)$ obtained in Guo--Tian--Zhou (Calc.\ Var.\ Partial Differential Equations, 2022). The proof in that paper relies in an essential way on the two-dimensional structure and does not directly apply in higher dimensions, whereas here we develop a different approximation scheme and functional setting adapted to the high-dimensional sphere-at-infinity concentration regime.

math.AP↗