Radial solutions for equations of Weingarten type

Abstract

In this paper we study the linear Weingarten equation defined by the fully non-linear PDE adivDu/root 1+|Du|(2)+bdetD(2)u/(1+|Du|(2))(2)=?(1/root 1+|Du|(2)) in a domain omega subset of R-2, where ?is an element of C-1([-1,1]) and a,b is an element of R. We approach the existence of radial solutions when omega is a disk of small radius, giving an affirmative answer when the PDE is of elliptic type. In the hyperbolic case we show that no radial solution exists, while in the parabolic case we find explicitly all the solutions. In the elliptic case we prove uniqueness and symmetry results concerning the Dirichlet problem of such equation.