Non-existence results for harmonic maps with potential

We establish Laplacian methods for harmonic maps with potential and apply these to obtain non-existence results. In particular, we obtain restrictions on the existence of maps from the 2-torus to the 2-sphere with certain potential. Applying our technique to harmonic maps (without potential) yields new proofs of important properties of harmonic maps (without potential) from surfaces. Finally, the Newtonian system associated to harmonic maps with potential from the 2-torus into the 2-sphere is studied and a discussion of solutions is given.