Exponential non-linearity in crystal surface models
Xiangsheng Xu · arXiv (Cornell University) · 2021
We consider the existence of a solution to the boundary value problem for the equation $-\mbox{div} \left(D( abla u) abla e^{-\mbox{div}\left(| abla u|^{p-2} abla u+\beta_0| abla u|^{-1} abla u\right)}\right) +a u=f$. This problem is derived from the mathematical modeling of crystal surfaces. The analytical difficulty is due to the fact that the smallest eigenvalue of the mobility matrix $D( abla u)$ is not bounded away from $0$ below and the inside operator is an exponential function composed with a linear combination of the p-Laplace operator and the 1-Laplace operator. Known existence results on problems related to ours either have to allow the possibility that the exponent in the equation be a measure or assume that data are suitably small in order to eliminate the possibility. In this paper we show the existence of a non-measure valued weak solution without any smallness assumption on the data. We achieve this by employing a power series expansion technique.