Normalized ground states for the NLS equation with combined nonlinearities
By Nicola Soave. Journal of Differential Equations, 2018
482 citations recorded by the source corpus.
We study existence and properties of ground states for the nonlinear Schrodinger equation with combined power nonlinearities \[ -\Delta u= \lambda u + \mu |u|^{q-2} u + |u|^{p-2} u \qquad \text{in $\mathbb{R}^N$, $N \ge 1$,} \] having prescribed mass \[ \int_{\mathbb{R}^N} |u|^2 = a^2. \] Under different assumptions on $q 0$ and $\mu \in \mathbb{R}$ we prove several existence and stability/instability results. In particular, we consider cases when \[ 2<q \le 2+ \frac{4}{N} \le p<2^*, \quad q \neq p, \] i.e. the two nonlinearities have different character with respect to the $L^2$-critical exponent. These cases present substantial differences with respect to purely subcritical or supercritical situations, which were already studied in the literature. We also give new criteria for global existence and finite time blow-up in the associated dispersive equation.
Subfields: Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/975b89da6e6ef0492b91c6d428ff0cbf2d8edd67
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Citations in the corpus
- Existence and asymptotic behavior of normalized ground states for Sobolev critical Schrödinger systems (2022, methodology): “We refer the reader to [7, 26, 27, 28, 29, 30, 39, 40, 43] for scalar equations, [8, 9, 10, 11, 14, 21, 22, 31] for systems of two equations, [35] for systems of k equations.”
- Normalized solution to the Schrödinger equation with potential and general nonlinear term: Mass super-critical case (2021, background): “For the non-potential case, we refer to [22,23].”
- 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): “In [36], Soave studied the normalized solutions for the nonlinear Schrödinger equation (1.”
- A global branch approach to normalized solutions for the Schrödinger equation (2021, unknown): “Let us just mention here [3, 5, 6, 10, 22, 17, 35].”
- Orbital stability of ground states for a Sobolev critical Schrödinger equation (2020, background): “This approach, that we shall follow here, is relevant from the physical point of view, in particular, since the L2 norm is a preserved quantity of the evolution and since the variational characterization of such solutions is often a strong help to analyze their orbital stability, see for example, [6”
- Standing waves to upper critical Choquard equation with a local perturbation: Multiplicity, qualitative properties and stability (2021, methodology): “12) The fiber map τ Ψu( ) is introduced by Jeanjean in [30] for the Schrödinger equation and is well studied by Soave in [31].”
- Normalized solutions to a class of Kirchhoff equations with Sobolev critical exponent (2021, methodology): “This process is quite different from that adopted in [27, 28] since the appearance of the nonlocal term ( ́ R3 |∇u|(2))∆u.”
- Normalized solutions to mass supercritical Schrödinger equations with negative potential (2021, background): “After [22], only recently more general autonomous equations and systems have been considered, also refining and developing this initial strategy, see [4, 6, 8, 5, 17, 9, 21, 34, 35] and references therein.”
- Multiple normalized solutions for a Sobolev critical Schrödinger-Poisson-Slater equation (2021, background): “In particular, compared to the works [19, 20, 32, 33, 36] we do not benefit from the property that γ−(c) ≥ 0 = supu∈Λ+(c)F(u).”
- Normalized solutions of mass supercritical Schrödinger equations with potential (2020, background): “Even the autonomous case has been dealt with only very recently in [28, 29].”