  In this paper, we study minimization problems among Bravais lattices for
finite energy per point. We prove that if a function is completely monotonic,
then the triangular lattice minimizes energy per particle among Bravais
lattices with density fixed for any density. Furthermore we give an example of
convex decreasing positive potential for which triangular lattice is not a
minimizer for some densities. We use Montgomery method presented in our
previous work to prove minimality of triangular lattice among Bravais lattices
at high density for some general potentials. Finally we deduce global
minimality among all Bravais lattices, i.e. without density constraint, of a
triangular lattice for some parameters of Lennard-Jones type potentials and
attractive-repulsive Yukawa potentials.
