  We study Sobolev spaces on the $n-$dimensional unit ball $B^n(1)$ endowed
with a parameter-depending Finsler metric $F_a$, $a\in [0,1],$ which
interpolates between the Klein metric $(a=0)$ and Funk metric $(a=1)$,
respectively. We show that the standard Sobolev space defined on the Finsler
manifold $(B^n(1),F_a)$ is a vector space if and only if $a\in [0,1).$
Furthermore, by exploiting variational arguments, we provide non-existence and
existence results for sublinear elliptic problems on $(B^n(1),F_a)$ involving
the Finsler-Laplace operator whenever $a\in [0,1).$
