Szulkin, A.

English

Institute of Mathematics AS CR, v. v. i.

EQUADIFF 7Andrzej SzulkinHamiltonian systems with periodic nonlinearitiesIn: Jaroslav Kurzweil (ed.): Equadiff 7, Proceedings of the 7th Czechoslovak Conferenceon Differential Equations and Their Applications held in Prague, 1989. BSB B.G.Teubner Verlagsgesellschaft, Leipzig, 1990. Teubner-Texte zur Mathematik, Bd. 118.pp. 131--134.Persistent URL: http://dml.cz/dmlcz/702354Terms of use:© BSB B.G. Teubner Verlagsgesellschaft, 1990Institute of Mathematics of the Academy of Sciences of the Czech Republic providesaccess to digitized documents strictly for personal use. Each copy of any part of thisdocument must contain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The CzechDigital Mathematics Library http://project.dml.czHAMILTONIAN SYSTEMS WITH PERIODIC NONLINEARITIES S Z U L K I N A;, S T O C K H O L M , Sweden 1. Abs trac t result Let M = E x Tk, where E is a real Hilbert space with an inner product (,) and Tk is the fc-torus. We will be concerned with a class of functionals $ G Cl(M, R ) of the form $(x,v) = \(Lx,x) — xp(x,v), where L and t/) satisfy the following hypotheses: (i) L : E —:• E is a bounded, linear and selfadjoint operator to which there corresponds an orthogonal decomposition E = E+ 0 E~ into L-invariant subspaces such that (Lx, x) is positive definite on F+ and negative definite on E~. (ii) The gradient of ifr, denoted VV>, is compact (in the sense that it maps bounded sets into compact ones). (iii) ~7%/>(M) is bounded, i.e., there exists a constant C such that | |V^(x , v)\\ < C V(x,v) G M. Recall that a differentiable functional $ is said to satisfy the Palais-Smale condition (PS) if each sequence (zn) such that $ ( z n ) is bounded and V $ ( z n ) —> 0 possesses a convergent subsequence. Note that it is an easy consequence of hypotheses (i)-(iii) that our functional $ satisfies (PS). Recall also that a set A C M is said to be of category k in M (denoted catjvf(A) = k) if k is the smallest integer such that A can be covered by k closed sets which are contractible to a point in M. If A = M, we will write cat (M) = catjw(M). Properties of the category may be found e.g. in [4, 9, 11, 12]. T h e o r e m 1. Suppose that the functional $ satisfies hypotheses (i)-(iii). Then $ possesses at least k + 1 distinct critical points. If E~ = {0}, $ is bounded below. So in this case Theorem 1 follows from a result by Palais [9] (because c a t ( M ) = ca t (T*) = k + 1 [11]). For finite dimensional E~ Theorem 1 was first proved by Chang [1, 2], A different argument, which we sketch below, has been proposed by Fournier and Willem [8]. The proof for d i r n E " < oo is due to the author [13]. Remark 1. (i) The conclusion of Theorem 1 remains valid if Tk is replaced by a compact manifold Vd such that cuplength(V d ) = k [1, 2, 8, 13]. (ii) The conclusion remains valid if L has a finite dimensional kernel E° and ip(x°, v) —> - o o (or ip(x°,v) -> -foo) as ||a:0|| -> oo, x° G E° [1, 2, 13]. If E~ = {0}, the proof of Theorem 1 is easy to obtain directly (without invoking Palais' theorem [9]) by using a standard argument [4, 12] based on the minimax characterization of critical values of $ as bj = inf sup $(x,v), 1 < j < k -f 1 catM(.4)>> (X|t,)eA 131 and the deformation lemma. Note that catjvf ({0} xTk) = cat (Tk) = k -f 1, so all bj are well defined and finite. If dim E~ > 0, the functional $ becomes unbounded below and the values bj defined above are equal to — oo. It is therefore necessary to employ a different argument . We will introduce two notions of relative category and indicate how they enter into the proof of Theorem 1 . A continuous mapping 77 : [0,1] x M —> M such that 77(0, z) — zVz G M will be called a deformation of M. Let A,N be two closed subsets of M. The set A is said to be of category k in M relative to jV, denoted catjv/,N(A) -= k, if k is the smallest integer such that A = Ao U Ai U ... U A*, where all Aj are closed, all Aj with j > 1 are contractible in M , and there exists a deformation 770 of M satisfying 770(1, Ao) C N and r}0(t,N) C JV Wt G [0,1]. If such k does not exist, catjVf,N(A) = 00. The above notion of relative category is due to Fournier and Willem [7, 8]. Our definition is a slight modification of theirs and may be found in [13]. Let D be a given class of deformations of M such tha t the trivial deformation rj(t, z) — z V(t,z) is in X>, and whenever 771,772 are in D, so is the deformation obtained by letting rji be followed by 772. The set A is said to be of category k in M relative to N and V, denoted c a*M N ( ^ ) — k/\i k is the smallest integer such that A — AQ U A\ U ... U At, where Aj and 770 are as in the preceding definition and 770 G V. If such k does not exist, cat j^ x(A) = 00. The above definition may be found in [13]. Suppose d i m E " < 00. Denote $ a = {z G M : $ ( z ) < a } , Tj = {A C M : A is closed and catjif $a(.<4) > j} and Cj = inf sup $ ( z ) , 1 < j < k -f 1. Then Cj > a (because catA.f,*„(A) = 0 whenever A C $a)- Denote MR = {(x, v) G M : \\x\\ < R}. It can be shown that if a is sufficiently small and R sufficiently large, then (*) catAf,*a(M/*) > c a t B x T f c | S x T f c ( B x Tk) > k + 1, where B = {x G E~ : \\x\\ < R} and S = dB (more precisely, (*) holds for a certain modified functional $ which has the same critical points as $ , cf. [8]). So Tj ^ 0 for 1 < j < k + 1 and the numbers Cj are well defined and finite. Now a s tandard argument shows that $ has at least k + 1 critical points. If dim E~ = 00, then B is contractible to 5 , so ca t# x T f c ) 5 x T f c (B x Tk) = 0 and the above argument fails (because all Tj may be empty). Therefore a further modification is needed in order to prove Theorem 1 in the most general case. Let V be the class of deformations which consists, roughly speaking, of solutions for 0 < t < 1 of initial value problems of the form ^ = -u;(T7)V(77), 77(0, x, v) = (x, v), where V is a certain pseudo-gradient vector field for $ , u? : M —* [0,1] is locally Lipschitz continuous, and u> = 0 in a neighbourhood of the set of critical points of $ ( the class V is in fact somewhat larger, see [13]). Then it can be shown that for R sufficiently large and a sufficiently small, ca t M ,* a (Mi j ) > c a t B x T f c j 5 x T f c ( B x Tk) > k + 1, where B = {x G E~ : \\x\\ < R} is a ball in a finite dimensional subspace E~ of E~. So replacing cat by c a t p in the definition of Cj we obtain the conclusion. 132 2 . H a m i l t o n i a n s y s t e m s Consider the Hamiltonian system of differential equations (HS) z = JHz(z,t), where z = (pyq) G R " x RN, H G C X ( R 2 N x R , R ) and (? o ) is the standard symplectic matrix. Assume H is 27r-periodic in t. Let H1/2(S111H.2N) = IT1/2 be the Sobolev space of 27r-periodic R 2 ^-valued functions z = 2^ ckelkt, where Ck G C2N and c-k = c*, ibez such that y>+ifci)|Ct|2<oo. kez Define Ф(z)=l [ (-JŻ-z)dt- [ H(z,t)dt. 2 Уo Jo It is known [10] that $ 6 C 1 ( H 1 / 2 , R ) , critical points of $ correspond to 27r-periodic solutions of (HS) and the gradient of / 0 H(z, t) dt is compact. Suppose H is 27r-periodic in all variables. Note that if z is a 27r-periodic solution of (HS), so are all functions z such that z — z G 27r Z2N (by periodicity of H). Two solutions z and z will be called geometrically distinct if z - z £2nZ2N. Let H1/2 = H+ 0 H° 0 H' be the decomposition corresponding to the positive, zero and negative part of the spectrum of —Jz. Denote E = H+ 0 H~. Then to each z £ H1/2 there corresponds a unique pair (x,v) such tha t x G E and v is the mean value of z modulo 27T. Clearly, v may be considered as an element of the torus T2N\ so (x, v) G E x T 2 N = M . Define $(x>v) = 7; [ (-Jx-x)dt- [ H(zyt)dt. 2 Jo Jo It is easy to see that $ G C1(M, R ) , critical points of $ correspond to 27r-periodic solutions of (HS) and, unlike for $ , distinct critical points correspond to solutions which are geometrically distinct . Since $ satisfies hypotheses (i)-(iii) of Section 1, we have the following T h e o r e m 2 [13]. Suppose that H is 2n-periodic in all variables. Then (HS) possesses at least 2N -f 1 geometrically distinct 2ir-periodic solutions. This result, which is known to imply an affirmative answer to one of Arnold's conjectures, has been first t ime proved by Conley and Zehnder for H G C2(K2N x R , R ) [5]. A different proof, also for H G C 2 , may be found in Chang [1, 2]. Using Remark 1 and a variant of the above argument, one can easily prove 133 T h e o r e m 3 [13]. Suppose that H € C*(R2N x R , R ) is 2iz-periodic in q and t, H(z,t) = |Bp-p-\-G(z, t), where B is a symmetric N x N-matrix and the derivative Gz is bounded. If the null space of B, N(B), is nontrivial, suppose also that G(p,q,t) —• -fco (or G(p,q,t) —• —oo) uniformly in t as \p\ —• oo, p £ N(B). Then (HS) possesses at least N + 1 geometrically distinct 2n-periodic solutions. Similar results, for nonlinearities of class C2, may be found in [3, 6]. R e m a r k 2. The proofs in [1-3, 5, 6] employ a finite dimensional reduction which requires that H € C2 (and Hxx be bounded) . On the other hand, our Theorem 1 allows one to avoid this reduction and therefore have H G C1. References 1. K . С Сhang, Infinite Dimensionаl Morse Theory аnd Its Applicаtions, Séminaire de Mathématiques Supérieures, Vol. 97, Les Presses de ľUniversité de Montréal, Montгéal, 1985. 2. K . С Сhang, Applicаtions of homoloдy theory to some proЫems in differentiаl equаtions, P r o c Sym. Pure Math., Vol. 45, Part 1, Amer. Math. S o c , Providence, R.I., 1986, pp. 253-262. 3. K . С Сhang, On the periodic nonlineаrity аnd the multiplicity of solutions, J . Nonl. Anal. 13 (1989), 527-537. 4. S.N. Сhow and J.K. Hale, Methods of Bifurcаtion Theory, Springer-Verlag, New York, 1982. 5. С . С Сonley and E. Zehnder, The Birkhoff-Lewis fixed point theorem аnd а conjecture of V.L Arnold, Invent. Math. 73 (1983), 33-49. 6. A. Fonda and J. Mawhin, Multiple periodic solutions of conservаtive systems with periodic noniineаrity, Preprint. ' 7. G. Fournieг and M. Willem, Multiple solutions of the forced douЫe pendulum equаtion, Preprint. 8. G. Fournier and M. Willem, Relаtive cаtegory аnd the cаlculus of vаriаtions, Pгeprint. 9. R.S. Palais, Criticаl point theory аnd the minimаx principle, Proc. Sym. Pure Math., Vol. 15, Ameг. Math. S o c , Providence, R.I., 1970, pp. 185-212. 10. P.H. Rabinowitz, Minimаx Methods in Criticаl Point Theory with Applicаtions to Dif-ferentiаl Equаtions, СBMS 65, Amer. Math. Soc, Providence, R.I., 1986. 11. J .T . Schwaгtz, Nonlineаr Functionаl Anаlysis, Gordon and Breach, New York, 1969. 12. A. Szulkin, Criticаl point theory of Ljusternik-Schnirelmаnn type аnd аpplicаtions to pаrtiаl differentiаl equаtions, Séminaire de Mathématiques Supéгieures, Vol. 107, Les Presses de ľUniversité de Montréal, 1989, pp. 35-96. 13. A. Szulkin, A relаtive cаtegory аnd аpplicаtions to criticаl point theory for strongly in-definite functionаls, J . Nonl. Anal., to appear. 134 

Hamiltonian systems with periodic nonlinearities

http://dml.cz/bitstream/handle/10338.dmlcz/702354/Equadiff_07-1990-1_25.pdf

Hamiltonian systems with periodic nonlinearities

Abstract

Similar works

Full text

Available Versions

Institute of Mathematics AS CR, v. v. i.