  We study the fractional Schr\"{o}dinger equations coupled with a neutral
scalar field
  $$
  (-\Delta)^s u+V(x)u=K(x)\phi u +g(x)|u|^{q-2}u, \quad x\in
\mathbb{R}^3,\qquad
  (I-\Delta)^t \phi=K(x)u^2, \quad x\in \mathbb{R}^3,
  $$ where $(-\Delta)^s$ and $(I-\Delta)^t$ denote the fractional Laplacian and
Bessel operators with $\frac{3}{4} <s<1$ and $0<t<1$, respectively. Under some
suitable assumptions for the external potentials $V$, $K$ and $g$, given
$q\in(1,2)\cup(2,2_s^*)$ with $2_s^*:= \frac{6}{3-2s}$, with the help of an
improved Fountain theorem dealing with a class of strongly indefinite
variational problems approached by Gu-Zhou [Adv. Nonlinear Stud., {\bf 17}
(2017), 727--738], we show that the system admits infinitely many nontrivial
solutions.
