  In this paper we study a crystal surface model first proposed by H.~Al Hajj
Shehadeh, R.V.~Kohn, and J.~Weare (2011 Physica D, 240,1771-1784). By seeking a
solution of a particular function form, we are led to a boundary value problem
for a fourth-order nonlinear elliptic equation. The mathematical challenge of
the problem is due to the fact that the degeneracy in the equation is directly
imposed by one of the two boundary conditions. An existence theorem is
established in which a meaningful mathematical interpretation of one of the
boundary conditions remains open. Our proof seems to suggest that this is
unavoidable. We also obtain self-similar solutions to the crystal surface model
which are positive and unbounded. This is in sharp contrast with the linear
biharmonic heat equation.
