Ground state sign-changing solutions for a class of Schr\"{o}dinger-Poisson-Kirchhoff type problems with a critical nonlinearity in $\mathbb{R}^{3}$

J. Korean Math. Soc. 2021 Vol. 58, No. 5, 1181-1209 https://doi.org/10.4134/JKMS.j200497 Published online May 31, 2021 Printed September 1, 2021

Aixia Qian, Mingming Zhang Qufu Normal University; Qufu Normal University

Abstract : In the present paper, we are concerned with the existence of ground state sign-changing solutions for the following Schr\"{o}dinger-Poisson-Kirchhoff system $$\left\{ \aligned &-(1\!+\!b\int_{\mathbb R^3}|\nabla u|^{2}dx)\triangle u \!+\! V(x)u \!+\! k(x) \phi u\!=\! \lambda f(x)u \!+\! |u|^{4}u, &\hbox{in}\ \ \mathbb{R}^{3},\\ & -\triangle \phi \!=\! k(x) u^{2}, & \hbox{in}\ \ \mathbb{R}^{3},\endaligned\right.$$ where $b>0$, $V(x)$, $k(x)$ and $f(x)$ are positive continuous smooth functions; $0<\lambda<\lambda_{1}$ and $\lambda_{1}$ is the first eigenvalue of the problem $-\triangle u + V(x)u = \lambda f(x)u$ in $H$. With the help of the constraint variational method, we obtain that the Schr\"{o}dinger-Poisson-Kirchhoff type system possesses at least one ground state sign-changing solution for all $b>0$ and $0<\lambda<\lambda_{1}$. Moreover, we prove that its energy is strictly larger than twice that of the ground state solutions of Nehari type.

Keywords : Schr\"{o}dinger-Poisson-Kirchhoff system, ground state sign-changing solution, critical nonlinearity, nonlocal term

MSC numbers : Primary 35J05, 35J20, 35J60

Supported by : This work was financially supported by the Shandong Science Foundation ZR2020MA005.