Asymptotic Sphere Concentration at Infinity for NLS with L^2 Constraint
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915669130346496 |
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| author | Guo, Qing Tian, Chongyang |
| author_facet | Guo, Qing Tian, Chongyang |
| contents | We consider the nonlinear Schrödinger equation$$-Δu + V(x)\,u = a\,u^p + μu \quad \text{in }\mathbb{R}^n,\qquad \int_{\mathbb{R}^n} u^2 = 1,$$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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_10512 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic Sphere Concentration at Infinity for NLS with L^2 Constraint Guo, Qing Tian, Chongyang Analysis of PDEs 35B38, 35Q55 We consider the nonlinear Schrödinger equation$$-Δu + V(x)\,u = a\,u^p + μu \quad \text{in }\mathbb{R}^n,\qquad \int_{\mathbb{R}^n} u^2 = 1,$$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. |
| title | Asymptotic Sphere Concentration at Infinity for NLS with L^2 Constraint |
| topic | Analysis of PDEs 35B38, 35Q55 |
| url | https://arxiv.org/abs/2512.10512 |