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

• DOI: 10.3934/math.2022922
• Publication Date: 2022-07-13T10:21:01Z
• Authors: Kun Cheng; Li Wang
• Keywords: disc's theorem; variational methods; QA1-939; nodal solutions; kirchhoff type problem; schrödinger-poisson system; 0101 mathematics; 01 natural sciences; Mathematics

<abstract><p>This paper is dedicated to studying the following Kirchhoff-Schrödinger-Poisson system:</p>

<p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{\begin{array}{ll} - \left(a+b \int_{ \mathbb{R}^3} |\nabla u|^2 dx \right) \Delta u+V(|x|) u+\lambda\phi u = K(|x|)f(u), &amp; x \in \mathbb{R}^{3}, \\ -\Delta \phi = u^2, &amp; x \in \mathbb{R}^{3}, \end{array}\right. \end{equation*} $\end{document} </tex-math></disp-formula></p>

<p>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\rightarrow 0 $ and $ \lambda \rightarrow 0 $.</p></abstract>