Hamiltonian systems: periodic and homoclinic solutions by variational methods

Transcription

1 Hamiltonian systems: periodic and homoclinic solutions by variational methods Thomas Bartsch Mathematisches Institut, Universität Giessen Arndtstr. 2, Giessen, Germany Andrzej Szulkin Department of Mathematics, Stockholm University Stockholm, Sweden Contents Introduction 2 1 Critical point theory Basic critical point theory Critical point theory for strongly indefinite functionals Periodic solutions Variational setting for periodic solutions Periodic solutions near equilibria Fixed energy problem Superlinear systems Asymptotically linear systems Spatially symmetric Hamiltonian systems Homoclinic solutions Variational setting for homoclinic solutions Existence of homoclinics Multiple homoclinic solutions Multibump solutions and relation to the Bernoulli shift References 67 1

2 Introduction The complex dynamical behavior of Hamiltonian systems has attracted mathematicians and physicists ever since Newton wrote down the differential equations describing planetary motions and derived Kepler s ellipses as solutions. Hamiltonian systems can be investigated from different points of view and using a large variety of analytical and geometric tools. The variational treatment of Hamiltonian systems goes back to Poincaré who investigated periodic solutions of conservative systems with two degrees of freedom using a version of the least action principle. It took however a long time to turn this principle into a useful tool for finding periodic solutions of a general Hamiltonian system (HS) { ṗ = Hq (p, q, t) q = H p (p, q, t) as critical points of the Hamiltonian action functional Φ(p, q) = p q dt H(p, q, t) dt defined on a suitable space of 2π-periodic functions (p, q) : R R N R N. The reason is that this functional is unbounded from below and from above so that the classical methods from the calculus of variations do not apply. Even worse, the quadratic form (p, q) p q dt has infinite-dimensional positive and negative eigenspaces. Therefore Φ is said to be strongly indefinite. For strongly indefinite functionals refined variational methods like Morse theory or Lusternik-Schnirelmann theory still do not apply. These were originally developed for the closely related problem of finding geodesics and extended to many other ordinary and partial differential equations, in particular to the second order Hamiltonian system q = V q (q, t) where the associated Lagrangian functional J(q) = 1 2 q 2 dt V (q, t) dt is not strongly indefinite. A major breakthrough was the pioneering paper [78] of Rabinowitz from 1978 who obtained for the first time periodic solutions of the first order system (HS) by the above 2

3 mentioned variational principle. Some general critical point theory for indefinite functionals was subsequently developed in the 1979 paper [19] by Benci and Rabinowitz. Since then the number of papers on variational methods for strongly indefinite functionals and on applications to Hamiltonian systems has been growing enormously. These methods are not restricted to periodic solutions but can also be used to find heteroclinic or homoclinic orbits and to prove complex dynamics. In fact, they can even be applied to infinite-dimensional Hamiltonian systems and strongly indefinite partial differential equations having a variational structure. The goal of this paper is to present an introduction to variational methods for strongly indefinite functionals like Φ and its applications to the Hamiltonian system (HS). The paper is divided into three parts. Part 1 is concerned with critical point theory, Part 2 with periodic solutions, and Part 3 with homoclinic solutions of (HS). We give proofs or sketches of proofs for selected basic theorems and refer to the literature for more advanced results. No effort is being made to be as general as possible. Neither did we try to write a comprehensive survey on (HS). The recent survey [81] of Rabinowitz in Volume 1A of the Handbook of Dynamical Systems facilitated our task considerably. We chose our topics somewhat complementary to those treated in [81] and concentrated on the first order system (HS), though a certain overlap cannot and should not be avoided. As a consequence we do not discuss second order systems nor do we discuss convex Hamiltonian systems where one can work with the dual action functional which is not strongly indefinite. One more topic which we have not included - though it has recently attracted attention of many researchers - is the problem of finding heteroclinic solutions by variational methods. These and many more topics are being treated in a number of well written monographs dealing with variational methods for Hamiltonian systems, in particular [1, 5, 33, 52, 69, 73, 8]. Further references can be found in these books and in Rabinowitz survey [81]. Naturally, the choice of the topics is also influenced by our own research experience. Restricting ourselves to variational methods we do not touch upon the dynamical systems approach to Hamiltonian systems which includes perturbation theory, normal forms, stability, KAM theory, etc. An introduction to these topics can be found for instance in the textbooks [47, 75]. Also we do not enter the realm of symplectic topology and Floer homology dealing with Hamiltonian systems on symplectic manifolds. Here we refer the reader to the monograph [74] and the references therein. We conclude this introduction with a more detailed description of the contents. In the first part of the paper we consider pertinent results in critical point theory. Particular emphasis is put on a rather simple and direct approach to strongly indefinite functionals. The second part is concerned with periodic solutions of (HS). We present a unified approach, via a finite-dimensional reduction in order to show the existence of one solution, and via a Galerkin-type method in order to find more solutions. Section 2.2 concerns the existence of periodic solutions near equilibria (Lyapunov-type results) and in Section 2.3 3

4 the fixed energy problem is considered (finding solutions of a priori unknown period which lie on a prescribed energy surface). The remaining sections consider the existence and the number of periodic solutions under growth conditions on the Hamiltonian and for spatially symmetric Hamiltonians. The third part deals with homoclinic solutions for (HS) with time-periodic Hamiltonian. Here we present a few basic existence and multiplicity results and discuss a relation to the Bernoulli shift and complicated dynamics. The proofs are more sketchy than in Part 2 because we did not want to enter too much into technicalities which are more complex than in the periodic case. Moreover, the subject of this part is still rapidly developing and has not been systematized in the same way as the periodic solution problem. 1 Critical point theory 1.1 Basic critical point theory Let E be a real Hilbert space with an inner product.,. and Φ a functional in C 1 (E, R). Via the Riesz representation theorem we shall identify the Fréchet derivative Φ (x) E with a corresponding element of E, and we shall write Φ (x), y rather than Φ (x)y. Our goal here is to discuss those methods of critical point theory which will be useful in our applications to Hamiltonian systems. In particular, although most of the results presented here can be easily extended to real Banach spaces, we do not carry out such extension as it will not be needed for our purposes. Recall that {x j } is said to be a Palais-Smale sequence (a (P S)-sequence in short) if Φ(x j ) is bounded and Φ (x j ). The functional Φ satisfies the Palais-Smale condition (the (P S)-condition) if each (P S)-sequence possesses a convergent subsequence. If Φ(x j ) c and Φ (x j ), we shall sometimes refer to {x j } as a (P S) c -sequence. Φ satisfies the Palais-Smale condition at the level c (the (P S) c -condition) if every (P S) c -sequence has a convergent subsequence. We shall frequently use the following notation: Φ c := {x E : Φ(x) c}, K c := {x E : Φ(x) = c, Φ (x) = }. One of the basic technical tools in critical point theory is the deformation lemma. Below we state a version of it, called the quantitative deformation lemma. It is due to Willem [11], see also [23], Theorem I.3.4 and [12], Lemma 2.3. A continuous mapping η : A [, 1] E, where A E, is said to be a a deformation of A in E if η(x, ) = x for all x A. Denote the distance from x to the set B by d(x, B). Lemma 1.1 Suppose Φ C 1 (E, R) and let c R, ε, δ > and a set N E be given. If (1.1) Φ (x) δ whenever d(x, E \ N) δ and Φ(x) c ε, 4

5 then there exists an ε (, ε), depending only on ε and δ, and a deformation η : E [, 1] E such that: (i) η(x, t) = x whenever Φ(x) c ε; (ii) η(φ c+ε \ N, 1) Φ c ε and η(φ c+ε, 1) Φ c ε N; (iii) The mapping t Φ(η(x, t)) is nonincreasing for each x E. Proof Since the argument is well known, we omit some details. A complete proof may be found e.g. in [23], Theorem I.3.4 or [12], Lemma 2.3. A mapping V : E \ K E is said to be a pseudo-gradient vector field for Φ if V is locally Lipschitz continuous and satisfies (1.2) V (x) 2 Φ (x), Φ (x), V (x) Φ (x) 2 for each x E \ K. It is well known and not difficult to prove that any Φ C 1 (E, R) has a pseudo-gradient vector field; see e.g. [23, 8, 12]. Let χ : E [, 1] be a locally Lipschitz continuous function such that { if Φ(x) c ε or d(x, E \ N) δ χ(x) = 1 if Φ(x) c ε/2 and d(x, E \ N) δ/2 and consider the Cauchy problem dη dt = 1 V (η(x, t)) δχ(η(x, t)), η(x, ) = x. 2 V (η(x, t)) Since the vector field above is locally Lipschitz continuous and bounded, η(x, t) is uniquely determined and continuous for each (x, t) E R. It is now easy to see that (i) is satisfied. Moreover, d (1.3) Φ(η(x, t)) = Φ (η(x, t)), dη δ dt dt 4 χ(η(x, t)) Φ (η(x, t)) according to (1.2). Hence also (iii) holds. Let x Φ c+ε \ N and < ε ε/2. In order to establish the first part of (ii) we must show that Φ(η(x, 1)) c ε. Since t η(x, t) x dη ds ds 1 2 δt, d(η(x, t), E \ N) δ/2 whenever t 1. We may assume Φ(η(x, 1)) c ε/2 (otherwise we are done). Then, according to (1.3) and the definition of χ, 1 d Φ(η(x, 1)) = Φ(x) + dt Φ(η(x, t)) dt Φ(x) δ 1 Φ (η(x, t)) dt c + ε δ 2 /4. 4 5

6 Hence Φ(η(x, 1)) c ε if we choose ε min{ ε/2, δ 2 /8}. In order to prove the second part of (ii) it remains to observe that if x Φ c+ε and η(x, t) / N, then d(η(x, t), E \ N) δ/2 and therefore again Φ(η(x, 1)) c ε. We emphasize that the constant ε is independent of the functional Φ and the space E as long as Φ satisfies (1.1). We shall make repeated use of this fact. It is easy to see that if Φ satisfies (P S) and N is a neighbourhood of K c, then there exist ε, δ > such that (1.1) holds. Next we introduce the concept of local linking, due to Li and Liu [6]. Let Φ C 1 (E, R) and denote the ball of radius r and center at the origin by B r. The corresponding sphere will be denoted by S r. The function Φ is said to satisfy the local linking condition at if there exists a subspace F E and α, ρ > such that F and F have positive dimension, (1.4) Φ on F B ρ, Φ α on F S ρ and (1.5) Φ on F B ρ, Φ α on F S ρ. We shall denote the inner product of x and y in R m by x y and we set x := (x x) 1/2. For a symmetric matrix B we denote the Morse index of the quadratic form corresponding to B by M (B). Theorem 1.2 Suppose Φ C 1 (R m, R) satisfies the local linking conditions (1.4) and (1.5) for some F R m. Then Φ has a critical point x with Φ( x) α in each of the following two cases: (i) There exists R > such that Φ < in R m \ B R ; (ii) Φ(x) = 1 2 Bx x + ψ(x), where ψ (x) = o( x ) as x, B is a symmetric invertible matrix and M (B) > dim F. Proof We first consider case (ii) which is more difficult. If there exists a critical point x with Φ( x) α, we are done. If there is no such point, then there exists a pseudogradient vector field V whose domain contains Φ α, and since Φ (x) is bounded away from as x is large, Φ (x) δ (where δ > ) whenever x Φ α. Hence the Cauchy problem dγ dt = V (γ(x, t)), has a solution for all x Φ α, t and (1.6) Φ(γ(x, t)) = Φ(x) + t γ(x, ) = x d Φ(γ(x, s)) ds α ds 6 t Φ (γ(x, s)) 2 ds δ 2 t.

7 Choose R > such that ψ (x) 1 2 λ x for all x R, where λ := {inf λ j : λ j is an eigenvalue of B}. Let R m = F + F, with F ± respectively being the positive and the negative space of B. For x R m write x = x + + x, x ± F ±. By (1.6) and the form of Φ, γ(x, T ) R for any x F S ρ provided T is large enough. Let S n and D n+1 be the unit sphere and the unit closed ball in R n+1. Recall that a space X is called l-connected if any mapping from S n to X, n l, can be extended to a mapping from D n+1 to X (cf. [84], Section 1.8). We want to show that the set Φ α {x R m : x R} is (k 1)-connected if k < M (B), possibly after choosing a larger R. This will imply in particular that any homeomorphic image of S k 1 contained in this set is contractible there. Let r(x, t) := (1 t)x + + x, t 1. Then r is a strong deformation retraction of Φ α {x R m : x R} onto F \B R. To see this, we only need to verify that r(x, t) Φ α for all x, t. Suppose first Bx + x Bx x. Then x + C x for some C; thus ψ((1 t)x + + x ) = o( x 2 ) as x and Φ(r(x, t)) = 1 2 (1 t)2 Bx + x Bx x + ψ((1 t)x + + x ) 1 4 Bx x + ψ((1 t)x + + x ) α if R is large enough. Let now Bx + x Bx x ; then x D x + and d dt Φ(r(x, t)) = Φ (r(x, t)) x + = Bx + x + ψ ((1 t)x + + x ) x +, again provided R is large enough. Hence in this case Φ(r(x, t)) Φ(x) α. Since F \ B R is homeomorphic to S l 1 [R, ), where l := M (B), F \ B R is (k 1)- connected for any k < l. It follows that so is the set Φ α {x R m : x R}. The set {γ(x, T ) : x F S ρ } is contained in Φ α {x R m : x R} and homeomorphic to S k 1, k < M (B). Hence it can be contracted to a point x in Φ α {x R m : x R}. Denote this contraction by γ, let D := F B ρ, D := D [, 1] and define a mapping f : D R m by setting x, s =, x D γ(x (1.7) f(x, s) :=, 2sT ), s 1, x 2 D = F S ρ } 1 γ (γ(x, T ), 2s 1), s 1, x 2 D x, s = 1, x D. It is clear from the construction that Φ(f(x, s)) whenever (x, s) D and < if x D. Let (1.8) Γ := {g C(D, R m ) : g D = f} and (1.9) c := inf max g Γ (x,s) D 7 Φ(g(x, s)).

8 We shall show that f( D) links F S ρ in the sense that if g Γ, then g(x, s) F S ρ for some (x, s) D. Assuming this, we see that the maximum in (1.9) is always α, and hence c α. We claim c is a critical value. Otherwise K c =, so (1.1) holds for g with N =, ε (, c) because Φ satisfies the (P S)-condition as a consequence of (ii). Let ε < ε and η be as in Lemma 1.1. Let g Γ be such that g(x, s) Φ c+ε for all (x, s) D. Since Φ(f(x, s)) if (x, s) D and η(x, 1) = x for x Φ, the mapping (x, s) η(g(x, s), 1) is in Γ. But this is impossible according to the definition of c because Φ(η(g(x, s), 1)) c ε for all (x, s) D. It remains to show that f( D) links F S ρ. Write x = x + x F F, D = F B ρ. For g Γ we consider the map G : (D D ) [, 1] Rm, G(x, s) = x g(x, s). If there is no linking, then G(x, s) for some g Γ and all x D, x D, s 1. For x D we have g(x, s) = f(x, s), hence Φ(g(x, s)) < and G(x, s) (because Φ(x ) ). It follows that G(x, s) when x (D D ) and G is an admissible homotopy for Brouwer s degree. Hence deg(g(., ), D D, ) = deg(g(., 1), D D, ). Since G(x, ) = x g(x, ) = x f(x, ) = x x, the degree on the left-hand side above is ( 1) dim F. On the other hand, f(x, 1) = x, where x is a point outside D D ; hence G(x, 1) for any x D D and the degree is. This contradiction completes the proof of (ii). In case (i) the argument is similar but simpler. Suppose there is no critical point x with Φ( x) α. Since Φ in R m \ B R, γ(x, T ) R for some T > and all x F S ρ. It is obvious that the set {γ(x, T ) : x F S ρ } (which is homeomorphic to a sphere of dimension m 2) can be contracted to a point in R m \ B R. Now we can proceed as above. Note only that Φ satisfies the (P S) c -condition because any (P S) c -sequence lies eventually in B R. We shall need the following extension of Theorem 1.2: Theorem 1.3 Suppose Φ C 1 (R m, R) satisfies the local linking conditions (1.4) and (1.5) for some F R m. If there exist subspaces F F F, F F, and R > such that Φ < on F \ B R and Φ F has no critical point x Φ α, then Φ ( x) = for some x with α Φ( x) max x F BR+1 Φ(x). Proof This time we obtain γ by solving the Cauchy problem dγ dt = χ(γ(x, t))v (γ(x, t)), γ(x, ) = x, 8

9 where χ C (R m, [, 1]) is such that χ = 1 on B R, χ = on R m \B R+1 and V : F Φ α F is a pseudogradient vector field for Φ F. Now we proceed as in the proof of case (i) above and obtain T such that γ(x, T ) F (B R+1 \ B R ) whenever x F S ρ. This set can be contracted to a point in F ( B R+1 \B R ), hence we obtain a map f C( D, F B R+1 ) as in (1.7). Since there exists g Γ C(D, F B R+1 ) where Γ is as in (1.8) it follows that c max Φ(g(x, s)) max Φ(x). (x,s) D x F B R+1 An infinite-dimensional version of the linking theorems (in a setting which corresponds to Theorems 1.2 and 1.3) may be found in [62]. However, we shall only make use of the finite-dimensional versions stated above. If the functional Φ is invariant with respect to a representation of some symmetry group, then Φ usually has multiple critical points. In order to exploit such symmetries, we introduce index theories. Let E be a Hilbert space and (1.1) Σ := {A = C O E : C is closed, O is open and A = A}. Intersections of an open and a closed set (of a topological space) are called locally closed. Thus Σ consists of the locally closed symmetric subsets of E. Let A Σ, A. The genus of A, denoted γ(a), is the smallest integer k such that there exists an odd mapping f C(A, R k \{}). If such a mapping does not exist for any k, then γ(a) := +. Finally, γ( ) =. Equivalently, γ(a) = 1 if A and if there exists an odd map A {+1, 1}; γ(a) k if A can be covered by k subsets A 1,..., A k Σ such that γ(a j ) 1. Proposition 1.4 The two definitions of genus given above are equivalent for A Σ. Proof If f : A R k \ {} is as in the first definition, then the sets A j := {x A : f j (x) }, j = 1,..., k, cover A, are open in A and A j = A j, hence A j Σ. The map f j / f j : A j {+1, 1} shows that γ(a j ) 1. Suppose γ(a) k in the sense of the second definition. Since A A j Σ, we may assume A j A, A = C O, A j = C j O j and C j C, O j O, where C, C j, O, O j are as in the definition of Σ. If f j : A j {+1, 1} is odd, we may extend it to a continuous map f j : O j R. This is a consequence of Tietze s theorem because A j is a closed subset of O j. Replacing f(x) by 1 (f(x) f( x)) we may assume that the extension is also odd. 2 Let π j : A [, 1], j = 1,..., k, be a partition of unity subordinated to the covering O 1,..., O k of A. Replacing π j (x) by 1(π 2 j(x) + π j ( x)) we may assume that all π j are even. Now the map f : A R k, f(x) = (π 1 (x)f 1 (x),..., π k (x)f k (x)) 9

10 is well defined, continuous, odd, and satisfies f(a) R k \ {}. The above definitions of genus do not need to coincide for arbitrary subsets A = A which are not locally closed. Proposition 1.5 Let A, B Σ. (i) If there exists an odd mapping g C(A, B), then γ(a) γ(b). (ii) γ(a B) γ(a) + γ(b). (iii) There exists an open neighbourhood N Σ of A such that γ(a) = γ(n). (iv) If A is compact and / A, then γ(a) <. (v) If U is an open bounded neighbourhood of in R l such that Ū Σ, then γ( U) = l. In particular, γ(s l 1 ) = l, where S l 1 is the unit sphere in R l. (vi) If X is a subspace of codimension m in E and γ(a) > m, then A X. (vii) If / A and i(a) 2, then A is an infinite set. A proof of this classical result may be found e. g. in [8, 85] if Σ contains only closed sets. This restriction is however not needed; see Proposition 1.7 below. Let G be a compact topological group. A representation T of G in a Hilbert space E is a family {T g } g G of bounded linear operators T g : E E such that T e = id (where e is the unit element of G and id the identity mapping), T g1 g 2 = T g1 T g2 and the mappping (g, x) T g x is continuous. T is an isometric representation if each T g is an isometry. A set A E is called T -invariant if T g A = A for all g G. When there is no risk of ambiguity we shall say A is G-invariant or simply invariant. The set will be called the orbit of x and O(x) := {T g x : g G} E G := {x E : T g x = x for all g G} the set of fixed points of the representation T. Obviously, E G is a closed subspace of E and O(x) = {x} if and only if x E G. Let (1.11) Σ := {A E : A is locally closed and T g A = A for all g G}. Note that the definition (1.11) of Σ coincides with (1.1) if G = Z/2 {1, 1} and T ±1 x = ±x. A mapping f : E R is said to be T -invariant (or simply invariant) if 1

11 f(t g x) = x for all g G and x E. If T and S are two (possibly different) representations of G in E and F, then a mapping f : E F is equivariant with respect to T and S (or equivariant) if f(t g x) = S g f(x) for all g G, x E. Finally, if f : E F, we set (1.12) f G (x) := S g 1f(T g x) dg, G where the integration is performed with respect to the normalized Haar measure. It is easy to see that f G is equivariant. As a special case, for G = Z/2 acting via the antipodal map on E and F we have f G (x) = 1 2 (f(x) f( x)), so f G is odd. If G acts trivially on F (i.e., S ±1 x = x) we obtain f G (x) = 1 2 (f(x) + f( x)), so f G is even. If Φ C 1 (E, R) is invariant with respect to an isometric representation T of G, then it is easy to see that Φ (T g x) = T g Φ (x) for all x E, g G. Hence x is a critical point of Φ if and only if so are all y O(x). The set O(x) will be called a critical orbit of Φ. In what follows we restrict our attention to isometric representations of G = Z/p, where p 2 is a prime, and G = S 1 = R/2πZ. If G = S 1, we do not distinguish between θ R and the corresponding element of G, and we may also identify this element with e iθ. The same applies for G = Z/p S 1, where we identify the elements of G with roots of unity, again represented as e iθ. Next we define an index i : Σ N { } for G = S 1 and G = Z/p, p a prime number. In the case p = 2 we recover the genus. For A Σ, A, we define i(a) = 1 if there exists a continuous map f : A G C \ {} such that f(t θ x) = e inθ f(x) for some n N and all x, θ (n/p / N if G = Z/p). And i(a) k if A can be covered by k sets A 1,..., A k Σ such that i(a j ) 1. If such a covering does not exist for any k, then i(a) := +. Finally, we set i( ) :=. We have a version of Proposition 1.4 for G = S 1. Proposition 1.6 If G = S 1, then i(a) is the smallest integer k for which there exists a mapping f C(A, C k \ {}) such that f(t θ x) = e inθ f(x) for some n N and all x, θ. The proof is similar to that of Proposition 1.4. Note only that (1.12) needs to be used and if f j (T θ x) = e in jθ f j (x), then f(x) = (f 1 (x) n/n 1,..., f k (x) n/n k ) where n is the least common multiple of n 1,..., n k. The corresponding version for G = Z/p, p an odd prime, requires spaces lying between C k \ {} and C k+1 \ {}; see [1, Proposition 2.9]. The above definition is due to Benci [17, 18] in the case G = S 1 and to Krasnoselski [56] for G = Z/p. Benci used in fact mappings f C(A, C k \ {}) as in Proposition 1.6. Let us also remark that a different, cohomological index, has been introduced by Fadell and Rabinowitz [36] for G = Z/2 and G = S 1, and by Bartsch [1, Example 4.5] for G = Z/p. While the geometrical indexes of Krasnoselski and Benci are much more elementary, the cohomological indexes have some additional properties (which will not be needed here). Since we only consider isometric representations, it is easy to see that the orthogonal complement Ẽ := (EG ) is invariant. In order to formulate the properties of the index 11

12 for G = S 1 and G = Z/p we set d G := 1 + dim G = { 1 for G = Z/p; 2 for G = S 1. Proposition 1.7 Suppose G = S 1 or G = Z/p, where p is a prime, and let A, B Σ. (i) If there exists an equivariant mapping g C(A, B), then i(a) i(b). (ii) i(a B) i(a) + i(b). (iii) There exists an open neighbourhood N Σ of A such that i(a) = i(n). (iv) If A is compact and A E G =, then i(a) <. (v) If U is an open bounded invariant neighbourhood of in a finite-dimensional invariant subspace X of Ẽ, then i( U) = 1 d G dim X. (vi) If X is an invariant subspace of then A (E G X). Ẽ with finite codimension and if i(a) > 1 d G codimẽ X, (vii) If A E G, then i(a) = +. If A E G = and i(a) 2, then A contains infinitely many orbits. Proof (i) Let i(b) = k < (otherwise there is nothing to prove) and B 1,..., B k be a covering of B as in the definition of the index i(b). Then g 1 (B 1 ),..., g 1 (B k ) is a covering of A as in the definition of i(a), hence i(a) k. (ii), (iii) and (iv) are obvious. (v) It follows easily from Proposition 1.4 or 1.6 that i( U) 1 d G dim X if G = Z/2 or G = S 1, respectively. In the Z/p-case for p 3 we may identify X with C l and take the covering A j := {z C l : z j, p arg(z j ) mod 2π}, B j := {z C l : z j, p arg(z j ) π mod 2π}, j = 1,..., k, of C l \ {} in order to see that i(c l \ {}) 2l = dim X. The reverse inequality is a consequence of the Borsuk-Ulam theorem. A proof for G = S 1 may be found in [73], Theorem 5.4, and for G = Z/p in [9]. (vi) Let Y be the orthogonal complement of X in Ẽ. Then Y is invariant and dim Y = codimẽ X. Suppose A (E G X) = and let f(x) = P Y x where P Y denotes the orthogonal projector onto Y. Then f : A Y \ {}. If G = Z/2 or G = S 1 this implies i(a) 1 d G dim Y by Propositions 1.4, 1.6, respectively. If G = Z/p, p 3, we identify Y with C m and write f = (f 1,..., f m ). It follows from the Peter-Weyl theorem (see [73], Theorem 5.1, where the case G = S 1 is considered) that f j (T θ x) = e injθ f(x) (n j mod 12

13 p). Let g j (x) := f j (x) n/n j, where n is the least common multiple of n 1,..., n k. Then g : A C m \ {} and g(t θ x) = e inθ g(x), so i(a) i(c m \ {}) 2m = codimẽ X, a contradiction. (vii) Suppose A E G and there exists a covering A 1,..., A k of A as in the definition. Then A j E G for some j. For each x A j E G we have f(t θ x) = f(x). So if f j (T θ x) = e inθ f j (x), with n as before, then f j (x) =. Thus there is no mapping f j : A j G C \ {} as required in the definition of index, hence i(a) = +. If A E G = and A consists of k orbits O(x 1 ),..., O(x k ), then we let n j 1 be the largest integer such that 2π/n j G and T 2π/nj x j = x j. If G = Z/p then all n j = 1. We define f : A G by setting f(t θ x j ) = e inθ, where n is the least common multiple of n 1,..., n k. It is easy to prove an equivariant version of the deformation lemma 1.1 for invariant functionals Φ : E R. One simply observes that if V is a pseudo-gradient vector field for Φ then (1.13) V G (x) := Tg 1 V (T g x) dg G is an equivariant pseudo-gradient vector field for Φ. Integrating V G as in the proof of Lemma 1.1 yields an equivariant deformation η. Theorem 1.8 Suppose f C 1 (S n 1, R) is invariant with respect to a representation of Z/p in R n without nontrivial fixed points. Then f has at least n Z/p-orbits of critical points. Theorem 1.9 Suppose f C 1 (S 2n 1, R) is invariant with respect to a representation of S 1 in R 2n without nontrivial fixed points. Then f has at least n S 1 -orbits of critical points. Proof (outline) The proofs of Theorems 1.8 and 1.9 are standard. Set M = S d G n 1 and suppose f has finitely many critical orbits O j = O(x j ), j = 1,..., k. We may assume that the critical values c j = f(x j ) are ordered: c 1 c k. Then using the properties of the index and an equivariant deformation lemma for functionals defined on manifolds one sees that i(f c j ) j. The result follows from i(m) = n. We present a simpler proof which works if f is locally Lipschitz continuous. Let η be the negative gradient flow of f on M and consider the sets A j := {x M : η(x, t) O j as t }. and j B j := A i for j =,..., k. i=1 13

14 Then B =, B k = M, and it is not difficult to see that B j 1 is an open subset of B j. Consequently all A j are locally closed. Using the flow η one constructs an equivariant map f j : A j O j. This implies i(a j ) i(o j ) = 1, and therefore n = i(m) = i(b k ) k. This proof does not need the equivariant deformation lemma, and it produces directly a covering of M as in the definition of the index. Remark 1.1 One can define index theories satisfying properties 1.7 (i)-(iv), (vii) for arbitrary compact Lie groups G. However, properties 1.7 (v), (vi) which are important for applications and computations, cannot be extended in general, except for a very restricted class of groups. This has been investigated in detail in [1]. In certain applications the representation of G in E is of a special form which allows to obtain similar results as above. In order to formulate this, call a finite-dimensional representation space V = R n of the compact Lie group G admissible if every equivariant map O V k 1, O V k a bounded open and invariant neighbourhood of in V k, has a zero on O. Clearly the antipodal action of Z/2 on R is admissible as are the nontrivial representations of Z/p or S 1 in R 2. Let E = j=1 E j be the Hilbert space sum of the finite-dimensional Hilbert spaces E j such that each E j is isomorphic to V as a representation space of G. For instance, E = L 2 (S 1, V ) with the representation of G given by (T g x)(t) = T g (x(t)) has this property. The same is true for subspaces like H 1 (S 1, V ) or H 1/2 (S 1, V ). For an invariant, locally closed set A E let i(a) = 1 if A and there exists a continuous equivariant map A SV = {v V : v = 1}. And let i(a) k if A can be covered by A 1,..., A k Σ with i(a) 1. Proposition 1.7 can be extended to this index theory. See [6, 16] for applications to Hamiltonian systems. 1.2 Critical point theory for strongly indefinite functionals As will be explained in Section 2.1, functionals naturally corresponding to Hamiltonian systems are strongly indefinite. This means that they are of the form Φ(z) = 1 Lz, z 2 ψ(z) where L : E E is a selfadjoint Fredholm operator with negative and positive eigenspace both infinite-dimensional, and the same is true for the Hessian Φ (z) of a critical point z of Φ. In order to study such functionals it will be convenient to use a variant of the Palais-Smale condition that allows a reduction to the finite-dimensional case and leads to simpler proofs. We shall also present two useful critical point theorems which apply when the Palais-Smale condition does not hold. These will be needed for the existence of homoclinic solutions. First we introduce certain sequences of finite-dimensional subspaces and replace the Palais-Smale condition by another one which is adapted to these sequences. Let {E n } n 1 be a sequence of finite-dimensional subspaces such that E n E n+1 for 14

15 all n and E = E n. n=1 Let P n : E E n denote the orthogonal projection. Then {x j } is called a (P S) -sequence for Φ (with respect to {E n }) if Φ(x j ) is bounded, each x j E nj for some n j, n j and P nj Φ (x j ) as j. Φ is said to satisfy the (P S) -condition if each (P S) - sequence has a convergent subsequence. It is easy to see that K c is compact for each c if (P S) holds. Indeed, let x j K c, then we can find n j j such that y j x j 1/j, Φ(y j ) c and P nj Φ (y j ), where y j := P nj x j. Hence {y j }, and therefore also {x j }, has a convergent subsequence. We shall repeatedly use the notation Φ n := Φ En and A n := A E n. Observe that Φ n(x) = P n Φ (x) for all x E n. The condition (P S) (in a slightly different form) has been introduced independently by Bahri and Berestycki [7, 8], and Li and Liu [6]. Lemma 1.11 If Φ satisfies (P S) and N is a neighbourhood of K c, then there exist ε, δ > and n 1 such that Φ n(x) δ whenever d(x, E \ N) δ, Φ(x) c ε and n n. Proof If the conclusion is false, then we find a sequence {x j } such that x j E nj for some n j j, d(x j, E \ N), Φ nj (x j ) c and Φ n j (x j ). Hence {x j } is a (P S) -sequence. Passing to a subsequence, x j x K c. However, since K c is compact, the sequence {x j } is bounded away from K c and therefore x / K c, a contradiction. Next we introduce the notion of limit index in order to deal with symmetric functionals. As in Section 1.1 we consider the groups G = Z/p, where p is a prime, or G = S 1 and their isometric representations in E. The group Z/2 always acts via the antipodal map (i.e., T ±1 x = ±x) so that obviously E G = {}. The reason for going beyond the usual index is that we need to distinguish between certain infinite-dimensional sets having i(a) = ; in particular, we need to compare different spheres of infinite dimension and codimension. Let {E n } be a sequence of subspaces as above and suppose in addition that each E n is G-invariant and E G E n for some n. Let {d n } be a sequence of integers and E := {E n, d n } n=1. The limit index of A Σ with respect to E, i E (A), is defined by i E (A) := lim sup(i(a n ) d n ). n 15

16 Clearly i E (A) = if A E G. The limit index, in a somewhat different form, has been introduced by Y. Q. Li [63], see also [92]. A special case is the limit genus, γ E (A). We note that i E (A) can take the values + or and if E n = E, d n = for all n, then i E (A) = i(a), and similarly for the genus. Remark 1.12 The limit index is patterned on the notion of limit relative category introduced by Fournier et al. in [42]. Recall that if Y is a closed subset of X, then a closed set A X is said to be of category k in X relative to Y, denoted cat X,Y (A) = k, if k is the least integer such that there exist closed sets A,..., A k X, A Y, which cover A, all A j, 1 j k, are contractible in X and there exists a deformation h : A [, 1] X with h(a, 1) Y and h(y, t) Y for all t [, 1]. If Y = (and A = ), then cat X (A) = cat X, (A) is the usual Lusternik-Schnirelman category of A in X. For X E and using the above notation for subsets of E, the limit relative category cat X,Y (A) is by definition equal to lim sup n cat Xn,Y n (A n ). Note that unlike for the limit index, the limit category is necessarily a nonnegative integer. Note also that if D is the unit closed ball and S its boundary in an infinite-dimensional Hilbert space, then cat D,S (D) = cat S (S) = while cat D,S (D) = cat S (S) = 1. Below we formulate some properties of i E which automatically hold for γ E. As before Ẽ is the orthogonal complement of E G. It follows from the invariance of E n that the dimension of Ẽ n = E n Ẽ is even except when G = Z/2. Recall the notation d G = 1 + dim G. Proposition 1.13 Let A, B Σ. (i) If for almost all n there exists an equivariant mapping g n C(A n, B n ), then i E (A) i E (B). (ii) i E (A B) i E (A) + i(b) if i E (A). (iii) Let l Z, R >. If Y is an invariant subspace of Ẽ such that dim Y n = (d n + l)d G for almost all n, then i E (Y S R ) = l. (iv) Let m Z. If X is an invariant subspace of Ẽ such that codim Ẽ n X n = (d n + m)d G for almost all n and if i E (A) > m, then A (E G X). Proof (i) It follows from (i) of Proposition 1.7 that i(a n ) d n i(b n ) d n. So passing to the limit as n we obtain the conclusion. (ii) i(a n B n ) d n i(a n ) d n + i(b n ) (i(a n ) d n ) + i(b). Now we can pass to the limit again. (iii) This follows from (v) of Proposition

17 (iv) There exists a number n such that E G E n, codimẽn X n = (d n + m)d G and i(a n ) > d n + m. So by (vi) of Proposition 1.7, A n (E G X n ) A (E G X). Recall that if x is a critical point of an invariant functional Φ, then so are all y O(x). We have the following results concerning the existence of critical orbits. Theorem 1.14 Suppose that Φ C 1 (E, R) is G-invariant, satisfies (PS)* and Φ() =. Moreover, suppose there exist numbers ρ >, α < β <, integers m < l, and invariant subspaces X, Y Ẽ such that: (i) E G E n for almost all n; (ii) codimẽn X n = (d n + m)d G and dim Y n = (d n + l)d G for almost all n; (iii) Φ Y Sρ β; (iv) Φ E G X α and Φ E G. Then Φ has at least l m distinct critical orbits O(x j ) such that O(x j ) E G =. The corresponding critical values can be characterized as c j = inf i E (A) j and are contained in the interval [α, β]. sup Φ(x), m + 1 j l, x A Proof It is clear that {A Σ : i(a) j + 1} {A Σ : i(a) j}, hence c m+1 c m+2... c l. According to (iii) of Proposition 1.13, i E (Y S ρ ) l, hence by (iii), c l β. Suppose i E (A) m + 1. Then A (E G X) by (iv) of Proposition 1.13 and it follows from (iv) that c m+1 α. Moreover, (iv) implies K cj E G =. Suppose c := c j =... = c j+p for some p. The proof will be complete if we can show that i(k c ) p + 1 (because either all c j are distinct and K cj, or i(k cj ) 2 for some j and K cj contains infinitely many orbits according to (vii) of Proposition 1.7). By (iii) of Proposition 1.7 there exists a neighbourhood N Σ such that i(n) = i(k c ), and for this N we may find ε, δ > and n 1 such that the conclusion of Lemma 1.11 holds. It follows from Lemma 1.1 that we can find an ε > such that for each n n there exists a deformation η n : E n [, 1] E n with η n (Φn c+ε, 1) Φc ε n N n. Moreover, using (1.13) we may assume that η n (., t) is equivariant for each t. So by (i) and (ii) of Proposition 1.13 and the definition of c, (1.14) j + p i E (Φ c+ε ) i E (Φ c ε N) i E (Φ c ε ) + i(n) < j + i(n). Hence i(k c ) = i(n) > p. Applying Theorem 1.14 to Φ we immediately obtain the following result which will be more convenient in our applications: 17

18 Corollary 1.15 Suppose that Φ C 1 (E, R) is G-invariant, satisfies (PS)* and Φ() =. Moreover, suppose there exist numbers ρ >, < α < β, integers m < l, and invariant subspaces X, Y Ẽ such that: (i) E G E n for almost all n; (ii) codimẽn X n = (d n + m)d G and dim Y n = (d n + l)d G for almost all n; (iii) Φ Y Sρ α; (iv) Φ E G X β and Φ E G. Then Φ has at least l m distinct critical orbits O(x j ) such that O(x j ) E G =. The corresponding critical values can be characterized as c j = sup i E (A) j and are contained in the interval [α, β]. inf Φ(x), m + 1 j l, x A Corollary 1.16 If the hypotheses of Theorem 1.14 or Corollary 1.15 are satisfied with l Z fixed and m Z arbitrarily small, then Φ has infinitely many geometrically distinct critical orbits O(x j ) such that O(x j ) E G =. Moreover, c j in Theorem 1.14 and c j in Corollary 1.15 as j. Proof It suffices to consider the case of Theorem The value c j is defined for all j l, j Z and since the sequence {c j } is nondecreasing, either c j and we are done, or c j c R as j. In the second case K c is nonempty and compact according to (P S). Let N Σ be a neighbourhood of K c such that i(n) = i(k c ) < and let ε > be as in Lemma 1.1. Since c + ε c j for some j and c ε < c j for all j l, we have (cf. (1.14)) j i E (Φ c+ε ) i E (Φ c ε ) + i(n) =, a contradiction. Remark 1.17 Proposition 1.13, Theorem 1.14 and Corollaries are valid if G = Z/2 and T ±1 x = ±x (i.e., Φ is even). For this G, i(a) is just the genus γ(a). If G = Z/p and p 3, then l m is necessarily an even integer. We conclude this part with a critical point theorem which needs tools from algebraic topology. 18

19 Theorem 1.18 Let M be a compact differentiable manifold and Φ : E M R a C 1 -functional defined on the product of the Hilbert space E and M. Suppose Φ satisfies (P S), there exist numbers ρ >, α < β γ and subspaces W, Y, where E = W Y, W n W, Y n Y, dim W n 1, such that: (i) Φ (W Sρ) M α; (ii) Φ Y M β; (iii) Φ (W B ρ) M γ. Then Φ possesses at least cupl(m) + 1 critical points. Here cupl(m) denotes the cuplength of M with respect to singular cohomology theory with coefficients in an arbitrary field. For a proof we refer to Fournier et al. [42]. The argument there uses the limit relative category (see Remark 1.12) and is in the spirit of Theorem In particular, the numbers c j are defined by minimaxing over sets A D with cat C,D (A) j, where (C, D) := (E M, (W S ρ) M). An important role is played by the inequality cat C,D ((W B ρ ) M) cupl(m) + 1. A related result can be found in [9]. For applications to homoclinic solutions one has to deal with functionals where neither the (PS)- nor the (PS) -condition holds. We present two abstract critical point theorems which are helpful in this case. The proofs involve again a reduction to a finite-dimensional situation. Theorem 1.19 Let E be a separable Hilbert space with the orthogonal decomposition E = E + E, z = z + + z, and suppose Φ C 1 (E, R) satisfies the hypotheses: (i) Φ(z) = 1 2 ( z+ 2 z 2 ) ψ(z) where ψ C 1 (E, R) is bounded below, weakly sequentially lower semicontinuous with ψ : E E weakly sequentially continuous; (ii) Φ() = and there are constants κ, ρ > such that Φ(z) > κ for every z S ρ E + ; (iii) there exists e E + with e = 1, and R > ρ such that Φ(z) for z M where M = {z = z + ζe : z X, z R, ζ }. Then there exists a sequence {z j } in E such that Φ (z j ) and Φ(z j ) c for some c [κ, m], where m := sup Φ(M)]. The theorem is due to Kryszewski and Szulkin [58]. Some compactness is hidden in condition (i) where the weak topology is used. In the applications the concentrationcompactness method, see [65], can sometimes be used in order to obtain an actual critical 19

20 point. Of course, if the Palais-Smale condition holds then there exists a critical point at the level c. Proof (outline) Let P ± : E E ± be the orthogonal projections. We choose a Hilbert basis {e k } k of E and define the norm { } u := max P + u, e k u,. 2 k The topology induced on E by this norm will be denoted by τ. On subsets {u E : P u R} this topology coincides with the weak strong product topology (E, w) (E +, ) on E. In particular, for a -bounded sequence {u j } in E we have u j u if and only if u j u with respect to. Given a finite-dimensional subspace F E +, -bounded subsets of E F are -precompact. We prove the theorem arguing indirectly. Suppose there exists α > with Φ (u) α for all u Φ m κ := {u E : κ Φ(u) m}. Then we construct a deformation h : I Φ m Φ m, I = [, 1], with the properties: (h 1 ) h : I Φ m Φ m is continuous with respect to the -topology on Φ m, and with respect to the τ-topology; (h 2 ) h(, u) = u for all u Φ m ; (h 3 ) Φ(h(t, u)) Φ(u) for all t I, u Φ m κ ; (h 4 ) each (t, u) I Φ m has a τ-open neighbourhood W such that the set {v h(s, v) : (s, v) W } is contained in a finite-dimensional subspace of E; (h 5 ) h(1, Φ m ) Φ κ. This leads to a contradiction as follows. Since M is τ-compact, by (h 1 ) and (h 4 ) there exists a finite-dimensional subspace F E containing the set {v h(s, v) : (s, v) I M}, hence h(i (M F )) F. Since k=1 h(i M) F Φ κ F \ (S ρ E + ) a standard argument using the Brouwer degree yields h(1, M F ) S ρ E +. (The sets F M and F S ρ E + link in F.) Now condition (ii) of the theorem implies h(1, M F ) Φ κ, contradicting (h 5 ). It remains to construct a deformation h as above. For each u Φ m κ we choose a pseudogradient vector w(u) E, that is w(u) 2 and Φ (u), w(u) > Φ (u) (this definition differs somewhat from (1.2)). By condition (i) of the theorem there exists a τ-open neighbourhood N(u) of u in E such that Φ (v), w(u) > Φ (u) for all v N(u) Φ m κ. 2

21 If Φ(u) < κ we set N(u) = {v E : Φ(v) < κ}. As a consequence of condition (i) of the theorem this set is τ-open. Let (π j ) j J be a τ-lipschitz continuous partition of unity of Φ m subordinated to the covering N(u), u Φ m. This exists because the τ-topology is metric. Clearly, the π j : E [, 1] are also -Lipschitz continuous. For each j J there exists u j Φ m with supp π j N(u j ). We set w j = w(u j ) and define the vector field f : Φ m E, f(u) := m κ α π j (u)w j. This vector field is locally Lipschitz continuous and τ-locally Lipschitz continuous. It is also τ-locally finite-dimensional. Thus we may integrate it and obtain a flow η : [, ) Φ m Φ m. It is easy to see that the restriction of η to [, 1] Φ m satisfies the properties (h 1 )-(h 5 ). Remark 1.2 A sequence {z j } is called a Cerami sequence if Φ(z j ) is bounded and (1 + z j )Φ (z j ). This definition has been introduced by Cerami in [22]. Note in particular that if {z j } is as above, then Φ (z j ), z j which does not need to be the case for an (a priori unbounded) (P S)-sequence. It has been shown in [59] that under the hypotheses of Theorem 1.19 a stronger conclusion holds: there exists a Cerami sequence {z j } such that Φ(z j ) c [κ, m]. The next result of this section deals with Z/p-invariant functionals Φ C 1 (E, R). As a substitute for the (P S)- or (P S) -condition we introduce the concept of (P S)-attractor. Given an interval I R, we call a set A E a (P S) I -attractor if for any (P S) c -sequence {z j } with c I, and any ε, δ > one has z j U ε (A Φ c+δ c δ ) provided j is large enough. Here U ε (F ) denotes the ε-neighbourhood of F in E. Theorem 1.21 Let E be a separable Hilbert space with an isometric representation of the group G = Z/p, where p is a prime, such that E /p = {}. Let E = E+ E, z = z + + z, be an orthogonal decomposition and E ± be Z/p-invariant. Let Φ C 1 (E, R) be a Z/p-invariant functional satisfying the following conditions: (i) Φ(z) = 1 2 ( z+ 2 z 2 ) ψ(z) where ψ C 1 (E, R) is bounded below, weakly sequentially lower semicontinuous with ψ : E E weakly sequentially continuous; (ii) Φ() = and there exist κ, ρ > such that Φ(z) > κ for every z S ρ E + ; (iii) there exists a strictly increasing sequence of finite-dimensional Z/p-invariant subspaces F n E + such that sup Φ(E n ) < where E n := E F n, and an increasing sequence of real numbers R n > with sup Φ(E n \ B Rn ) < inf Φ(B ρ ); j J (iv) for any compact interval I (, ) there exists a (P S) I -attractor A inf{ z + w + : z, w A, z + w + } >. such that 21

22 Then Φ has an unbounded sequence {c j } of positive critical values. Proof (outline) Let τ be the topology on E introduced in the proof of Theorem For c R we consider the set M(c) of maps g : Φ c E satisfying: (P 1 ) g is τ-continuous and equivariant; (P 2 ) g(φ a ) Φ a for all a inf Φ(B ρ ) 1 where ρ is from condition (ii); (P 3 ) each u Φ c has a τ-open neigbourhood W E such that the set (id g)(w Φ c ) is contained in a finite-dimensional linear subspace of E. Let i be the Z/p-index from Section 1.1 and set i (c) := min g M(c) i(g(φc ) S ρ E + ) N { }. Clearly i is nondecreasing and i (c) = for c κ where κ is from (ii). i is a kind of pseudoindex in the sense of Benci s paper [18]. Now we define the values c k := inf{c > : i (c) k}. One can show that i (c) is finite for every c R and can only change at a critical level of Φ. In order to see the latter, given an interval [c, d] without critical values one needs to construct maps g M(d) with g(φ d ) Φ c. Such a map can be obtained as time-1-map of a deformation as in the proof of Theorem Of course one has to make sure that the deformation is equivariant which is the case if the vector field is equivariant. This can be easily achieved, see (1.13). Given a finite-dimensional subspace F n E + from condition (iii) one next proves that i (c) dim F n for any c Φ(E F n ). This is a consequence of the properties of the index stated in Proposition 1.7. No extension of the index to infinite dimensions is needed. Details of the proof, of a slightly more general result in fact, can be found in [12] for p = 2 and in [13] for p an odd prime. 2 Periodic solutions 2.1 Variational setting for periodic solutions In this section we reformulate the problem of existence of 2π-periodic solutions of the Hamiltonian system (2.1) ż = JH z (z, t) in terms of the existence of critical points of a suitable functional and we collect some basic facts about this functional. When looking for periodic solutions of (2.1) we shall always assume that the Hamiltonian H = H(z, t) satisfies the following conditions: 22

23 (H 1 ) H C(R 2N R, R), H z C(R 2N R, R 2N ) and H(, t) ; (H 2 ) H is 2π-periodic in the t-variable; (H 3 ) H z (z, t) c(1 + z s 1 ) for some c > and s (2, ). We note that it causes no loss of generality to assume H(, t). Occasionally we shall need two additional conditions: (H 4 ) H zz C(R 2N R, R 4N 2 ); (H 5 ) H zz (z, t) d(1 + z s 2 ) for some d > and s (2, ). Clearly, (H 5 ) implies (H 3 ). Let E := H 1/2 (S 1, R 2N ) be the Sobolev space of 2π-periodic R 2N -valued functions (2.2) z(t) = a + (a k cos kt + b k sin kt), a, a k, b k R 2N k=1 such that k=1 k( a k 2 + b k 2 ) <. Then E is a Hilbert space with an inner product (2.3) z, w := 2πa a + π k(a k a k + b k b k ), where a k, b k are the Fourier coefficients of w. It is well known that the Sobolev embedding E L q (S 1, R 2N ) is compact for any q [1, ) (see e.g. [2]) but z E does not imply z is bounded. There is a natural action of R on L q (S 1, R 2N ) and E given by time translation: k=1 (T θ z)(t) := z(t + θ) for θ, t R. Since the functions z are 2π-periodic in t, T induces an isometric representation of G = S 1 R/2πZ. In the notation of Section 1.1, we have O(z 1 ) = O(z 2 ) if and only if z 2 (t) = z 1 (t + θ) for some θ and all t R. Let and Φ(z) := 1 2 ( Jż z) dt ψ(z) := H(z, t) dt. H(z, t) dt Proposition 2.1 If H satisfies (H 1 )-(H 3 ), then Φ C 1 (E, R) and Φ (z) = if and only if z is a 2π-periodic solution of (2.1). Moreover, ψ is completely continuous in the sense that ψ (z j ) ψ (z) whenever z j z. If, in addition, H satisfies (H 4 ) and (H 5 ), then Φ C 2 (E, R) and ψ (z) is a compact linear operator for each z. 23

24 Proof We only outline the argument. The details may be found e.g. in [8], Appendix B or [12], Appendix A and Lemma Although the results in [12] concern elliptic partial differential equations, the proofs are easy to adapt to our situation. Let s = s/(s 1) be the conjugate exponent. By (H 3 ), H z : L s (S 1, R 2N ) L s (S 1, R 2N ) is a continuous mapping, and using this one shows that ψ C 1 (E, R) and ψ (z), w = H z (z, t) w dt. Moreover, it follows by the compact embedding of E into L s (S 1, R 2N ) that ψ is completely continuous. Since Jż w = ż Jw, the bilinear form (z, w) ( Jż w) dt is (formally) selfadjoint. According to (2.2) and (2.3), ( Jż w) dt = π k( Jb k a k + Ja k b k ), k=1 hence this form is continuous in E and the quadratic form z ( Jż z) dt is of class C 1. Now it is easy to see that Φ (z) = if and only if z is a 2π-periodic solution of (2.1). Moreover, by elementary regularity theory, z C 1 (S 1, R 2N ). If (H 4 ) and (H 5 ) are satisfied, then, referring to the arguments in [8], [12] again, we see that ψ C 2 (E, R) and ψ (z)w, y = H zz (z, t)w y dt. Since ψ is completely continuous, ψ (z) is a compact linear operator. Note that complete continuity of ψ implies weak continuity of ψ (i.e., ψ(z j ) ψ(z) whenever z j z). Remark 2.2 If system (2.1) is autonomous, i. e. H = H(z), then Φ(T θ z) = Φ(z) for all θ R. Thus Φ is T -invariant. Two 2π-periodic solutions z 1, z 2 of an autonomous system are geometrically distinct if and only if O(z 1 ) O(z 2 ). When H = H(z), we shall write H (z) instead of H z (z). Let z(t) = a k cos kt ± Ja k sin kt. Then ( Jż z) dt = ±2πk a k 2 = ± z 2. 24