ON THE EXISTENCE OF SOLUTIONS OF A k-HESSIAN EQUATION AND ITS CONNECTION WITH SELF-SIMILAR SOLUTIONS

Let alpha, beta be real parameters and let a > 0. We study radially symmetric solutions of S-k(D(2)v) + alpha v + beta xi center dot del v = 0, v > 0 in R-n, v(0) = a, where the dot means the usual scalar product in R-n and S-k(D(2)v) denotes the k-Hessian operator of v. For beta > 0 and alpha <= beta(n-2k)/k with k < n/2, we prove the existence of a unique solution to this problem. We also prove existence and properties like strict convexity of the solutions of the above equation for other ranges of the parameters alpha and beta, which are valid for all 1 <= k <= n. These results are then applied to construct different types of explicit solutions, in self-similar forms, of a related evolution equation. In particular, for the heat equation, we find a new family of self-similar solutions which blow up in finite time. These solutions are represented as power series, called Kummer function.