Homoclinic solutions for a class of second order Hamiltonian systems

Abstract

Key words Hamiltonian system, homoclinic solution, variational method MSC (2010) 34C37, 58E30, 37J45In this paper, we study homoclinic solutions for the nonperiodic second order Hamiltonian systemswhere L is unnecessarily coercive or uniformly positively definite, and W (t, u) is only locally defined near the origin with respect to u. Under some general conditions on L and W , we show that the above system has infinitely many homoclinic solutions near the origin. Some related results in the literature are extended and generalized.

Multiplicity of Solutions for Fractional Hamiltonian Systems under Local Conditions

“…The proof is motivated by the argument in [17]. We will modify and extend W to an appropriate  W and show for the associated modified functional I the existence of a sequence of solutions converging to zero in ∞ L norm, therefore to obtain infinitely many solutions for the original problem.…”