Abstract
Assuming that the solution is bounded from one-side, by Bernstein-type arguments, on \((M^{2},g),\) we prove the local gradient estimates for a type of fully nonlinear equation from conformal geometry.
Similar content being viewed by others
References
Chen, S.S.: Local estimates for some fully nonlinear elliptic equations. Int. Math. Res. Not. 55, 3403–3425 (2005)
Guan, P., Lin, C.S., Wang, G.: Local gradient estimates for quotient equations in conformal geometry. Int. J. Math. 18(4), 349–361 (2007)
Guan, P., Wang, G.: Local estimates for a class of fully nonlinear equations arising from conformal geometry. Int. Math. Res. Not. 26, 1413–1432 (2003)
Li, A., Li, Y.Y.: On some conformally invariant fully nonlinear equations. Comm. Pure Appl. Math. 56(10), 1416–1464 (2003)
Li, Y.Y., Lu, H., Lu, S.: A Liouville theorem for Möbius invariant equations. Peking Math. J. 6(2), 609–634 (2023)
Li, Y.Y., Lu, H., Lu, S.: On the \(\sigma _{2}\)-Nirenberg problem on \(\mathbb{S} ^{2}\). J. Funct. Anal. 283(10), Paper No. 109606, 50 pp. (2022)
Li, Y.Y.: Local gradient estimates of solutions to some conformally invariant fully nonlinear equations. Comm. Pure Appl. Math. 62(10), 1293–1326 (2009). (C. R. Math. Acad. Sci. Paris 343(4), 249–252 (2006))
Sheng, W.M., Trudinger, N., Wang, X.J.: The Yamabe problem for higher order curvatures. J. Differ. Geom. 77(3), 515–553 (2007)
Wang, X.J.: A priori estimates and existence for a class of fully nonlinear elliptic equations in conformal geometry. Chin. Ann. Math. Ser. B 27(2), 169–178 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
W. Wei is partially supported by the Natural Science Foundation of Jiangsu Province BK20220755 and the National Natural Science Foundation of China 12201288.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Wei, W. Local gradient estimates for a type of fully nonlinear equations. Arch. Math. (2024). https://doi.org/10.1007/s00013-024-01992-0
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00013-024-01992-0