Homoclinic solutions for singular Hamiltonian systems without the strong force condition

Abstract

We consider the existence of homoclinic orbits at the origin of a Hamiltonian system q ¨ + V ′ ( q ) = 0 in R N ( N ≥ 3 ) where V has a strict global maximum at q = 0 and a singularity at a point e ≠ 0 , namely V ( q ) → − ∞ as q → e . We establish via variational methods the existence of a generalized homoclinic orbit q ˜ that may enter the singularity e without assuming the strong force condition of Gordon. Moreover when V ∼ − 1 / | q − e | α ( 0 < α < 2 ) near e , we give a bound for the number of collisions of q ˜ based on the Morse index of approximated solutions. As a consequence we obtain that q ˜ is classical (non-collision) orbit for α ∈ ] 1 , 2 [ and enters the singularity e at most one time in R if α ∈ ] 0 , 1 ] .