# An application of Leray-Schauder degree theory for Nirenberg problem on the 2-sphere

I'm studying the article "The scalar curvature equation on 2- and 3-spheres" by Chang, Gursky and Yang and I'm particulary interested in the 2-sphere case.

They prove that if $$K:S^2\rightarrow \mathbb{R}$$ is a positive smooth function satisfying some nondegeneracy condition, then $$K$$ can be obtained as Gaussian curvature from a conformal change of metric.

The idea of the proof is to consider the one parameter family of maps: $$K_s=sK+(1-s)$$ and looking for zeroes of the maps $$\psi_s(w)=w-\Delta^{-1}\Psi_s(w)$$

where $$\Psi_s(w)=1-K_se^{2w}$$.

They apply the Leray-Schauder degree theory and some apriori estimates on the $$C^{2,\alpha}$$ norm of zeroes of $$\psi_s$$ for $$s\geq s_0$$, for any given $$s_0>0$$

It is not clear to me, however, how to apply degree theory in this context. More precisely, from what I know, Leray-Schauder degree theory can by applied to maps $$Id-K:\overline{U}\rightarrow X$$ where $$X$$ is a Banach space, $$U$$ is a bounded open set in $$X$$, and $$K$$ is a compact map such that $$0\notin(I-K)(\partial U)$$

In the case at hand, it is not clear me how to choose $$X$$ and $$U$$.

The authors of the article consider the set $$X=\{w\in C^{2,\alpha}(S^2)| \frac{1}{4\pi}\int Ke^{2w}=1\}$$ and the bounded set $$\Omega_C=\{w\in X| |w|_{C^{2,\alpha}}< C\}$$ where $$C$$ is given by the apriori estimates.

• $$\Delta^{-1}\Psi_s(w)$$ is not well defined if $$w\in X$$, since for this to make sense I need that $$\frac{1}{4\pi}\int K_se^{2w}=1$$