On the $\sigma_2$-Nirenberg problem on $\mathbb{S}^2$

We establish theorems on the existence and compactness of solutions to the $\sigma_2$-Nirenberg problem on the standard sphere $\mathbb S^2$. A first significant ingredient, a Liouville type theorem for the associated fully nonlinear M\"obius invariant elliptic equations, was established in an earlier paper of ours. Our proof of the existence and compactness results requires a number of additional crucial ingredients which we prove in this paper: A Liouville type theorem for the associated fully nonlinear M\"obius invariant degenerate elliptic equations, a priori estimates of first and second order derivatives of solutions to the $\sigma_2$-Nirenberg problem, and a B\^ocher type theorem for the associated fully nonlinear M\"obius invariant elliptic equations. Given these results, we are able to complete a fine analysis of a sequence of blow-up solutions to the $\sigma_2$-Nirenberg problem. In particular, we prove that there can be at most one blow-up point for such a blow-up sequence of solutions. This, together with a Kazdan-Warner type identity, allows us to prove $L^\infty$ a priori estimates for solutions of the $\sigma_2$-Nirenberg problem under some simple generic hypothesis. The higher derivative estimates then follow from classical estimates of Nirenberg and Schauder. In turn, the existence of solutions to the $\sigma_2$-Nirenberg problem is obtained by an application of the by now standard degree theory for second order fully nonlinear elliptic operators.