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Article Dans Une Revue Journal of Differential Equations Année : 2022

Non-homogeneous Gagliardo-Nirenberg inequalities and application to a biharmonic non-linear Schrödinger equation

Résumé

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.
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Dates et versions

hal-03336068 , version 1 (06-09-2021)

Identifiants

Citer

Antonio Fernández, Louis Jeanjean, Rainer Mandel, Mihai Mariş. Non-homogeneous Gagliardo-Nirenberg inequalities and application to a biharmonic non-linear Schrödinger equation. Journal of Differential Equations, 2022, 330, pp.1-65. ⟨10.1016/j.jde.2022.04.037⟩. ⟨hal-03336068⟩
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