  This paper deals with some mathematical models arising in the theory of
epitaxial growth of crystal. We focalize the study on a stationary problem
which presents some analytical difficulties. We study the existence of
solutions. The central model in this work is given by the following fourth
order elliptic equation, $$\begin{array}{rclll} \Delta^2 u=\text{det} \left(D^2
u \right) &+&\lambda f, \quad & x\in \Omega\subset\mathbb{R}^2\\
\hbox{conditions on} &\quad& & \partial \Omega. \end{array} $$ The framework to
study the problem deeply depends on the boundary conditions.
