The two dimensional Lp Minkowski problem and nonlinear equations with negative exponents

In this paper we present various existence results for nonlinear differential equations related to the Lp Minkowski problem in the plane and the one dimensional conformal curvature problem. In particular, the supercritical case (that is p<−2) for the Lp Minkowski problem with sign-changing data is addressed. A new geometric sharp inequality is also obtained.