Characterization of ground-states for a system of $M$ coupled semilinear Schr\"odinger equations and applications

TL;DR

This paper investigates ground-states for coupled semilinear Schrödinger equations, providing existence, characterization, and applications to inequalities and blowup phenomena in the critical case.

Contribution

It introduces a variational approach to characterize ground-states and applies these results to inequalities and dynamic behaviors of the system.

Findings

Established existence and characterization of ground-states.

Derived the optimal constant for the vector-valued Gagliardo-Nirenberg inequality.

Analyzed global existence, concentration, and blowup profiles in the critical case.

Abstract

We focus on the study of ground-states for the system of $M$ coupled semilinear Schr\"odinger equations with power-type nonlinearities and couplings. General results regarding existence and characterization are derived using a variational approach. We show the usefulness of such a characterization in several particular cases, including those for which uniqueness of ground-states is already known. Finally, we apply the results to find the optimal constant for the vector-valued Gagliardo-Nirenberg inequality and we study global existence, $L^2$-concentration phenomena and blowup profile for the evolution system in the $L^2$-critical power case.