Asian Journal of Mathematics

Volume 20 (2016)

Number 4

Harmonic maps with potential from $\mathbb{R}^2$ into $S^2$

Pages: 597 – 628



Ruiqi Jiang (College of Mathematics and Econometrics, Hunan University, Changsha, China; and Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China)


We study the existence problem of harmonic maps with potential from $\mathbb{R}^2$ into $S^2$. For a specific class of potential functions on $S^2$, we give the sufficient and necessary conditions for the existence of equivariant solutions of this problem. As an application, we generalize and improve the results on the Landau-Lifshitz equation from $\mathbb{R}^2$ into $S^2$ in [7] due to Gustafson and Shatah.


harmonic maps with potential, Pohozaev identity, Landau-Lifshitz

2010 Mathematics Subject Classification

Primary 58E20. Secondary 35J20, 35J60.

Published 1 November 2016