On the asymptotically cubic fractional Schrödinger–Poisson system

Abstract

In this paper, we consider the following fractional Schrödinger–Poisson system: $\begin{array}{r}\left\{\begin{array}{ll}(-\mathrm{\Delta }{)}^{s}u+u+K(x)\phi u=Q(x\left)f\right(u)& \mathrm{i}\mathrm{n}{\mathbb{R}}^3,\\ (-\mathrm{\Delta }{)}^t\phi =K(x)u^2& \mathrm{i}\mathrm{n}{\mathbb{R}}^3,\end{array}\right.\end{array}$ where $s\in (\frac{3}{4},1)$, $t\in (0,1)$ are two fixed constants, K, Q are continuous, Q is positive, K can be vanishing at infinity, and f is a continuous and asymptotically cubic nonlinearity. Via the variational methods, we show that this system has a positive solution.

Keywords:

• Fractional Schrödinger–Poisson system
• asymptotically cubic
• Nehari manifold

MR(2010):

• 35J15
• 35J20
• 35J50

Acknowledgments

The authors are grateful to the reviewer for her/his valuable comments upon which the paper was revised. The authors would like to thank Professor F. Zhao for his fruitful discussions and his helpful suggestions during the preparation of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by The National Natural Sciences Foundation of People's Republic of China under Grant No. 11861072 and Chinese Postdoctoral Science Foundation.