In this paper the following system of elliptic equations is considered: \[ \begin{cases} -\Delta u+\lambda \phi u=|u|^{p-1}u & \text{ in }\Omega,\\ -\Delta \phi=u^2 & \text{ in }\Omega,\\ u, \phi=0 & \text{ on } \partial \Omega,\end{cases} \] where \(p\in (1,5)\) and \(\lambda>0\) are real parameters, and \(\Omega\subset \mathbb{R}^3\) is a smooth bounded domain. The existence and multiplicity of nontrivial solutions are studied. Regarding positive solutions \((u,\phi)>(0,0)\), the main results are as follows:
1) \(1<p<2\). For \(\lambda\) small, there are at least two positive solutions. For \(\lambda>>1\), there is no positive solution.
2) \(p=2\). For \(\lambda\) small, there is at least one positive solution. For \(\lambda>>1\), there is no positive solution.
3) \(2<p<5\). For \(\lambda\) small, there is at least one positive solution. For almost every \(\lambda>>1\), there is at least one positive solution.