Suppose $u\in H_0^1(\Omega)$ is the weak solution of the following equations: $$ \begin{cases} -\left(a^{ij}(x)u_i\right)_j +b^iu_i=\sin u(x),x\in \Omega\\ u(x)=0,x\in \partial\Omega \end{cases} $$ where $\partial \Omega\in C^{\infty},a^{ij}=a^{ij}\in C^{\infty}(\bar{\Omega})$ is uniformly elliptic ,$b^i\in C^{\infty}(\bar{\Omega})$.Show that $$ u\in C^{\infty}(\bar{\Omega}) $$
I want to use $H^k$-regularity theorey to prove it, so I need to prove if $u\in H^{m+1}(\Omega)$ then $\sin u(x)\in H^{m}(\Omega)$.However, what I can deal with is the cases when $m=1$.