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 $. We study radially symmetric solutions of$ \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 the dot means the usual scalar product in $ \mathbb{R}^n $ and $ S_k(D^2v) $ denotes the $ k $-Hessian operator of $ v $. For $ \beta>0 $ and $ \alpha\leq\frac{\beta(n-2k)}{k}\;\;\mbox{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\leq k\leq 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.