Radial solutions of the Dirichlet problem for a class of quasilinear elliptic equations arising in optometry

Abstract

This paper deals with the quasilinear elliptic problem − div∇u∕1+|∇u|2+a(x)u=b(x)∕1+|∇u|2inB,u=0on∂B,where B is an open ball in RN, with N≥2, and a,b∈C1(B¯) are given radially symmetric functions, with a(x)≥0 in B. This class of anisotropic prescribed mean curvature equations appears in the description of the geometry of the human cornea, as well as in the modeling theory of capillarity phenomena for compressible fluids. Unlike all previous works published on these subjects, existence and uniqueness of solutions of the above problem are here analyzed in the case where the coefficients a,b are not necessarily constant and no sign condition is assumed on b.