The paper deals with the semilinear elliptic Dirichlet boundary problem \[ \begin{cases} -\Delta u=f(x,u)\quad & \text{in }\Omega,\\ u=0\quad &\text{on } \partial \Omega,\end{cases} \tag{1} \] where \(\Omega\subset R^d\) \((d\geq 1)\) is a bounded smooth domain and \(f:\overline\Omega\times R\to R\) is a Carathéodory function. Throughout this paper the authors assume that there are positive constant \(C_1\) and \(f_0\in L^q(\Omega)\) (real valued) such that \(|f(x,t) |\leq C_1|t|^{p-1} +f_0(x)\), and \(f(\cdot,0)\in L^\infty (\Omega)\) for all \(t\in R\) and a.e. \(x\in\Omega\), where \(p\in(2,{2d\over d-2})\) for \(d\geq 3\), \(p\in (2,+\infty)\) for \(d=1,2\). By using both reduction method and the minimax methods the authors obtain the existence and multiplicity results of solutions of (1).