Nodal solutions for the Kirchhoff-Schrödinger-Poisson system in ${\mathbb{R}}^3$

This paper is dedicated to studying the following Kirchhoff-Schrödinger-Poisson system:

$$\{\begin{array}{ll}-\left(a+b{\int }_{{\mathbb{R}}^3}|\mathrm{\nabla }u{|}^{2}dx\right)\mathrm{\Delta }u+V(|x|)u+\lambda \varphi u=K(|x|)f(u),& x\in {\mathbb{R}}^3,\\ -\mathrm{\Delta }\varphi =u^2,& x\in {\mathbb{R}}^3,\end{array}$$

where $V,K$ are radial and bounded away from below by positive numbers. Under some weaker assumptions on the nonlinearity $f$, we develop a direct approach to establish the existence of infinitely many nodal solutions $\{u_k^{b,\lambda }\}$ with a prescribed number of nodes $k$, by using the Gersgorin disc's theorem, Miranda theorem and Brouwer degree theory. Moreover, we prove that the energy of $\{u_k^{b,\lambda }\}$ is strictly increasing in $k$, and give a convergence property of $\{u_k^{b,\lambda }\}$ as $b\to 0$ and $\lambda \to 0$.