Standing waves and global well-posedness for the 2d Hartree equation with a point interaction

Abstract

We study a class of two-dimensional nonlinear Schrödinger equations with point-like singular perturbation and Hartree non-linearity. The point-like singular perturbation of the free Laplacian induces appropriate perturbed Sobolev spaces that are necessary for the study of ground states and evolution flow. We include in our treatment both mass sub-critical and mass critical Hartree non-linearities. Our analysis is two-fold: we establish existence, symmetry, and regularity of ground states, and we demonstrate the well-posedness of the associated Cauchy problem in the singular perturbed energy space. The first goal, unlike other treatments emerging in parallel with the present work, is achieved by a nontrivial adaptation of the standard properties of Schwartz symmetrization for the modified Weinstein functional. This produces, among others, modified Gagliardo-Nirenberg type inequalities that allow to efficiently control the non-linearity and obtain well-posedness by energy methods. The evolution flow is proved to be global in time in the defocusing case, and in the focusing and mass sub-critical case. It is also global in the focusing and mass critical case, for initial data that are suitably small in terms of the best Gagliardo-Nirenberg constant.

Keywords:

• Hartree equation
• point-like singular perturbation of the Laplacian
• Green function
• Weinstein functional
• Schwartz re-arrangement

MSC SUBJECT CLASSIFICATION (2010):

• 35A15
• 35Q41
• 35Q55
• 47J30
• 47J35
• 81Q10
• 81Q80

Acknowledgments

The last two authors gratefully acknowledge the kind hospitality of V.G. at the Department of Mathematics of the University of Pisa, where a large part of this work was carried on.

Additional information

Funding

Partially supported by the Italian National Institute for Higher Mathematics – INdAM (V.G., A.M., R.S.), the project “Problemi stazionari e di evoluzione nelle equazioni di campo non-lineari dispersive” of GNAMPA – Gruppo Nazionale per l’Analisi Matematica (V.G.), the PRIN project no. 2020XB3EFL of the MIUR – Italian Ministry of University and Research (V.G.), the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences (V.G.), the Top Global University Project at Waseda University (V.G.), and the Alexander von Humboldt Foundation (A.M.).