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

In this paper, we show some nonexistence results of radial solutions for the following Minkowski curvature problems in an exterior domain: cases \ -div (φ(∇ v(x)))=k(x)f(v(x)), x∈Ω,\\ \ v=0\ on \ ∂Ω, limx→∞v(x)=0\\ cases for R sufficiently large, where φ(s)=s√1-s² for s∈ R with s²<1, Ω=∈ RN:\ |x| > R\, N≥3 is an integer, |·| denotes the Euclidean norm on RN, R is a positive parameter, f:R is an odd and locally Lipschitz continuous function and k ∈ C¹(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.