Existence of solutions for 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 \begin{equation*} S_k(D^2v)+\alpha v+\beta \xi\cdot\nabla v=0,\, v>0\;\; \mbox{in}\;\; \mathbb{R}^n,\; v(0)=a, \end{equation*} where $S_k(D^2v)$ denotes the $k$-Hessian operator of $v$. For $\alpha\leq\frac{\beta(n-2k)}{k}\;\;\mbox{and}\;\;\beta>0$, we prove the existence of a unique solution to this problem, without using the phase plane method. We also prove existence and properties of the solutions of the above equation for other ranges of the parameters $\alpha$ and $\beta$. These results are then applied to construct different types of explicit solutions, in self-similar forms, to a related evolution equation. In particular, for the heat equation, we have found a new family of self-similar solutions of type II which blows up in finite time. These solutions are represented as a power series, called the Kummer function.