Summary: The purpose of this paper is to study the following nonlinear problem involving the \(p(x)\)-Laplace operator: \[\begin{cases}-\Delta_{p(x)}u=\lambda f(x,u)\qquad&\text{in }\Omega,\\ u>0&\text{in }\Omega,\\ u=0&\text{on }\partial\Omega,\end{cases}\tag{\(P_\lambda\)}\] where \(\Omega\subset \mathbb R^N\) (\(N\ge 2\)) is a bounded domain with \(C^2\)-boundary, \(\lambda\) is a positive parameter, \(p(x)\) and \(f(x,u)\) are assumed to satisfy the assumptions (H1)-(H4) in the introduction. We employ variational techniques in order to show the existence of a number \(\Lambda\in (0,\infty)\) such that problem (\(P_\lambda\)) admits two solutions for \(\lambda\in (0,\Lambda)\), one solution for \(\lambda=\Lambda\) and no solutions for \(\lambda>\Lambda\).