  The manufacturing of crystal films lies at the heart of modern
nanotechnology. How to accurately predict the motion of a crystal surface is of
fundamental importance. Many continuum models have been developed for this
purpose, including a number of PDE models, which are often obtained as the
continuum limit of a family of kinetic Monte Carlo models of crystal surface
relaxation that includes both the solid-on-solid and discrete Gaussian models.
In this paper we offer an analytical perspective into some of these models. To
be specific, we study the existence of a weak solution to the boundary value
problem for the equation $ - \Delta e^{-\mbox{div}\left(|\nabla u|^{p-2}\nabla
u\right)}+au=f$, where $p>1, a>0$ are given numbers and $f$ is a given
function. This problem is derived from a crystal surface model proposed by
J.L.~Marzuola and J.~Weare (2013 Physical Review, E 88, 032403). The
mathematical challenge is due to the fact that the principal term in our
equation is an exponential function of a p-Laplacian. Existence of a
suitably-defined weak solution is established under the assumptions that
$p\in(1,2], \ N\leq 4$, and $f\in W^{1,p}$. Our investigations reveal that the
key to our existence assertion is how to control the set where
$-\mbox{div}\left(|\nabla u|^{p-2}\nabla u\right)$ is $\pm\infty$.
