Sign-changing solutions for a modified quasilinear Kirchhoff–Schrödinger–Poisson system via perturbation method

Abstract

In this paper, we consider the following quasilinear Kirchhoff–Schrödinger–Poisson system: $\begin{array}{r}\{\begin{array}{l}{\displaystyle -(a+b{\int }_{{\mathbb{R}}^3}|\mathrm{\nabla }u{|}^2\mathrm{d}x)\mathrm{\Delta }u}\\ +V(x)u-{\displaystyle \frac{1}{2}}u\mathrm{\Delta }(u^2)+\varphi u=g(u),& x\in {\mathbb{R}}^3,\\ -\mathrm{\Delta }\varphi =u^2,& x\in {\mathbb{R}}^3,\end{array}\end{array}$ where a, b are positive constants, $V(x)$ is a smooth potential function and g is an appropriate nonlinear function. To overcome the technical difficulties caused by the quasilinear term, the perturbation method of adding 4-Laplacian operator is adapted to consider the perturbation problem, so that the corresponding functional has both smoothness and compactness in the appropriate space. Moreover, when g satisfies the appropriate hypotheses, a sign-changing solution $u_0$ of above problem can be obtained by taking advantage of constraint variational method, the quantitative deformation lemma and approximation technique, which has precisely two nodal domains.

Keywords:

• Quasilinear
• Kirchhoff–Schrödinger–Poisson system
• perturbation method
• sign-changing solutions

AMS Subject Classifications:

• 35A15
• 35J10
• 35J60

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

J. Zhang was supported by the Natural Science Foundation of Inner Mongolia Autonomous Region (No. 2019MS01004), Inner Mongolia Autonomous Region university scientific research project (No. NJZY18021) and the National Natural Science Foundation of China (No. 11962025). C. Ji was partially supported by National Natural Science Foundation of China (No. 12171152) and Natural Science Foundation of Shanghai (No. 20ZR1413900).