Some fourth order nonlinear elliptic problems related to epitaxial growth

Abstract: 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.