Existence results for nonlinear fourth-order elliptic boundary value problems

This paper is concerned with the existence of a 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. We firstly present an existence result under a growth condition of $f(x,\,\xi ,\,\eta )$ on ξ and η, then use this result and a truncating function technique to establish an upper and lower solution theorem for the existence without monotonicity hypothesis of f. Using the upper and lower solution theorem we obtain an existence result of positive solution.