Some results for a supercritical Schrödinger-Poisson type system with $(p,q)$-Laplacian

In this work, we focus our attention on the existence of nontrivial solutions to the following supercritical Schrödinger-Poisson type system with $(p,q)$-Laplacian:

$$\{\begin{array}{ll}-{\mathrm{\Delta }}_{p}u-{\mathrm{\Delta }}_{q}u+\varphi |u{|}^{q-2}u=f\left(x,u\right)+\mu |u{|}^{s-2}u& \text{in}\mathrm{\Omega },\\ -\mathrm{\Delta }\varphi =|u{|}^q& \text{in}\mathrm{\Omega },\\ u=\varphi =0& \text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}$$

where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded smooth domain, $\mu >0,N>1$, and $-{\mathrm{\Delta }}_{\mathrm{\wp }}\phi =div(|\mathrm{\nabla }\phi {|}^{\mathrm{\wp }-2}\mathrm{\nabla }\phi )$, with $\mathrm{\wp }\in \{p,q\}$, is the homogeneous $\mathrm{\wp }$-Laplacian. $1<p<q<\frac{q^{\ast }}{2}$, $q^{\ast }:=\frac{Nq}{N-q}<s$, and $q^{\ast }$ is the critical exponent to $q$. The proof is accomplished by the Moser iterative method, the mountain pass theorem, and the truncation technique. Furthermore, the $(p,q)$-Laplacian and the supercritical term appear simultaneously, which is the main innovation and difficulty of this paper.