Weak solutions of the boundary value problem \[ \Delta u+f(u)-\varepsilon =0 \quad \text{in }\Omega,\qquad u=0 \quad \text{on }\partial\Omega, \] are studied, where \(\Omega \) is a smooth bounded region in \(\mathbb{R}^{N}\), the function \(f\in C^{1}(\mathbb{R})\) satisfies \(f(0)=0\) and \(\varepsilon \neq 0\) with \(|\varepsilon |\) is taken to be sufficiently small. Thus, \(f(u)-\varepsilon \) is nonzero at \(u=0\). The leading term \(f\) is assumed to be both subcritical and superlinear. Using the same line of arguments as in A. Castro, J. Cossio and J. M. Neuberger [Rocky Mountain J. Math. (to appear)] the existence of at least four nontrivial solutions is proved: the first one \(\omega _{0}\) is the small negative solution in \(\Omega \), the second one \(\omega _{1}^{-}<0\) in \(\Omega \), the third one \(\omega _{1}^{+}\) has a nontrivial positive part in \(\Omega \), and the fourth one \(\omega _{2}\) changes sign in \(\Omega \), i.e. \(\omega _{2}\) has nontrivial both positive and negative parts in \(\Omega \).