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

TL;DR

This paper proves the existence of solutions to the nonlinear Schrödinger equation where the mass concentrates on spheres at infinity, revealing a new high-dimensional concentration phenomenon distinct from classical point or fixed-compact-set concentration.

Contribution

It introduces a novel approximation scheme and analysis for high-dimensional sphere-at-infinity concentration in NLS with L^2 constraint, extending previous 2D results.

Findings

Solutions with mass on diverging spheres exist in all dimensions n≥2.

The concentration set escapes to infinity, unlike classical fixed-set concentration.

The approach combines finite-dimensional reduction with blow-up analysis and Pohozaev identities.

Abstract

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.