Computations of critical groups and applications to some differential equations at resonance

In this paper, we first consider the computations of the critical groups of a functional with generalized Ahmad–Lazer–Paul type conditions both at zero and infinity. Then the abstract results are applied to investigate some new cases of the existence of nontrivial solutions of the second order Hamiltonian systems and elliptic boundary value problems, which may be resonant both at zero and infinity.