Positive and sign-changing least energy solutions for a fractional Schrödinger–Poisson system with critical exponent

ABSTRACT

In this paper, we study the following fractional Schrödinger–Poisson system (FSP) $\begin{array}{rl}& (-\mathrm{\Delta }{)}^{s}u+u+K(x)\phi u=h(x)|u{|}^{p-2}u+|u{|}^{2_s^{\ast }-2}u,\mathrm{in}{\mathbb{R}}^3,\\ & (-\mathrm{\Delta }{)}^t\phi =K(x)u^2,\mathrm{in}{\mathbb{R}}^3,\end{array}$(FSP) where $s\in (\frac{3}{4},1),t\in (0,1)$ are two fixed constants, $2_s^{\ast }:=6/(3-2s)$ is the fractional critical exponent in dimension 3. Under some certain assumptions on non-negative functions $K\left(x\right)$ and $h\left(x\right)$, we obtain the existence of a positive and a sign-changing least energy solution for $\left(\mathrm{FSP}\right)$ via variational methods. Moreover, we show that the energy of the sign-changing least solution is strictly larger than twice of the least energy.

KEYWORDS:

• Fractional Schrödinger–Poisson system
• sign-changing solutions
• variational methods

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35Q40
• 35J50
• 58E05

Acknowledgments

We should like to thank the anonymous referee for his/her careful readings of our manuscript and the useful comments made for its improvement.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by National Natural Science Foundation of China (NSFC) [11771385, 11671026], China.