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

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