Harmonic Maps with Potential from ${\mathbb{R}}^2$ into $S^2$

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 \cite{G_S} due to Gustafson and Shatah.