Infinitely many solutions for a class of fractional Schrodinger equations coupled with neutral scalar field

We study the fractional Schrödinger equations coupled with a neutral scalar field $$(-\mathrm{\Delta }{)}^{s}u+V(x)u=K(x)\phi u+g(x)\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u,x\in {\mathbb{R}}^3,(I-\mathrm{\Delta }{)}^t\phi =K(x)u^2,x\in {\mathbb{R}}^3,$$ where $(-\mathrm{\Delta }{)}^s$ and $(I-\mathrm{\Delta }{)}^t$ 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 $q\in (1,2)\cup (2,2_s^{\ast })$ with $2_s^{\ast }:=\frac{6}{3-2s}$, 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.