Geophysics and Stuart vortices on a sphere meet differential geometry

Abstract

We prove new existence criteria relevant for the non-linear elliptic PDE of the form $\Delta_{S^2} u=C-he^{u}$, considered on a two dimensional sphere $S^2$, in the parameter regime $2\leq C<4$. We apply this result, as well as several previously known results valid when $C<2$, to discuss existence of solutions of a particular PDE modelling ocean surface currents.