Infinitely many fast homoclinic orbits for a class of superquadratic damped vibration systems

Abstract

. In this paper, we consider the following damped vibration system ¨ u ( t )+ q ( t ) ˙ u ( t ) − L ( t ) u ( t )+ ∇ W ( t , u ( t )) = 0 , ∀ t ∈ R , where q ∈ C ( R , R ) , L ∈ C ( R , R N 2 ) is a symmetric matrix valued function and W ( t , x ) ∈ C 1 ( R × R N , R ) . We prove the existence of infinitely many fast homoclinic solutions for the system when Q ( t ) = (cid:82) t 0 q ( s ) ds → + ∞ as | t | → ∞ , L is neither coercive nor uniformly positive definite and W ( t , x ) is superquadratic at infinity in the second variable but does not satisfy the well-known superquadratic growth conditions like the Ambrosetti-Rabinowitz or the Fei’s conditions.