Existence and Nonexistence of Solutions of Minkowski-Curvature Problems in Exterior Domains

ABSTRACT In this paper, we show some nonexistence results of radial solutions for the following Minkowski curvature problems in an exterior domain: cases \ -div ( (v (x) ) ) =k (x) f (v (x) ), x, \\ \ v=0\ on \, ₗv (x) =0\\ cases for R sufficiently large, where (s) =s1-s^{2} for s R with s²1, =\x{ R^{N}: \ |x| R\}, N3 is an integer, || denotes the Euclidean norm on R^N, R is a positive parameter, f: R is an odd and locally Lipschitz continuous function and k C^1 (R^+, \ R^+) with R^+= (0, +). We also apply the fixed-point index theory to establish the existence of positive radial solutions of the above problems for R sufficiently small.