  We investigate the Ericksen-Leslie system for the Oseen-Frank model with
unequal Frank elastic constants in $\mathbb{R}^3$. To generalize the result of
Hineman-Wang \cite{HW}, we prove existence of solutions to the Ericksen-Leslie
system with initial data having small $L^3_{uloc}$-norm. In particular, we use
a new idea to obtain a local $L^3$-estimate through interpolation inequalities
and a covering argument, which is different from the one in \cite{HW}.
Moreover, for uniqueness of solutions, we find a new way to remove the
restriction on the Frank elastic constants by using the rotation invariant
property of the Oseen-Frank density. We combine this with a method of
Li-Titi-Xin \cite{LTX} to prove uniqueness of the $L^3_{uloc}$-solutions of the
Ericksen-Leslie system assuming that the initial data has a finite energy.
