Multiplicity and concentration of solutions for a fractional Schrödinger–Poisson system with sign-changing potential

Abstract

This paper is concerned with the following fractional Schrödinger–Poisson system: $\{\begin{array}{ll}{\displaystyle (-\mathrm{\Delta }{)}^{\alpha }u+V_{\lambda }(x)u+\mu \varphi u=f(x,u)+\beta (x)|u{|}^{\nu -2}u}& \text{in}{\mathbb{R}}^3,\\ (-\mathrm{\Delta }{)}^t\varphi =u^2& \text{in}{\mathbb{R}}^3,\end{array}$where $\mu >0$ is a parameter, $\alpha ,t\in (0,1)$, $\nu \in (1,2)$, $2\alpha +2t>3$, $V_{\lambda }$ is allowed to be sign-changing and f is an indefinite function. We require that $V_{\lambda }(x)=\lambda V^{+}(x)-V^{-}(x)$ with $V^{+}(x)$ having a potential well Ω whose depth is controlled by λ and $V^{-}(x)\ge 0$ for all $x\in {\mathbb{R}}^3$. Under some suitable assumptions on $V_{\lambda }(x)$ and $f(x,u)$, we prove that the above system has at least two nontrivial solutions. Moreover, the phenomenon of concentration of the solutions is also explored.

Keywords:

• Fractional Schrödinger–Poisson system
• multiplicity
• concentration
• variational methods

2010 Mathematics Subject Classifications:

• 35B38
• 35J60

Acknowledgments

The authors thank the referees for carefully reading the manuscript, giving valuable comments and suggestions to improve the results as well as the exposition of this paper.

Disclosure statement

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

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [Grant No. 12001114, 12071486] and the Guangdong Basic and Applied Basic Research Foundation [Grant No. 2019A1515110275].