  We give new results of the phase transition of dilute colloidal solutions of
rod-like molecules in dimension $D \geq 3$. For the low concentration of
particles in a carrier fluid, we prove that the isotropic phase is the unique
solution to the Doi-Onsager model with the general potential kernel. In
addition, we present the regime of the bifurcation of nematic phases in the
class of axially symmetric solutions. Our method is based on a generalization
of the classical Leray-Schauder degree we developed for this problem.
