Multiple normalized solutions for a Sobolev critical Schrödinger equation

By Louis Jeanjean, Thanh Trung Le. Mathematische Annalen, 2020

242 citations recorded by the source corpus.

We study the existence of standing waves, of prescribed L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-norm (the mass), for the nonlinear Schrödinger equation with mixed power nonlinearities i∂tϕ+Δϕ+μϕ|ϕ|q-2+ϕ|ϕ|2∗-2=0,(t,x)∈R×RN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} i \partial _t \phi + \Delta \phi + \mu \phi |\phi |^{q-2} + \phi |\phi |^{2^* - 2} = 0, \quad (t, x) \in \mathbb {R}\times \mathbb {R}^N, \end{aligned}$$\end{document}where N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \ge 3$$\end{document}, ϕ:R×RN→C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi : \mathbb {R}\times \mathbb {R}^N \rightarrow \mathbb {C}$$\end{document}, μ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu > 0$$\end{document}, 2<q<2+4/N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2< q < 2 + 4/N $$\end{document} and 2∗=2N/(N-2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^* = 2N/(N-2)$$\end{document} is the critical Sobolev exponent. It was proved in Jeanjean et al. (Orbital stability of ground states for a Sobolev critical Schrödinger equation, 2020) that, for small mass, ground states exist and correspond to local minima of the associated Energy functional. It was also established that despite the nonlinearity is Sobolev critical, the set of ground states is orbitally stable. Here we prove that, when N≥4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \ge 4$$\end{document}, there also exist standing waves which are not ground states and are located at a mountain-pass level of the Energy functional. These solutions are unstable by blow-up in finite time. Our study is motivated by a question raised by Soave (J Funct Anal 279(6):108610, 2020).

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/230583786be106cb0565c6403e4b6582928494f1

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Citations in the corpus

Existence and asymptotic behavior of normalized ground states for Sobolev critical Schrödinger systems (2022, background): “Then the following estimates hold; proofs can be found in [30, 40].”
Normalized solutions for a Schrödinger equation with critical growth in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{documen (2021, unknown): “While in [29] a multiplicity result is established, where the second solution is not a ground state.”
A global branch approach to normalized solutions for the Schrödinger equation (2021, background): “We also refer to [19, 35, 39] for the treatment of nonlinearities which are Sobolev critical.”
Normalized solutions to a class of Kirchhoff equations with Sobolev critical exponent (2021, methodology): “N μ p and q classifications of solutions references N ≥ 1 μ > 0 2 < q < p ≤ 2 + 4 N a globla minimizer [7, 27] N ≥ 1 μ<0 2 < q ≤ 2 + 4 N < p < 2∗ a Mountain Pass solution [27] N ≥ 1 μ > 0 2 < q < 2 + 4 N < p < 2∗ a local minimizer; a Mountain Pass solution [27] N ≥ 3 μ > 0 2 < q < 2 + 4 N , p = 2∗ a”
Standing waves to upper critical Choquard equation with a local perturbation: Multiplicity, qualitative properties and stability (2021, methodology): “We combine the methods used in [27] and [29] to prove Theorem 1.”