Normalized ground states for general pseudo-relativistic Schrödinger equations

ABSTRACT

In this paper, we consider the pseudo-relativistic type Schrödinger equations with general nonlinearities. By studying the related constrained minimization problems, we obtain the existence of ground states via applying the concentration-compactness principle. Then some properties of the ground states have been discussed, including regularity, symmetry and etc. Furthermore, we prove that the set of minimizers is a stable set for the initial value problem of the equations, that is, a solution whose initial data is near the set will remain near it for all time.

Keywords:

• Pseudo-relativistic Schrödinger equation
• ground state
• concentration-compactness
• regularity
• orbital stability

2010 Mathematics Subject Classifications:

• 35Q55
• 35B35
• 35A15

Acknowledgments

The authors are grateful to the referees for their careful reading and helpful suggestions and comments. Haijun Luo is supported by the NSFC (Grant No. 11901182) and the Fundamental Research Funds of the Central Universities (No. 531118010205). Dan Wu is supported by the NSFC (Grant No. 11901183), NSF of Hunan Province (No. 2017JJ3028) and the Fundamental Research Funds of the Central Universities (No. 531118040104).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Haijun Luo is supported by the National Natural Science Foundation of China (NSFC) [grant number 11901182] and the Fundamental Research Funds of the Central Universities [No. 531118010205]. Dan Wu is supported by the National Natural Science Foundation of China (NSFC) [grant number 11901183], Natural Science Foundation of Hunan Province [No. 2017JJ3028] and the Fundamental Research Funds of the Central Universities [No. 531118040104].