  Let M be a compact Riemannian manifold of dimension n. The k-curvature, for
k=1,2,..n, is defined as the k-th elementary symmetric polynomial of the
eigenvalues of the Schouten tenser. The k-Yamabe problem is to prove the
existence of a conformal metric whose k-curvature is a constant. When k=1, it
reduces to the well-known Yamabe problem. Under the assumption that the metric
is admissible, the existence of solutions to the k-Yamabe problem was recently
proved by Gursky and Viaclovsky for k>n/2. In this paper we prove the existence
of solutions for the remaining cases k <n/2, k=n/2, assuming that the equation
is variational.
