Studies on a system of nonlinear Schr\"odinger equations with potential and quadratic interaction

Abstract

In this work, we study the existence of various classes of standing waves for a nonlinear Schr\"odinger system with quadratic interaction, along with a harmonic or partially harmonic potential. We establish the existence of ground-state normalized solutions for this system, which serve as local minimizers of the associated functionals. To address the difficulties raised by the potential term, we employ profile decomposition and concentration-compactness principles. The absence of global energy minimizers in critical and supercritical cases leads us to focus on local energy minimizers. Positive results arise in scenarios of partial confinement, attributed to the spectral properties of the associated linear operators. Furthermore, we demonstrate the existence of a second normalized solution using Mountain-pass geometry, effectively navigating the difficulties posed by the nonlinear terms. We also explore the asymptotic behavior of local minimizers, revealing connections with unique eigenvectors of the linear operators. Additionally, we identify global and blow-up solutions over time under specific conditions, contributing new insights into the dynamics of the system.