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Zbl 1230.37079
Sun, Juntao; Chen, Haibo; Nieto, Juan J.
Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. (English)
[J] J. Math. Anal. Appl. 373, No. 1, 20-29 (2011). ISSN 0022-247X

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 $|t|\rightarrow\infty$. \par The authors prove that problem (1) has infinitely many homoclinic orbits under the following conditions: {\parindent=8mm \item{(H$_1$)} $L \in C(R,R^{N{^2}})$ 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 $$(L(t)x,x)\geq l(t)|x|^2,\ l(t)\rightarrow\infty\ \text {as}\ |t|\rightarrow\infty.$$ \item{(H$_2'$)} $W(t,x)=a(t)|x|^r$, 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. \par} 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. \par Theorem 1.2 in this paper generalizes the result in [{\it Z. Zhang} and {\it 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 $$a\in L^2(R, R)\cap L^{\frac{2}{2-r}}(R, R).$$
[Chun-Lei Tang (Chongqing)]

MSC 2000:: *37J45 Periodic, homoclinic and heteroclinic orbits, etc.

Keywords: homoclinic solutions; Hamiltonian systems; subquadratic; variational methods

Citations: Zbl 1173.34330

Cited in: Zbl 1237.37044

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