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

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?

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.
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