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: Nabla symbol

<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), & x \in \mathbb{R}^{3}, \\ -\Delta \phi u^2, \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 nonlinearity f $, we develop a direct approach establish existence of infinitely many nodal solutions \{u_k^{b, \lambda}\} with prescribed number nodes k using Gersgorin disc's theorem, Miranda theorem Brouwer degree theory. Moreover, prove that energy strictly increasing in give convergence property as b\rightarrow 0 \lambda \rightarrow $.</p></abstract>