  Let $(M,F)$ be a $C^\infty$ Finsler manifold, $p\geq 1$ a real number, $k$ a
positive integer and $H_k^p (M)$ a certain Sobolev space determined by a
Finsler structure $F$.
  Here, it is shown that the set of all real $C^{\infty}$ functions with
compact support on $M$ is dense in the Sobolev space $H_1^p (M)$.
  This result permits to approximate certain solution of Dirichlet problem
living on $H_1^p (M)$ by $C^ \infty$ functions with compact support on $(M,F)$.
  Moreover, let $W \subset M$ be a regular domain with the $C^r$ boundary
$\partial W$, then the set of all real functions in $C^r (W) \cap C^0
(\overline W)$ is dense in $H_k^p (W)$, where $k\leq r$. This work is an
extension of some density theorems of T. Aubin on Riemannian manifolds.
