Existence and convexity of local solutions to degenerate Hessian equations

In this work, we prove the existence of convex solutions to the following k-Hessian equationSk[u]=K(y)g(y,u,Du) in the neighborhood of a point (y0,u0,p0)∈Rn×R×Rn, where g∈C∞,g(y0,u0,p0)>0, K∈C∞ is nonnegative near y0, K(y0)=0 and Rank(Dy2K)(y0)≥n−k+1.