  We study uniaxial solutions of the Euler-Lagrange equations for a Landau-de
Gennes free energy for nematic liquid crystals, with a fourth order bulk
potential, with and without elastic anisotropy. In the elastic isotropic case,
we show that (i) all uniaxial solutions of the Euler-Lagrange equations, with a
director field of certain symmetry, necessarily have the radial-hedgehog
structure modulo an orthogonal transformation, (ii) the "escape into third
dimension" director cannot correspond to a purely uniaxial solution of the
Landau-de Gennes Euler-Lagrange equations and we do not use artificial
assumptions on the scalar order parameter and (iii) there are no non-trivial
uniaxial solutions that have $\mathbf{e}_z$ as an eigenvector. In the elastic
anisotropic case, we prove that all uniaxial solutions of the corresponding
Euler-Lagrange equations, with a certain symmetry, are strictly of the
radial-hedgehog type, i.e. the elastic anisotropic case enforces the
radial-hedgehog structure (or the degree $+1$-vortex structure) more strongly
than the elastic isotropic case and the associated partial differential
equations are technically far more difficult than in the elastic isotropic
case.
