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Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. (English) Zbl 1230.37079

Consider the second order Hamiltonian system

u ¨(t)-L(t)u(t)+W(t,u(t))=0( HS ,)

where LC(R,R N ) is a symmetric matrix valued function and WC 1 (R×R N ,R). A nonzero solution u of (HS) is said to be homoclinic (to 0) if u(t)0 as |t|.

The authors prove that problem (1) has infinitely many homoclinic orbits under the following conditions:

LC(R,R N 2 ) is a symmetric and positively definite matrix for all tR and there exists a continuous function l:RR such that l(t)>0 for all tR and

(L(t)x,x)l(t)|x| 2 ,l(t)as|t|·

W(t,x)=a(t)|x| r , where a:RR + is a continuous function such that

aL 2 2-r (R,R)

and 1<r<2 is a constant.

In fact, in Theorem 1.2 the condition respect to a is not sufficient and the condition that a is positive is also used in the proof of the Lemma 3.1.

Theorem 1.2 in this paper generalizes the result in [Z. Zhang and R. Yuan, Nonlinear Anal., Theory Methods Appl. 71, No. 9, A, 4125–4130 (2009; Zbl 1173.34330)], in which a is a positive continuous function such that

aL 2 (R,R)L 2 2-r (R,R)·

37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
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