Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system with $p$-Laplacian

In this paper, we consider the following Schrödinger-Poisson system with $p$-laplacian \begin{equation} \begin{cases} -Δ_{p}u+V(x)|u|^{p-2}u+φ|u|^{p-2}u=f(u)\qquad&x\in$\mathbb{R}$^{3},\newline -Δφ=|u|^{p}&x\in$\mathbb{R}$^{3}. \end{cases}\notag \end{equation} We investigate the existence of multiple sign-changing solutions. By using the method of invariant sets of descending flow, we prove that this system has infinitely many sign-changing solutions. This system is new one coupled by Schrödinger equation of $p$-laplacian with a Poisson equation. Our results complement the study made by Zhaoli Liu, Zhiqiang Wang, Jianjun Zhang (Annali di Matematica Pura ed Applicata, 195(3):775-794(2016)).