Rigidity results on Liouville equation

Abstract: We give a complete classification of solutions bounded from above of the Liouville equation $$-\Delta u=e^{{2u}\quad\mbox{in}\quad} {\mathbf{R}}<sup>2.$$ More generally, solutions in the class $$N:={ u:\limsup_{z\to\infty} u(z)/\log|z|:=k(u)&lt;\infty}$$ are described. As a consequence, we obtain five rigidity results. First, $k(u)$ can take only a discrete set of values: either $k=-2$, or $2k$ is a non-negative integer. Second, $u\to-\infty$ as $z\to\infty$, if and only if $u$ is radial about some point. Third, if $u$ is symmetric with respect to $x$ and $y$ axes and $u_x&lt;0,\; u_y&lt;0$ in the first quadrant then $u$ is radially symmetric. Fourth, if $u$ is concave and bounded from above, then $u$ is one-dimensional. Fifth, if $u$ is bounded from above, and the diameter of ${\mathbf{R}}<sup>2$ with the metric $e<sup>{2u}\delta$ is $\pi$, where $\delta$ is the Euclidean metric, then $u$ is either radial about a point or one-dimensional. In addition, we extend the concavity rigidity result on Liouville equation in higher dimensions.