Some non-homogeneous Gagliardo-Nirenberg inequalities and application to a biharmonic non-linear Schr\"odinger equation

TL;DR

This paper develops new non-homogeneous Gagliardo-Nirenberg inequalities and applies them to analyze the existence and non-existence of minimizers for a biharmonic nonlinear Schrödinger equation with fixed mass.

Contribution

It introduces a novel method for proving non-homogeneous Gagliardo-Nirenberg inequalities and applies this to establish optimal existence results for minimizers in different mass regimes.

Findings

Existence of minimizers in mass-subcritical and mass-critical cases.

Non-existence of global minimizers in mass supercritical case.

Identification of a threshold mass for local minimizers.

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.