Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape

We prove existence and uniqueness of classical solutions of the anisotropic prescribed mean curvature problem egin{equation*} { m -div}left({ abla u}/{sqrt{1 + | abla u|^2}} ight) = -au + {b}/{sqrt{1 + | abla u|^2}}, ext{ in } B, quad u=0, ext{ on } partial B, end{equation*} where $a,b>0$ are given parameters and $B$ is a ball in ${mathbb R}^N$. The solution we find is positive, radially symmetric, radially decreasing and concave. This equation has been proposed as a model of the corneal shape in the recent papers [13,14,15,18,17], where however a linearized version of the equation has been investigated.