Perturbed S1-symmetric hamiltonian systems

Keywords--Pala is-Smale condition, Critical point theory, Hamiltonian systems, Perturbation from symmetry, Multiple periodic solutions. 1. I N T R O D U C T I O N In this paper, in the spirit of [1], we want to investigate the effect of perturbing the S l symmet ry of a general class of Hamiltonian systems. Studied around 1980 by Bahri and Berestycki in [2], the problem of finding multiple periodic solutions of nonsymmetr ic systems of type (T ij (s) = 5ij) a/e = Ds, V ( v ) + ~e, g = l , . . . , n , (1.1) where ~ c L2(S 1, Nn), have also been considered by Rabinowitz in [3] via techniques of classical critical point theory. In order to find weak solutions to (1.1), he looked for critical points of the smooth (C 1) action L : H I ( S 1 , N n) --* ~ defined by /02 /0 ~ ( ~ ) = 2 J0 I#l~ dr V (~) dr ~ ~ dT. On the other hand, the action of a mechanical system with n degree of freedom, in general, may be represented by quasi-linear functionals L : H I ( S 1, R n) --* N of the type 1/02 /0 f/ L(')') = ~ TiJ (V)~i'~j dr V(7) dr ~" 3' dr, (1.2) i,j=l 0893-9659/01/$ see front matter (~) 2001 Elsevier Science Ltd. All rights reserved. Typeset by A~-R~X PII: S0893-9659 (00)00164-6 376 M. SQUASSINA where {TiJ(s)} is the symmetric positive definite quadratic form of kinetic energy and V is the potential energy. If ~ -0, clearly for each 7 E HI(SI,I~ n) we have V ~ 6 ]~, L(To'~) = L('~), (ToT)(T) = 7(T + ~q), (SLsymmetry) . If ~ # 0, the Sl-symmetry drops and the associated evolution system is given by 1 Ds~TiJ (7);h~/j = Ds~ V(7) + ~t, (1.3) + i = l i , j=l for ~ = 1 , . . . , n. Now, since LI ( s 1, N n) C_ H-1($1, R'~), (1.2) is a smooth functional and we shall apply the techniques of classical critical point theory [3-5]. Recently, some papers have been published about the existence of weak solutions to quasilinear elliptic systems subjected to perturbation from Z2-symmetry. ( L ( 7 ) = L(7)). See [6,7]. On the other hand, to my knowledge, little is known for gLsymmetries in case of the quasi-linear functional (1.2). Throughout the paper, we shall consider the following assumptions. (i) TiJ(.) • CI([~ n) N L°Z(N n) and DsTiJ(.) e L°°(N n) for each i , j = 1 , . . . ,n. Moreover, ~ T i J ( s ) ~ i ~ j >_ v [~[2, (v > 0), (1.4) i ,j= l for each (s, ~) E R 2n. (ii) V E CI(N '~) and there exist bl,b2, R > 0 and a , p > 2 such that V(s) <_ bl + b21sl °, (1.5) >R o < V(s) < s . (1 .6 ) for each s E R n. Finally, there exists 0 El0, # --2[ such that