Extremal functions in Poincare-Sobolev inequalities for functions of bounded variation

If Omega subset of R-n is a smooth bounded domain and q is an element of (0, n/n-1) we consider the Poincare-Sobolev inequality c(integral(Omega)vertical bar u vertical bar n/n-1)(1-1/n) <= integral(Omega)vertical bar Du vertical bar, for every u is an element of BV(Omega) such that integral(Omega)vertical bar u vertical bar(q-1)u = 0. We show that the sharp constant is achieved. We also consider the same inequality on an n-dimensional compact Riemannian manifold M. When n >= 3 and the scalar curvature is positive at some point, then the sharp constant is achieved. In the case n >= 2, we need the maximal scalar curvature to satisfy some strict inequality.