On rigidity of the steady Ericksen-Leslie system

We study solutions, with scaling-invariant bounds, to the steady simplified Ericksen-Leslie system in $\mathbb{R}^n\setminus \{0\}$. When $n=2$, we construct and classify a class of self-similar solutions. When $n\ge 3$, we establish the rigidity asserting that if $(u,d)$ satisfies a scaling-invariant bound with a small constant, then $u\equiv 0$ and $d=$ constant for $n\geq 4$ or $u$ is a Landau solution and $d=$ constant for $n=3$. Such a smallness condition can be weaken when $n=4$ or the solutions are self-similar.