Partial regularity for an exponential PDE in crystal surface models

We study the regularity properties of a weak solution to the boundary value problem for the equation $-\mathrm{\Delta }\rho +au=f$ in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$, where $\rho=e^{-\mbox{div}\left(|\nabla u|^{p-2}\nabla u+\beta_0|\nabla u|^{-1}\nabla u\right)}$. This problem is derived from the mathematical modeling of crystal surfaces. It is known that the exponent term can exhibit singularity. In this paper we obtain a partial regularity result for the weak solution. It asserts that there exists an open subset ${\mathrm{\Omega }}_0\subset \mathrm{\Omega }$ such that $\mathrm{\mid }\mathrm{\Omega }\setminus {\mathrm{\Omega }}_0\mathrm{\mid }=0$ and the exponent term is locally bounded in ${\mathrm{\Omega }}_0$. Furthermore, if $x_0\in \Omega\setminus\Omega_0$,then$\rho$vanishesof$N+2-\varepsilon$orderat$x_0$foreach$\varepsilon\in(0,2)$. Our results reveal that the exponent term behaves well if it stays away from negative infinity.