Multi-peak solutions to fractional nonlinear Schrödinger equation with general nonlinearity

Abstract

In this paper we study the following fractional nonlinear Schrödinger equation ${\epsilon }^{2s}(-\mathrm{\Delta }{)}^{s}v+V(x)v=K(x)f(v),x\in {\mathbb{R}}^N$ where ε is a positive parameter, $s\in (0,1)$, N>2s, $V(x)$ and $K(x)$ are positive continuous functions. Using the variational techniques, we construct a family of positive solutions under general conditions on f. Moreover, we show that the solutions concentrate around the isolated components of positive local minima of the corresponding ground energy function as ε tends to zero.

Keywords:

• Fractional Schrödinger equations
• multi-peak solutions
• variational method

AMS SUBJECT CLASSIFICATIONS:

• 35A15
• 35J60
• 35B38

Acknowledgements

The authors would like to express sincere thanks to the anonymous referees for their carefully reading this paper and valuable useful comments.

Disclosure statement

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

Additional information

Funding

This work was supported by National Natural Science Foundation of China [grant numbers 11601234, 12001274,11571176] and Natural Science Foundation of the Jiangsu Higher Education Institutions of China [grant number 19KJB110014] and Natural Science Foundation of Jiangsu Province of China [grant number BK20160571].