This paper is concerned with the existence of a positive solution of the nonlinear fourth-order elliptic boundary value problem \( \left \{ \textstyle\begin{array}{l} {\Delta}^{2} u = f(x,\,u,\,\Delta u),\qquad x\in \Omega , \\ u=\Delta u=0, \qquad x\in \partial \Omega , \end{array}\displaystyle \right . \) where Ω is a bounded smooth domain in $\mathbb{R}^{N}$ , $f: \overline{\Omega}\times \mathbb{R}^{+}\times \mathbb{R}^{-}\to \mathbb{R}^{+}$ is a continuous function. Under two inequality conditions of $f(x,\,\xi ,\,\eta )$ when $|(\xi ,\,\eta )|$ is small and large, an existence result of positive solutions is obtained. The inequality conditions is related to the principal eigenvalue $\lambda _{1}$ of the Laplace operator −Δ with the boundary condition $u|_{\partial \Omega}=0$ . The discussion is based on the fixed-point index theory in cones.
