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Some non-homogeneous Gagliardo-Nirenberg inequalities and application to a biharmonic non-linear Schr\'odinger equation
- Source :
- Journal of Differential Equations 330 (2022), 1-65
- Publication Year :
- 2020
-
Abstract
- 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.<br />Comment: Final version. The article will appear in Journal of Differential Equations 328 (2022), pp. 1-65
Details
- Database :
- arXiv
- Journal :
- Journal of Differential Equations 330 (2022), 1-65
- Publication Type :
- Report
- Accession number :
- edsarx.2010.01448
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1016/j.jde.2022.04.037