On the deeply negative range p<-n-1 for Lp-Minkowski problem

In this paper, we study the Lp-Minkowski problem in the deeply negative range p<=-n-1. Two long standing problems concerning solvability and uniqueness were considered. For the critical case p=-n-1, insolvability of the equation for some positive smooth function f has been observed by Jian-Lu-Wang [25] (see also [29] by Lu). The first main purpose of this paper is to show a same result holds for some Holder function f which is positive outside two polar of S^n in the deeply negative case p<-n-1. When considering the uniqueness, we have obtained the existence of non-constant positive smooth solution for f\equiv1, in the deeply negative case p<-2n-5. Our result for higher dimensional case generalizes a famous nonuniqueness result by Andrew for planar case n=1 and p<-7.Comment: revised version