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arXiv · 2010.01448

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

Antonio J. Fernández·Louis Jeanjean·Rainer Mandel·Mihai Mariş
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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.

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Antonio J. Fernández, Louis Jeanjean, Rainer Mandel, Mihai Mariş. 2020-10-04. Some non-homogeneous Gagliardo-Nirenberg inequalities and application to a biharmonic non-linear Schr\"odinger equation. https://doi.org/10.1016/j.jde.2022.04.037

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