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Existence of infinitely many homoclinic orbits in Hamiltonian systems. (English) Zbl 1237.37044

In this paper the authors consider the following second-oder Hamiltonian system

u ¨-L(t)u(t)+W(t,u(t))=0(1)

where t,u N , L: N×N and W:× N . A nonzero solution u of (1) is said to be homoclinic (to 0) if u(t)0 as |t|.

The authors deal with the existence and multiplicity of homoclinic solutions under the assumption that W(t,x) is indefinite sign and subquadratic as |x| by using the genus properties in critical theory. The results in this paper extends the results in [Z. Zhang and R. Yuan, Nonlinear Anal., Theory Methods Appl. 71, No. 9, A, 4125–4130 (2009; Zbl 1173.34330)] and [J. Sun, H. Chen and J. J. Nieto, J. Math. Anal. Appl. 373, No. 1, 20–29 (2011; Zbl 1230.37079)]. The main results are the following.

Theorem 1.1 Assume that L and W satisfy the following conditions:

LC(, N×N ) is a definite symmetric matrix for all t and there exists a constant β>0 such that

(L(t)x,x)β|x| 2

for all (t,x)× N ;

There exist two constants 1<r 1 <r 2 <2 and two functions a 1 ,a 2 L 2 2-r 1 (,[0,+)) such that

|W(t,x)|a 1 (t)|x| r 1

for all (t,x)× N ,|x|1, and

|W(t,x)|a 2 (t)|x| r 2

for all (t,x)× N ,|x|1;

There exist two functions bL 2/(2-r 1 ) (,[0,+)) and φC([0,+),[0,+)) such that


for all (t,x)× N , where φ(s)=O(s r 1 -1 ) as s0 + ;

There exist an open set J and two constants r 3 (1,2) and η>0 such that

W(t,x)η|x| r 3

for all (t,x)J× N , |x|1.

Then Problem (1) has at least one nontrivial homoclinic orbit.

Theorem 1.2 Assume that L and W satisfy (L), (W 1 ), (W 2 ), (W 3 ) and that W(t,-x)=W(t,x) for all t and x N . Then Problem (1) has infinitely many homoclinic orbits.

37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
34C37Homoclinic and heteroclinic solutions of ODE