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A prescribed anisotropic mean curvature equation modeling the corneal shape: a paradigm of nonlinear analysis. (English) Zbl 1374.35187

Summary: In this paper we survey, complete and refine some recent results concerning the Dirichlet problem for the prescribed anisotropic mean curvature equation \[ \mathrm{-div}\bigg{(}\nabla u/\sqrt{1 + |\nabla u|^2}\bigg{)} = -au + {b}/\sqrt{1 + |\nabla u|^2}, \] in a bounded Lipschitz domain \(\Omega \subset \mathbb{R}^N\), 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.

MSC:

35J93 Quasilinear elliptic equations with mean curvature operator
35J25 Boundary value problems for second-order elliptic equations
35J62 Quasilinear elliptic equations
35J67 Boundary values of solutions to elliptic equations and elliptic systems
35B09 Positive solutions to PDEs
35B51 Comparison principles in context of PDEs
35J20 Variational methods for second-order elliptic equations
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