Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case

By Nicola Soave. Journal of Functional Analysis, 2019

469 citations recorded by the source corpus.

We study existence and properties of ground states for the nonlinear Schrodinger equation with combined power nonlinearities \[ -\Delta u= \lambda u + \mu |u|^{q-2} u + |u|^{2^*-2} u \qquad \text{in $\mathbb{R}^N$, $N \ge 3$,} \] having prescribed mass \[ \int_{\mathbb{R}^N} |u|^2 = a^2, \] in the \emph{Sobolev critical case}. For a $L^2$-subcritical, $L^2$-critical, of $L^2$-supercritical perturbation $\mu |u|^{q-2} u$ we prove several existence/non-existence and stability/instability results. This study can be considered as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions, and seems to be the first contribution regarding existence of normalized ground states for the Sobolev critical NLSE in the whole space $\mathbb{R}^N$.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/eb8233cafc61fe3f656141594e074dfa191c0447

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Citations in the corpus

A global branch approach to normalized solutions for the Schrödinger equation (2021, background): “We also refer to [20, 36, 40] for the treatment of nonlinearities which are Sobolev critical.”
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 of mass supercritical Schrödinger equations with potential (2020, background): “Even the autonomous case has been dealt with only very recently in [28, 29].”
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.”
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 Schrödinger equations with critical Sobolev exponent and mixed nonlinearities (2021, background): “Ψu(t) = t(2) 2 ‖∇u‖(2)2 − μtq q ‖u‖q − t(2) ∗ 2∗ ‖u‖(2) ∗ 2∗ , as that in [49].”
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.”
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, result): “This paper is motivated by [14, 15, 36, 38] which considered normalized solutions to the Schrödinger equation with mixed nonlinearities −∆u = λu+ |u|2∗−2u+ μ|u|u, x ∈ R .”
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): “For the general case q ∈ (2, 2∗), we would like to mention Soave [37], where the existence of ground states was obtained by imposing that μa(1−γq )q < α, where α = α(N , q) is a specific constant that”