Ground state and sign-changing solutions for fractional Schrödinger–Poisson system with critical growth

ABSTRACT

In this paper, we study the following fractional Schrödinger–Poisson system with critical growth $\begin{array}{rl}& (-\mathrm{\Delta }{)}^{s}u+u+K(x\left)\phi \right(x)u=a(x)|u{|}^{p-2}u+|u{|}^{2_s^{\ast }-2}u\mathrm{i}\mathrm{n}{\mathbb{R}}^3,\\ & (-\mathrm{\Delta }{)}^t\phi =K(x)u^2\mathrm{i}\mathrm{n}{\mathbb{R}}^3,\end{array}$ where $p\in (4,2_s^{\ast })$, $s\in \left(\frac{3}{4},1\right)$, $t\in (0,1)$, 2s+2t>3 and $2_s^{\ast }=\frac{6}{3-2s}$. Under some suitable assumptions on $K\left(x\right)$ and $a\left(x\right)$, the ground state solutions and sign-changing solutions are obtained.

COMMUNICATED BY:

• D. Apushkinskaya

KEYWORDS:

• Fractional Schrödinger–Poisson system
• critical growth
• ground state solutions
• sign-changing solutions

AMS SUBJECT CLASSIFICATIONS:

• 35B38
• 35R11

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) grant 11501403 and fund program for the Scientific Activities of Selected Returned Overseas Professionals in Shanxi Province.