  We study the bifurcation diagram of the Onsager free-energy func- tional for
liquid crystals with orientation parameter on the sphere in three dimensions.
In particular, we concentrate on a very general class of two- body interaction
potentials including the Onsager kernel. The problem is reformulated as a
non-linear eigenvalue problem for the kernel operator, and a general method to
find the corresponding eigenvalues and eigenfunctions is presented. Our main
tools for this analysis are spherical harmonics and a special algorithm for
computing expansions of products of spherical harmon- ics in terms of spherical
harmonics. We find an explicit expression for the set of all bifurcation
points. Using a Lyapunov-Schmidt reduction, we derive a bifurcation equation
depending on five state variables. The dimension of this state space is further
reduced to two dimensions by using the rotational symmetry of the problem and
the theory of invariant polynomials. On the basis of these result, we show that
the first bifurcation occurring in case of the Onsager interaction potential is
a transcritical bifurcation and that the corresponding solution is uniaxial. In
addition, we prove some global proper- ties of the bifurcation diagram such as
the fact that the trivial solution is the unique local minimiser for high
temperatures, that it is not a local minimiser if the temperature is low, the
boundedness of all equilibria of the functional and that the bifurcation
branches are either unbounded or that they meet another bifurcation branch.
