On nonlinear fractional Schrödinger equations with Hartree-type nonlinearity

Abstract

In the present paper we deal with a nonlinear fractional Schrödinger equation in of the form

where is a parameter, , , V(x) is a continuous potential, and is the Riesz potential. By applying variational methods and the Ljusternik–Schnirelmann category theory, under some suitable conditions on V(x) we prove the existence and multiplicity of solutions for the above equation when the parameter is sufficiently small. Moreover, the concentration behavior of solutions for the fractional Schrödinger equation is also explored.

Keywords:

• Fractional Schrödinger equation
• Hartree-type nonlinearity
• existence and multiplicity
• concentration behavior

AMS Subject Classifications:

• 35A01
• 35B25
• 35J60

Acknowledgements

The authors thank the referee for recommending Refs. [7,18–21] and for suggesting improvements of the presentation.

Notes

No potential conflict of interest was reported by the authors.