Variational proof of the existence of periodic orbits in the Anisotropic Kepler problem The anisotropic Kepler problem is a model of the motion of free electrons on a $ n $ type semiconductor with, and is known to be a non-integrable Hamiltonian system. Many approximate periodic solutions have been found by numerical calculation \cite{numerical-solution}, but they have not been rigorously proved to exist. In this paper, we first show that the action functional of the anisotropic Kepler problem has a minimizer under a fixed region condition with boundary conditions on a vertical half-line. Next, we identify the smallest collision trajectory that satisfies the same boundary conditions. It is shown that the collision solution does not become the minimizer by constructing an orbit with an action functional smaller than this collision orbit by local deformation. This holds for any mu \in (0,1). Reversibility allows the periodic orbit to be constructed from the obtained minimizer.