Existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems

In this paper we study the following nonperiodic second order Hamiltonian system − u ¨ ( t ) + L ( t ) u ( t ) = ∇ u R ( t , u ) , ∀ t ∈ R , where L ( t ) may not be uniformly positive definite for all t ∈ R , and R ( t , u ) is asymptotically quadratic in u as | u | → ∞ . By applying a generalized linking theorem in Bartsch and Ding (2006) [8], we obtain the existence and multiplicity of homoclinic orbits for the above system.