Existence and asymptotical behavior of the minimizer of Hartree type equation with periodic potentials

ABSTRACT

In the present paper, we consider Hartree type equation with periodic potential. On the one hand, we prove the existence and nonexistence of the minimizer for the energy functional by using concentration-compactness lemma. On the other hand, we prove the energy estimates and describe the asymptotical behavior of the minimizer

COMMUNICATED BY:

• V. Moroz

KEYWORDS:

• Hartree type equation
• Periodic potential
• Energy estimate
• Variational methods

AMS SUBJECT CLASSIFICATIONS:

• 35Q55
• 35J61
• 35J20
• 35Q60
• 49J40
• 35B32

Acknowledgements

The authors thank the referee's thoughtful reading of details of the paper and nice suggestions to improve the results.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by Natural Science Foundation of China [grant numbers 11571140, 71704066], Natural Science Foundation for Outstanding Young Scholars of Jiangsu Province [grant number BK20160063], the Six big talent peaks project in Jiangsu Province [grant number XYDXX-015], Natural Science Foundation of Jiangsu Province [grant numbers BK20150478, BK20170542], the Major Projects of the National Social Science Foundation [grant number 18VSJ038], and Jiangsu University foundation [grant numbers 11JDG117, 14JDG070].