A prescribed anisotropic mean curvature equation modeling the corneal shape: a paradigm of nonlinear analysis

In this paper we survey, complete and refine some recent results concerning the Dirichlet problem for the prescribed anisotropic mean curvature equation egin{equation*} { m -div}left({ abla u}/{sqrt{1 + | abla u|^2}} ight) = -au + {b}/{sqrt{1 + | abla u|^2}}, end{equation*} in a bounded Lipschitz domain $Omega subset RR^N$, with $a,b>0$ parameters. This equation appears in the description of the geometry of the human cornea, as well as in the modeling theory of capillarity phenomena for compressible fluids. Here we show how various techniques of nonlinear functional analysis can successfully be applied to derive a complete picture of the solvability patterns of the problem.