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- Publisher
- Springer Journals
- Copyright
- Copyright © Springer Nature Switzerland AG 2020
- ISSN
- 1424-3199
- eISSN
- 1424-3202
- DOI
- 10.1007/s00028-020-00564-3
- Publisher site
- See Article on Publisher Site
Abstract
We study the asymptotic behavior of constraint minimizers for the energy functional related to the mass-critical fractional nonlinear Schrödinger equation with a subcritical perturbation I(a)=infE(ϕ):ϕ∈Hs(RN),‖ϕ‖L22=a,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} I(a) = \inf \left\{ E(\phi ) \ : \ \phi \in H^s({\mathbb {R}}^N), \Vert \phi \Vert ^2_{L^2} =a \right\} , \end{aligned}$$\end{document}where N≥1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N\ge 1$$\end{document}, 0<s<1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0<s<1$$\end{document}, a>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a>0$$\end{document} and E(ϕ):=12‖(-Δ)s/2ϕ‖L22-N2(N+2s)‖ϕ‖L2+4sN2+4sN-1α+2‖ϕ‖Lα+2α+2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} E(\phi ) := \frac{1}{2} \Vert (-\Delta )^{s/2} \phi \Vert ^2_{L^2} {- \frac{N}{2(N+2s)} \Vert \phi \Vert ^{2+\frac{4s}{N}}_{L^{2+\frac{4s}{N}}}} -\frac{1}{\alpha +2} \Vert \phi \Vert ^{\alpha +2}_{L^{\alpha +2}} \end{aligned}$$\end{document}with 0<α<4sN\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0<\alpha <\frac{4s}{N}$$\end{document}. We first show that minimizer for I(a) blows up as a↗a∗:=‖Q‖L22\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a \nearrow a^*:= \Vert Q\Vert ^2_{L^2}$$\end{document}, where Q is the unique positive radial solution to the elliptic equation (-Δ)sQ+Q-|Q|4sNQ=0.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} (-\Delta )^s Q + Q - |Q|^{\frac{4s}{N}} Q=0. \end{aligned}$$\end{document}We then give a detailed description of the blow-up behavior of minimizers as a↗a∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a \nearrow a^*$$\end{document}.
Journal
Journal of Evolution Equations – Springer Journals
Published: Feb 15, 2020