Existence and compactness of conformal metrics on the plane with unbounded and sign-changing Gaussian curvature

Abstract

We show that the prescribed Gaussian curvature equation in $\mathbb{R}^2$ $$-\Delta u= (1-|x|^p) e^{2u},$$ has solutions with prescribed total curvature equal to $\Lambda:=\int_{\mathbb{R}^2}(1-|x|^p)e^{2u}dx\in \mathbb{R}$, if and only if $$p\in(0,2) \qquad \text{and} \qquad (2+p)\pi\le\Lambda<4\pi$$ and prove that such solutions remain compact as $\Lambda\to\bar{\Lambda}\in[(2+p)\pi,4\pi)$, while they produce a spherical blow-up as $\Lambda\uparrow4\pi$.