Mixed-dispersion Schr\"odinger equations and Gagliardo-Nirenberg inequalities: equivalence between ground states and optimizers

TL;DR

This paper investigates the relationship between ground states of a mixed-dispersion nonlinear Schrödinger equation and optimizers of Gagliardo-Nirenberg inequalities, establishing their equivalence and strengthening existing existence results.

Contribution

It establishes the equivalence between energy and action ground states and links ground states to Gagliardo-Nirenberg inequality optimizers in a mixed-dispersion setting.

Findings

Identified the critical mass value dividing existence and nonexistence of ground states.

Proved the equivalence between energy and action ground states.

Connected ground states with optimizers of mixed Gagliardo-Nirenberg inequalities.

Abstract

We study a nonlinear Schr\"odinger equation with mixed dispersion in the mass competition regime, namely mass-supercritical for the Laplacian and mass-subcritical for the Bilaplacian. In this setting, the existence of a critical value of the mass $c_\varepsilon$, which divides existence and nonexistence of energy ground state solutions, was established in [Bonheure, Cast\'eras, dos Santos, Nascimento, SIAM J. Math. Anal. 50 (2018)]. In this work, we strengthen these results by investigating the relationship between the energy ground states with critical mass, and the optimizers of mixed Gagliardo-Nirenberg-type inequalities. Moreover, we discuss the equivalence between energy and action ground states solutions.