Acta mathematica scientia,Series A ›› 2026, Vol. 46 ›› Issue (4): 1428-1442.

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Nontrivial Solution for the Kirchhoff Type Quasilinear Schrödinger-Poisson Systems with Indefinite Potentials

Guzhen Huang1(), Li Wang1,*(), Jixiu Wang2()   

  1. 1 School of Science, East China Jiaotong University, Nanchang 330013
    2 School of Artificial Intelligence, Jianghan University, Wuhan 430074
  • Received:2025-12-29 Revised:2026-02-05 Online:2026-08-26 Published:2026-06-10
  • Contact: Li Wang E-mail:huangguzhen0428@163.com;wangli.423@163.com;wangjixiu127@aliyun.com
  • Supported by:
    NSFC(12161038);Jiangxi Provincial Natural Science Foundation(20232BAB201009);Science and Technology Project of Education Department of Jiangxi Province(GJJ2400901)

Abstract:

In this paper, we investigate the Kirchhoff type Quasilinear Schrödinger-Poisson systems:

$\begin{align*} \begin{cases} -\left(a+b\int_{\mathbf{R}^{3}}|\nabla u|^{2}\mathrm{d} x\right)\Delta u + V(x)u + \phi u = f(x, u), & x \in \mathbf{R}^{3}, \\ -\Delta\phi - \varepsilon^{4}\Delta_{4}\phi = u^{2}, & x \in \mathbf{R}^{3}, \end{cases} \end{align*}$

where the potential $V$ is indefinite, leading the Schrödinger operator $ -\Delta + V $ exhibit a finite-dimensional negative space. By Morse theory, we prove the existence of nontrivial solutions to this system. It also discusses the asymptotic behavior of the solution as $\varepsilon \to 0$ and $b \to 0$ separately.

Key words: quasilinear Schr?inger-Poisson systems, local linking, morse theory

CLC Number: 

  • O175.23
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