Multiple periodic solutions for lattice dynamical systems with superquadratic potentials

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:255

Article

Ma, Shiwang¹ 

[1] Nankai Univ, Sch Math Sci, Tianjin 300071, Peoples R China

关键词: An infinite lattice of particles;    Superlinear;    Indefinite;    Infinitely many solutions;   

DOI  :  10.1016/j.jde.2013.07.010

来源: Elsevier

【 摘 要 】

In this paper, we consider one dimensional lattices consisting of infinitely many particles with nearest neighbor interaction. The autonomous dynamical system is described by the following infinite system of second order differential equations (sic)(i) = Phi'(i-1)(q(i-1) - q(i)) - Phi'(i) (q(i+1)), i is an element of Z, where Phi(i) denotes the interaction potential between two neighboring particles and q(i)(t) is the state of the i-th particle. Supposing Phi(i) is superquadratic at infinity, for all T > 0, we obtain a nonzero T-periodic solution of finite energy which may be nonconstant in some range of period. If in addition Phi(i)(x) is even in x, we also obtain infinitely many geometrically distinct solutions for any period T > 0. In particular, a prescribed number of geometrically distinct nonconstant periodic solutions is obtained for some range of period. Since the functional associated to the above system is invariant under the actions of the non-compact group Z and the continuous compact group S1 under our assumptions, in order to prove our results, we need to extend the abstract critical point theorem about strongly indefinite functional developed by Bartsch and Ding [Math. Nachr. 279 (2006) 1267-1288] to a more general class of symmetry. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

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