The dimensional estimates of exponential growth solutions to uniformly elliptic equations of non-divergence form

In this paper, we demonstrate that, for general second order uniformly elliptic equations of non-divergence form in unbounded cylinders with zero boundary condition, the space of solutions with a given exponential growth is of finite dimension. In particular, we generalize a result of Hang and Lin (1999) [4] in the non-divergence's setting by using a different approach. This allows us to consider the general lower order terms. Furthermore, a sharp estimate of the dimension is established by using the mean value inequality.