A Liouville theorem for conformal Gaussian curvature type equations in $${{\mathbb{R}}^2}$$

Abstract

In this paper, we obtain a Liouville type theorem for a class of elliptic equations including the conformal Gaussian curvature equation $$-\Delta u=K(x)e^{2u}\quad {\rm in}\,\, {\mathbb{R}}^2,$$ where K(x) is a Holder continuous function in \({{\mathbb{R}}^2}\) that does not have a fixed sign near infinity. The main tool in our approach is an asymptotic formula for the solution at infinity and the method of moving planes. We also show how our Liouville theorem can be used to obtain a priori bound for solutions of the prescribing Gaussian curvature equation in S2, namely $$\Delta\, u+K(x)e^{2u}=1\, {\rm in}\, S^2,$$ where K(x) is Holder continuous and nonnegative in S2 but vanishes on a set with nonempty interior, a case left open in previous research.