An asymptotically linear non-cooperative elliptic system with lack of compactness

In this paper we consider an asymptotically linear, non-cooperative elliptic system at resonance. The functional associated to the system is strongly indefinite and so the Morse index of each critical point is infinite. To overcome this difficulty, we will use Morse theory for strongly indefinite functionals developed by Abbondandolo. We shall prove that, if the system has a non-resonant solution, the 'Morse index' of which is different from the 'Morse index' at infinity, then there exists another solution.