Journal of the
Korean Mathematical Society JKMS

In this paper, we consider the following Schr\"{o}dinger-Poisson system: $$ \left\{ \begin{array}{ll} -\triangle {u}+ u+\lambda\phi u =\mu f(u)+|u|^{p-2}u, & \mbox{in}\,\,\Omega, \\ -\triangle {\phi}=u^2, & \mbox{in}\,\, \Omega,\\ \phi=u=0, & \mbox{on}\,\, \partial\Omega, \end{array} \right. $$ where $\Omega$ is a smooth and bounded domain in $\R^3$, $p\in (1,6]$, $\lambda,\mu$ are two parameters and $f:\R\rightarrow\R$ is a continuous function. Using some critical point theorems and truncation technique, we obtain three multiplicity results for such a problem with subcritical or critical growth.

Keywords: Schr\"{o}dinger-Poisson system, subcritical growth, critical growth, variational methods

MSC numbers: 35J61, 35J66, 35J57