  In this paper, we consider the following Schr\"odinger-Poisson system with
$p$-laplacian \begin{equation}
  \begin{cases}
  -\Delta_{p}u+V(x)|u|^{p-2}u+\phi|u|^{p-2}u=f(u)\qquad&x\in\mathbb{R}^{3},\newline
  -\Delta\phi=|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 Schr\"odinger 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)).

Analysis of PDEs