*Accepted Paper*

**Inserted:** 15 dec 2022

**Journal:** Vietnam J. Math.

**Year:** 2021

**Doi:** 10.1007/s10013-021-00540-5

**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 $\Lambda\uparrow4\pi$.\\