  We investigate the solution landscapes of the Onsager free-energy functional
with different potential kernels, including the dipolar potential, the
Maier--Saupe potential, the coupled dipolar/Maier--Saupe potential, and the
Onsager potential. A uniform sampling method is implemented for the
discretization of the Onsager functional, and the solution landscape of the
Onsager functional is constructed by using the saddle dynamics coupled with
downward/upward search algorithms. We first compute the solution landscapes
with the dipolar and Maier--Saupe potentials, for which all critical points are
axisymmetric. For the coupled dipolar/Maier--Saupe potential, the solution
landscape shows a novel non-axisymmetric critical point, named as tennis, which
exists for a wide range of parameters. We further demonstrate various
non-axisymmetric critical points in the Onsager functional with the Onsager
potential, including square, hexagon, octahedral, cubic, quarter, icosahedral,
and dodecahedral states. The solution landscape provides an efficient approach
to show the global structures as well as the bifurcations of critical points,
which can not only verify the previous analytic results but also propose
several conjectures based on the numerical findings.
