Morse homology for asymptotically linear Dirac equations on compact manifolds

Abstract

We give constructions and calculations of Morse (co)homologies for asymptotically linear Dirac equations on compact manifolds which are non-resonance at infinity, or resonance at infinity and satisfy Landesman-Lazer type conditions. For the non-resonance case, we show that the homology depends only on the homotopy class of the linear part at infinity, while for the resonance case, the (co)homology is isomorphic with the Morse (co)homology of the restriction of the perturbation term to the degenerate space at infinity. From these, (co)homologies are explicitly calculated. Applications to the existence of solutions of asymptotically linear Dirac equations is also given.