Abstract

Let ?, ? be real parameters and let a > 0. We study radially symmetric solutions of Sk(D2v) + ?v + ?? · ?v = 0, v > 0 in Rn, v(0) = a, where the dot means the usual scalar product in Rn and Sk(D2v) denotes the k-Hessian operator of v. For ? > 0 and ? ? ?(n?k2k) 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 ? and ?, 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. © 2025 American Institute of Mathematical Sciences. All rights reserved.