The boundary value problem \(\Delta u =g(x,u) = h(x)\) in a bounded regular open set \(\Omega \subset \mathbb{R} ^n\), \(u=0\) on \(\partial\Omega\) is considered when \(h(x)\) is a nonnegative function on \(\Omega\) and the nonlinearity \(g\) “crosses” the first eigenvalue of \(-\Delta\) in \(\Omega\) with zero boundary condition. Conditions are presented under which the above problem has at least two solutions for every \(h\in C^{0,\alpha}(\overline{\Omega})\), \(h\geq 0\), \(h\neq 0\) in \(\Omega\). In the case \(n=1\) the conditions can be weakened.