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 equation S-k[u] = K(y) g(y, u, Du) in the neighborhood of a point (y(0), u(0), p(0)) is an element of R-n x R x R-n, where g is an element of C-infinity, g(y(0), u(0), p(0)) > 0, K is an element of C-infinity is nonnegative near y(0), K (y(0)) = 0 and Rank ((DyK)-K-2)(y(0)) >= n - k + 1. (C) 2018 Elsevier Inc. All rights reserved.