Logarithmic Bose-Einstein condensates with harmonic potential

TL;DR

This paper investigates the existence, explicit form, and stability of ground state solutions for the logarithmic Schrödinger equation with harmonic potential, using variational methods and compactness arguments.

Contribution

It provides the first explicit computation of ground states and proves their orbital stability for this class of equations.

Findings

Existence of ground state solutions established.

Explicit Gausson-type solutions computed.

Orbital stability of solutions demonstrated.

Abstract

In this paper, by using a compactness method, we study the Cauchy problem of the logarithmic Schr\"{o}dinger equation with harmonic potential. We then address the existence of ground states solutions as minimizers of the action on the Nehari manifold. Finally, we explicitly compute ground states (Gausson-type solution) and we show their orbital stability.