Existence and extendibility of rotationally symmetric graphs with a prescribed higher mean curvature function in Euclidean and Minkowski spaces

In this paper we investigate the existence of rotationally symmetric entire graphs (resp. entire spacelike graphs) with prescribed k  -th mean curvature function in Euclidean space Rn+1Rn+1 (resp. Minkowski spacetime Ln+1Ln+1). As a previous step, we analyze the associated homogeneous Dirichlet problem on a ball, which is not elliptic for k>1k>1, and then we prove that it is possible to extend the solutions. Moreover, a sufficient condition for uniqueness is given in both cases.