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On the σ2-Nirenberg problem on [formula omitted].
- Source :
-
Journal of Functional Analysis . Nov2022, Vol. 283 Issue 10, pN.PAG-N.PAG. 1p. - Publication Year :
- 2022
-
Abstract
- We establish theorems on the existence and compactness of solutions to the σ 2 -Nirenberg problem on the standard sphere S 2. A first significant ingredient, a Liouville type theorem for the associated fully nonlinear Möbius 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 Möbius invariant degenerate elliptic equations, a priori estimates of first and second order derivatives of solutions to the σ 2 -Nirenberg problem, and a Bôcher type theorem for the associated fully nonlinear Möbius invariant elliptic equations. Given these results, we are able to complete a fine analysis of a sequence of blow-up solutions to the σ 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 ∞ a priori estimates for solutions of the σ 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 σ 2 -Nirenberg problem is obtained by an application of the by now standard degree theory for second order fully nonlinear elliptic operators. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00221236
- Volume :
- 283
- Issue :
- 10
- Database :
- Academic Search Index
- Journal :
- Journal of Functional Analysis
- Publication Type :
- Academic Journal
- Accession number :
- 158885871
- Full Text :
- https://doi.org/10.1016/j.jfa.2022.109606