  We investigate the two-dimensional Hubbard model on the triangular lattice
with anisotropic hopping integrals at half filling. By means of a self-energy
functional approach, we discuss how stable the non-magnetic state is against
magnetically ordered states in the system. We present the zero-temperature
phase diagram, where the normal metallic state competes with magnetically
ordered states with $(\pi, \pi)$ and $(2\pi/3, 2\pi/3)$ structures. It is shown
that a non-magnetic Mott insulating state is not realized as the ground state,
in the present framework, but as a meta-stable state near the magnetically
ordered phase with $(2\pi/3, 2\pi/3)$ structure.
