Classical solutions of singular Monge-Ampère equations in a ball
The authors deal with the existence, uniqueness and regularity of convex, negative, radially symmetric classical solutions to the Dirichlet problem \[ \begin{gathered} \text{det}(D^2u)= \psi(x,-u)\quad\text{in }\mathbb{B},\\ u= 0\quad\text{on }\partial\mathbb{B},\end{gathered} \] where \((D^2 u)\) is the Hessian of \(u\), \(\mathbb{B}\subset\mathbb{R}^N\), \(N\geq 1\), is the unit ball with boundary \(\partial\mathbb{B}\), the function \(\psi: \mathbb{B}\times (0,\infty)\to [0,\infty)\) is continuous and radially symmetric in the first variable, that is, \(\psi(x,t):= \psi(|x|,t)\), where \(|x|\) is the Euclidean norm of \(x\). The main interest of the authors is when \(\psi\) is singular at \(|x|= 1\) and/or \(t= 0\). The authors employ fixed point arguments and the shooting method.
scientific article; zbMATH DE number 2159787
scientific article