  We study equilibrium configurations in spherical droplets of nematic liquid
crystal with strong radial anchoring, within the Landau-de Gennes theory with a
sixth-order bulk potential. The sixth-order potential predicts a bulk biaxial
phase for sufficiently low temperatures, which the conventional fourth-order
potential cannot predict. We prove the existence of a radial hedgehog solution,
which is a uniaxial solution with a single isotropic point defect at the
droplet centre, for all temperatures and droplet sizes, and prove that there is
a unique radial hedgehog solution for moderately low temperatures, but not deep
in the nematic phase. We numerically compute critical points of the Landau-de
Gennes free energy with the sixth order bulk potential, with rotational and
mirror symmetry, and find at least two competing stable critical points: the
biaxial torus and split core solutions, which have biaxial regions around the
centre, for low temperatures. The size of the biaxial regions increases with
decreasing temperature. We also compare the properties of the radial hedgehog
solution with the fourth-order and sixth-order potentials respectively, in
terms of the Morse indices as a function of the temperature and droplet radius;
the role of the radial hedgehog solution as a transition state in switching
processes; and compare the bifurcation plots with temperature, with the fourth-
and sixth-order potentials. Overall, the sixth-order potential has a
stabilising effect on biaxial critical points and a de-stabilising effect on
uniaxial critical points and we discover an altogether novel bulk biaxial
critical point of the Landau-de Gennes energy with the sixth-order potential,
for which the bulk biaxiality is driven by the sixth-order potential.
