Infinitely many sign-changing solutions for the nonlinear Schr\"odinger-Poisson system with $p$-Laplacian

TL;DR

This paper proves the existence of infinitely many sign-changing solutions for a coupled Schr"odinger-Poisson system involving the p-Laplacian, using invariant sets of descending flow, expanding the understanding of such nonlinear systems.

Contribution

It introduces a new coupled Schr"odinger-Poisson system with p-Laplacian and establishes the existence of infinitely many sign-changing solutions using a novel method.

Findings

Proved existence of infinitely many sign-changing solutions.

Developed a new approach using invariant sets of descending flow.

Extended previous results to a p-Laplacian coupled system.

Abstract

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)).