Ground state solutions for the non-linear fractional Schrödinger–Poisson system

ABSTRACT

In this paper, we study the existence of ground state solutions for the non-linear fractional Schrödinger–Poisson system $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{(-\mathrm{\Delta })}^{s}u+V(x)u+\varphi u={|u|}^{p-1}u,\hfill & \text{in}{\mathbb{R}}^3\text{,}\hfill \\ {(-\mathrm{\Delta })}^s\varphi =u^2,\hfill & \text{in}{\mathbb{R}}^3\text{,}\hfill \end{array}\right.\end{array}$$

where $2<p<2_s^{\ast }-1=\frac{3+2s}{3-2s}$, $s\in (\frac{3}{4},1)$. Under certain assumptions on V, the existence of a nontrivial ground state solution $(u,\varphi )$ is established using a monotonicity trick and global compactness Lemma. As its supplementary results, we prove some nonexistence results in the case of $1<p\le 2$ and $p=2_s^{\ast }-1$.

KEYWORDS:

• Fractional Schrödinger–Poisson system
• Pohozaev identity
• monotonicity trick
• ground state solution

AMS SUBJECT CLASSIFICATIONS:

• 35B09
• 35B38
• 35R11

Acknowledgements

The author wishes to express their gratitude to the anonymous referee for their careful reading and valuable suggestions.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by NSFC [grant number 11501403].