# Some non-homogeneous Gagliardo–Nirenberg inequalities and application to a biharmonic non-linear Schrödinger equation

Fernández, Antonio J.; Jeanjean, Louis; Mandel, Rainer; Mariş, Mihai

##### Abstract:
We study the standing waves for a fourth-order Schrödinger 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 $\textit{mass-subcritical}$ and $\textit{mass-critical}$ cases. In the $\textit{mass super-critical}$ 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 $μ_0\in (0,+\infty)$, our results on "best" local minimizers are also optimal.

 Zugehörige Institution(en) am KIT Institut für Analysis (IANA)Sonderforschungsbereich 1173 (SFB 1173) Publikationstyp Forschungsbericht/Preprint Publikationsmonat/-jahr 10.2020 Sprache Englisch Identifikator ISSN: 2365-662X KITopen-ID: 1000124276 Verlag KIT, Karlsruhe Umfang 46 S. Serie CRC Preprint ; 2020/27 Projektinformation SFB 1173/2 (DFG, DFG KOORD, SFB 1173/2 2019) Externe Relationen Siehe auch Schlagwörter Gagliardo-Nirenberg inequalities, biharmonic NLS, variational methods
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