Are there any researches on Liouville's equation $\Delta u=K e^{ u}$ when $K<0$?

Question

Are there any researches on Liouville's equation $\Delta u=K e^{ u}$ when $K<0$?

I have seen many papers on Liouville's equation $\Delta u=K e^{ u}$ when $K>0$, such as enter link description here

or the theorem

If $\bar{M}$ is a connected, compact 2-manifold with nonempty boundary $\partial M, g$ a Riemannian metric on $\bar{M}$, and $K \in C^{\infty}(\bar{M})$ a given function satisfying $$ K(x) \leq 0 \text { on } M, $$ then there exists $u \in C^{\infty}(\bar{M})$ such that the metric $g^{\prime}=e^{2 u} g$ conformal to $g$ has Gauss curvature $K$. Given any $v \in C^{\infty}(\partial M)$, there is a unique such $\mathrm{u}$ satisfying $u=v$ on $\partial M$.

which can be transformed to be solving a PDE in the form of $\Delta u=K e^{ u}$ when $K>0$.

But I hardly saw researches on Liouville's equation $\Delta u=K e^{ u}$ when $K<0$ What's the difficulty of this and are there any researches on this problem?

Answer 1

The theorem you stated can be true only for genus zero (that is for the sphere), if $K(x)<0$ at some point $x$); this follows from the Gauss Bonnet theorem that integral of the curvature $-K$ is equal to the Euler characteristic. It is called the Nierenberg problem, and the complete answer is not known.

For the conditions of solvability of your equation in this case, see

Kazdan, Jerry L.; Warner, F. W. Curvature functions for compact 2-manifolds. Ann. of Math. (2) 99 (1974), 14–47.

For recent surveys, see https://arxiv.org/pdf/1411.5743.pdf and https://arxiv.org/pdf/1707.02938.pdf

To obtain such metrics on surfaces of higher genus, one has to permit conic singularities of the metric. This is a hot research topic nowadays, and there is a recent survey.