Least energy sign-changing solutions for Kirchhoff-Schrödinger-Poisson system on bounded domains

Abstract

We investigate the following nonlinear system $ \begin{cases} -(a+ b\int _{\Omega}|\nabla u|^{2}dx)\Delta u+\phi u = \lambda u+\mu|u|^{2}u, \; \ x\in\Omega, \\ -\Delta\phi = u^{2}, \; \ x\in\Omega, \\ u = \phi = 0, \; \ x\in \partial\Omega, \end{cases} $ with $ a, b > 0 $, $ \lambda, \mu\in\mathbb{R} $, and $ \Omega\subset \mathbb{R}^{3} $ is bounded with smooth boundary. Let $ \lambda_{1} > 0 $ be the first eigenvalue of $ (-\Delta u, H^{1}_{0}(\Omega)) $. We get that for certain $ \widetilde{\mu} > 0 $ there exists at least one least energy sign-changing solution for the above system if $ \lambda < a\lambda_{1} $ and $ \mu > \widetilde{\mu} $. In addition, we remark that the nonlinearity $ \lambda u+\mu|u|^{2}u $ does not satisfy the growth conditions.

Keywords

• kirchhoff type equation
• schrödinger-poisson problem
• sign-changing solutions
• nehari manifold
• the growth conditions