Abstract: Overcoming the difficulty arising from the fact that the linearized operators of the elliptic k-Hessian ones do not satisfy the Maximum principle and employing Nash-Moser iteration, we prove the existence of C^2+α local solutions of k-Hessian equation when the nonhomogeneous term f ∈ C^α changes sign or is nonnegative. Of course there exists C^∞ local solution if f ∈ C^∞. The technique is that, for the solution to the linearized equation, we prefer at first to prove its uniqueness from which the existence of solution, together with the higher regularity and a priori estimates of solutions, follows.

Key words: k-Hessian equations, Local solution, Nash-Moser iteration