Elliptic problems on the ball endowed with Funk-type metrics

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