Some non-homogeneous Gagliardo-Nirenberg inequalities and application to a biharmonic non-linear Schrödinger equation
| Autor: | Antonio J. Fernández, Louis Jeanjean, Rainer Mandel, Mihai Mariş |
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| Přispěvatelé: | Department of Mathematical Sciences, University of Bath, University of Bath [Bath], Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Karlruhe Institute of Technology, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematical Sciences [Bath], Université Bourgogne Franche-Comté [COMUE] (UBFC), Karlsruhe Institute of Technology (KIT), Institut Universitaire de France (IUF), Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.) |
| Rok vydání: | 2020 |
| Předmět: |
Gagliardo-Nirenberg inequalities
global and local minimization Primary 35J35 biharmonic NLS with mixed dispersion 35J91 Applied Mathematics Mathematics::Analysis of PDEs 35Q55. Secondary 35J10 35J30 35J61 standing waves Functional Analysis (math.FA) Mathematics - Functional Analysis Global and local minimization Mathematics - Analysis of PDEs Mathematics - Classical Analysis and ODEs 35Q51 Classical Analysis and ODEs (math.CA) FOS: Mathematics [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] [MATH]Mathematics [math] Biharmonic non-linear Schrödinger equation with mixed dispersion 49J40 Analysis Analysis of PDEs (math.AP) |
| Zdroj: | Journal of Differential Equations Journal of Differential Equations, 2022, 330, pp.1-65. ⟨10.1016/j.jde.2022.04.037⟩ |
| ISSN: | 0022-0396 1090-2732 |
| DOI: | 10.48550/arxiv.2010.01448 |
| Popis: | We study the standing waves for a fourth-order Schr\"odinger equation with mixed dispersion that minimize the associated energy when the $L^2-$norm (the \textit{mass}) } is kept fixed. We need some non-homogeneous Gagliardo-Nirenberg-type inequalities and we develop a method to prove such estimates that should be useful elsewhere. We prove optimal results on the existence of minimizers in the {\it mass-subcritical } and {\it mass-critical } cases. In the { \it mass supercritical} case we show that global minimizers do not exist, and we investigate the existence of local minimizers. If the mass does not exceed some threshold $ \mu_0 \in (0,+\infty)$, our results on "best" local minimizers are also optimal. Comment: Final version. The article will appear in Journal of Differential Equations 328 (2022), pp. 1-65 |
| Databáze: | OpenAIRE |
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