On sign-changing solutions for quasilinear Schrödinger-Poisson system with critical growth

Abstract

In this paper, we consider the following quasilinear Schrödinger-Poisson system with critical growth: $\left\{\begin{array}{ll}-\mathrm{\Delta }u+V\left(x\right)u-\frac{1}{2}u\mathrm{\Delta }\left(u^2\right)+\varphi u=|u{|}^{4}u+\mu g\left(u\right),& x\in {\mathbb{R}}^3,\\ -\mathrm{\Delta }\varphi =u^2,& x\in {\mathbb{R}}^3,\end{array}\right.$ where $\mu >0$, $V\left(x\right)$ is a smooth potential function and g is a appropriate nonlinear function. For the sake of overcoming the technical difficulties caused by the quasilinear term, we shall apply the perturbation method by adding a 4-Laplacian operator to consider the perturbation problem. Moreover, when g satisfying suitable assumptions and sufficiently large μ, we take advantage of constraint variational method, the quantitative deformation lemma, Moser iteration and approximation technique to obtain a least-energy sign-changing solution $u_0$, which has precisely two nodal domains.

Keywords:

• Quasilinear Schrödinger-Poisson
• perturbation method
• sign-changing solutions

AMS Subject Classifications:

• 35A15
• 35J10
• 35J60

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

J. Zhang was supported by the Natural Science Foundation of Inner Mongolia Autonomous Region [grant number 2019MS01004] and the National Natural Science Foundation of China [grant number 11962025]. S. Liang was supported by the Foundation for China Postdoctoral Science Foundation [grant number 2019M662220], Scientific research projects for Department of Education of Jilin Province, China [grant number JJKH20210874KJ].