Standing waves with prescribed mass for Schrodinger equations with competing Van Der Waals type potentials

TL;DR

This paper studies standing wave solutions with fixed mass for Schrödinger equations involving Van Der Waals potentials, establishing existence, properties, and dynamics of solutions in a non-relativistic bosonic atom model.

Contribution

It introduces a new constrained minimization approach to prove the existence of two normalized solutions with distinct energy levels for the Schrödinger equation with Van Der Waals potentials.

Findings

Existence of two normalized solutions: a local minimizer and a global minimizer.

The global minimizer is located farther from the origin than the local minimizer.

Relations between ground state and least action solutions, with some dynamical and scattering results.

Abstract

We investigate standing waves with prescribed mass for a class of Schrodinger equations with competing Van Der Waals type potentials, arising in a model of non-relativistic bosonic atoms and molecules. By developing an approach based on a direct minimization of the energy functional on a new constrained manifold, we establish the existence of two normalized solutions for the corresponding stationary problem. One is a local minimizer with positive level and the other one is a global minimizer with negative level. Moreover, we find that the global minimizer is farther away from the origin than the local minimizer. Finally, we explore the relations between the ground state solution and the least action solution, and some dynamical behavior and scattering results are presented as well.