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Infinitely many homoclinic orbits for the second order Hamiltonian systems with general potentials. (English) Zbl 1200.37054

The authors consider the second order Hamiltonian system

(HS)-u ¨(t)+L(t)u(t)=R(t,u),t,

where L(t)C(, N 2 ) is a N×N symmetric matrix valued function and R(t,u)C 1 (× N ,). They use the variant fountain theorem to prove that the system (HS) possesses infinitely many homoclinic orbits, assuming that L(t) satisfies the coercive condition and R(t,u) satisfies the conditions that

(R 1 ) R(t,u)=F(t,u)+G(t,u)andF,GC 1 (× N ,) are even in u;

(R 2 ) There exist σ,δ(1,2),c 1 >0,c 2 >0,c 3 >0 such that

c 1 |u| σ F u (t,u)uc 2 |u| σ +c 3 |u| δ

for all (t,u)× N ;

(R 3 ) There exist ρ2 and c 4 >0 such that |G u (t,u)|c 4 (1+|u| p-1 ) for all (t,u)× N , moreover, lim u0 G u (t,u) |u|=0 uniformly for t;

(R 4 ) G(t,u)0 and lim |u| G u (t,u) |u|=+ uniformly for all t·

Conditions (R 2 )–(R 4 ) imply that F(t,u) is sub-quadratic and G(t,u) is super-quadratic, which generalize the sub-quadratic condition that

(R) 0<u·R(t,u)γR(t,u),(t,u)× N {0}, where γ(1,2).

Reviewer’s remark: In the proof of Lemma 3.3, the authors say that “Set w n :=u n u n , then w n =1, w n w,w n + w + ,w n 0 w 0 and w n - w - ”. It seems that this conclusion is incorrect.

MSC:
37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
34C37Homoclinic and heteroclinic solutions of ODE
37C29Homoclinic and heteroclinic orbits
58E05Abstract critical point theory
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