Ground state solutions for the periodic fractional Schrödinger–Poisson systems with critical Sobolev exponent

Abstract

We study the fractional Schrödinger–Poisson system with critical Sobolev exponent

$$\left\{\begin{array}{cc}{\left(-\mathrm{\Delta }\right)}^{s}u+V\left(x\right)u+\varphi u=f\left(x,u\right)+K\left(x\right)|u{|}^{2_s^{\ast }-2}u\hfill & \text{ in }{ℝ}^3,\hfill \\ {\left(-\mathrm{\Delta }\right)}^t\varphi =u^2\hfill & \text{ in }{ℝ}^3,\hfill \end{array}\right.$$

where ${\left(-\mathrm{\Delta }\right)}^{\alpha }$ denotes the fractional Laplacian of order $\alpha =s,t\in \left(0,1\right)$; $V\left(x\right)$, $f\left(x,u\right)$ and $K\left(x\right)$ are $1$-periodic in the $x$-variables; $2_s^{\ast }=6∕\left(3-2s\right)$ is the fractional critical Sobolev exponent in dimension $3$. Under some weaker conditions on $f$, we prove the existence of ground state solutions for such a system via the mountain pass theorem in combination with the concentration-compactness principle. Our results are new even for $s=t=1$.