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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 $\ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0\tag{HS},$ where $$L\in C(R,R^N)$$ is a symmetric matrix valued function and $$W \in C^1(R\times R^N, R)$$. A nonzero solution $$u$$ of (HS) is said to be homoclinic (to 0) if $$u(t)\rightarrow 0$$ as $$|t|\rightarrow\infty$$.
The authors prove that problem (1) has infinitely many homoclinic orbits under the following conditions:
(H
$$L \in C(R,R^{N{^2}})$$ is a symmetric and positively definite matrix for all $$t\in R$$ and there exists a continuous function $$l:R\rightarrow R$$ such that $$l(t)>0$$ for all $$t\in R$$ and $(L(t)x,x)\geq l(t)|x|^2,\;l(t)\rightarrow\infty\;\text{as}\;|t|\rightarrow\infty.$
(H
$$W(t,x)=a(t)|x|^r$$, where $$a:R\rightarrow R^+$$ is a continuous function such that $a\in L^{\frac{2}{2-r}}(R,R)$ 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 $a\in L^2(R, R)\cap L^{\frac{2}{2-r}}(R, R).$

##### MSC:
 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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