Existence of solutions for the fractional $p＆q$-Laplacian equation with nonlocal Choquard reaction

We consider the following class of fractional $p＆q$-Laplacian differential equation with Choquard term:

$$\{\begin{array}{ll}(-\mathrm{\Delta }{)}_p^{s}u+(-\mathrm{\Delta }{)}_q^{s}u+V(x)(|u{|}^{p-2}u+|u{|}^{q-2}u)+{\int }_{{\mathbb{R}}^N}g(x)|u{|}^{r}dx={\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{k(u(x))K(u(y))}{|x-y{|}^{\alpha }}dxdy,& x\in {\mathbb{R}}^N,\\ u\in W_V^{s,p}({\mathbb{R}}^N)\bigcap W_V^{s,q}({\mathbb{R}}^N),& x\in {\mathbb{R}}^N,\end{array}$$

where $s\in (0,1),2\le p\le r\le q<N/s,0<\alpha <N$, $(-\mathrm{\Delta }{)}_m^s$ with $m\in \{p,q\}$ is the fractional $m$-Laplacian operator, $g(x):{\mathbb{R}}^N\to \mathbb{R}$, by introducing a potential term function to restore compactness in the corresponding spaces. Using variational techniques and inequalities such as Hardy–Littlewood–Sobolev, we ensure the geometric conditions of the mountain pass theorem in order to show the existence of solutions.