Asymptotically linear Dirac-harmonic maps into flat tori

Abstract

This paper is concerned with the existence and compactness problems of perturbed Dirac-harmonic maps into flat tori. We consider two types of asymptotically linear perturbations, non-resonance and resonance cases. For the non-resonance case, we show that the set of solutions in a given free homotopy class is compact and there exist at least (n+1)-distinct solutions in each free homotopy class, where n is the dimension of the tori. For the resonance case, we prove a similar existence result under Landsman-Lazer type conditions.