# Existence of solutions for the Schrödinger-Kirchhoff-Poisson systems with a critical nonlinearity

Boundary Value Problems

, 2016:210

Recent Advances in PDE and Their Applications

Abstract

This paper is concerned with the following Schrödinger-Kirchhoff-Poisson system: $$\textstyle\begin{cases} -a+b\int {\Omega}| abla u|^{2}\,\mathrm{d}x\triangle u+\lambda\phi u=\eta fx,u+u^{5}, and\mbox{in } \Omega, \\ -\triangle\phi=u^{2}, and\mbox{in } \Omega , \\ u=\phi=0, and\mbox{on } \partial\Omega, \end{cases}$$ where \a\geq0\, \b>0\ and \\eta,\lambda>0\, \\Omega\subset R^{3}\ is a bounded smooth domain. With the help of the variational methods, the existence of a non-trivial solution is obtained.Keywordsmountain pass theorem variational methods Schrödinger-Kirchhoff-Poisson system non-trivial solution MSC35B38 35G99  Download fulltext PDF

Author: Liuyang Shao - Haibo Chen