Periodic solutions for second-order even and noneven Hamiltonian systems

In this paper, we consider the second-order Hamiltonian system


Introduction
Consider the second-order Hamiltonian system where N is a positive integer, V : R N → R is a potential function, and V denotes the gradient of V .In 1978, Rabinowitz [27] proved that, for any T > 0, system (1.1) possesses a nonconstant T-periodic solution under the following assumptions: (V 3) ((AR)-condition) there exist constants μ > 2 and r 0 > 0 such that 0 < μV (x) ≤ V (x), x , |x| ≥ r 0 .
Since the minimal period of this solution may be T/k for some positive integer k, Rabinowitz conjectured that system (1.1) possesses a T-periodic solution with minimal period T under assumptions (V 1)-(V 3).This is the so-called Rabinowitz minimal periodic solution conjecture.Since then, this conjecture has been studied by many mathematicians [1, 10, 12-15, 20-24, 34].If V is convex and superquadratic, then its Fenchel conjugate function is subquadratic.Then the dual variational functional attains its minimum energy at some point, which corresponds to a T-periodic solution with minimal period T. Using this fact, Ambrosetti and Mancini [1] studied the following second-order Hamiltonian system: -ẍ = Qx + V (x). (1.2) If Q is positive definite and V is convex, using the Clark dual, they proved that for any T > 0, system (1.2) possesses a sequence of solutions v σ with minimal period T, where σ = 2π/T > ω n , and ω n denotes the eigenvalues of Q.Moreover, v σ → 0 as T → 2π/ω n , whereas v σ L ∞ → ∞ as T → 0. For more results on the assumption, we refer to [1,12,13].
Releasing the convexity assumption, many mathematicians assumed that V (x) is twice continuously differentiable.Then we can define a Maslov-type index and prove an iterative formula.This formula can be used to estimate the minimal period of periodic solutions.This method was firstly introduced by Long [21], who studied second-order even Hamiltonian system (1.1) under assumptions (V 2), (V 3), and (V 1 ) V ∈ C 2 (R N , R), and V (x) ≥ 0 for all x ∈ R N ; (V 4) V (x) is even, i.e., V (-x) = V (x) for all x ∈ R N .He proved that system (1.1) possesses a T-periodic solution with minimal period T or T/3 when V satisfies (V 1 ) and (V 2) -(V 4).For more results for even Hamiltonian systems, we refer to [14,15] and references therein.For a second-order noneven Hamiltonian system, Long showed that system (1.1) possesses a T-periodic solution with minimal periodic T/k for some integer k satisfying 1 ≤ k ≤ n + 2 [22] or 1 ≤ k ≤ n + 1 [23].For more results for noneven Hamiltonian systems, we refer to [20] and references therein.
There is a third-type condition, the global (AR)-condition: (V 3 ) there exists a constant θ > 1 such that For V satisfying (V 1 ), (V 3 ), and (V 4), Xiao [34] proved the existence of a T-periodic solution with minimal period T. In 2020, Xiao and Shen [33] generalized (V 3 ) and assumed that (V 2 ) Using the Nehai manifold method and a disturbed technique, they still proved the existence of a T-periodic solution with minimal period T under assumptions (V 1), (V 2 ), and (V 4)-(V 7).As is well known, the Nehari manifold method [26] can be used to study the existence of ground state solutions to partial differential equations [2-4, 6, 16-19, 25, 28-31]) and periodic solutions to ordinary differential equations and difference equations [33].To use the Nehari manifold method, we need to build a homomorphism between the Nehari manifold and a suitable subspace.To do this, we need to introduce some monotonicity assumptions on F to prove the following inequality: where F(x, u) is nonlinear term, and f (x, u) = ∇ u F(x, u), s ∈ R + and u, v ∈ R (u, v ∈ R N , respectively).Those monotonicity hypotheses are divided into two cases: the lowdimensional and high-dimensional cases.In the low-dimensional case, Szulkin and Weth [29] introduced the following assumption: (S) the map u → f (x, u)/|u| is strictly increasing on (-∞, 0) and (0, ∞).They proved inequality (1.3) and built the homomorphism mentioned above.For more results on this direction, we refer to [8,17,18,29,30].In the high-dimensional case, the proof of (1.3) is more complicated.In 2015, Bartsch and Mederski [3] introduced the following assumptions: . If, in addition, F(x, u) = F(x, v), then the strict inequality holds; (BM2) F is convex with respect to u.Then they also built the homeomorphism.For more results in this direction, we refer to [3,5,7,9,25,31,33,37].
The rest part of this paper is divided into two parts.In Sect.2, we study system (1.1) with an even potential functional V (x).In Sect.3, we study system (1.1) with noneven potential functional V (x).

The even case
Given T > 0, let S T = R/(TZ).The Solobev space H 1 is defined as where ẋ is the weak derivative of x.The space H 1 is equipped with the usual norm and the corresponding inner product where | • | and (•, •) denote the standard norm and inner product in R N respectively.The variational functional corresponding to system (1.1) is According to Lemma 2.1 in [33], since V satisfies (V 1 ), (V 6), and (V 9), ϕ is continuously differentiable on H 1 , and ) * is compact.By the Fourier series theory, for any x ∈ H 1 , we have . . .Then we define a the following subspace E of H 1 : Obviously, E is a closed subspace of H 1 .Define the inner product •, • on E by setting which induces a new norm • on E as follows: It is well known that • 1 and • are equivalent norms on E.Moreover, by the Sobolev embedding theorem there exists M i > 0 such that where Restricted to E, ϕ can be rewritten as Obviously, ϕ is invariant by translations of Z 2 .
Lemma 2.1 [36] Critical points of ϕ restricted to E are critical points of ϕ on the whole space H 1 , which correspond to periodic solutions of system (1.1).
According to the lemma, the critical points of ϕ correspond to T-periodic solutions of system (1.1), but not certainly with minimal period T. Observing that the lower the energies of the solutions, the larger the minimal periods [32], we work on a manifold of E. The critical point of ϕ with least energy on such a manifold gives rise to a solution of system (1.1) with minimal period T.
Define the Nehari manifold Hence M contains all nontrivial critical points of ϕ.
From Lemma 2.2 we have the following lemma.
For any x ∈ E \ {0}, we define where R + := [0, +∞).Obviously, by Lemma 2.3, s x x is the unique critical point on ϕ| Ê(x) .Then we have s x x ∈ Ê(x) and ϕ (s x x), s x x = 0. Hence The above discussion yields the following lemma.

Lemma 2.4
Suppose that all assumptions of Theorem 1.1 hold.Then for any x ∈ E \ {0}, the set M ∩ Ê(x) consists of precisely one point m(x), which is the unique global maximum of ϕ| Ê(x) .
Proof First, (V 2 ) and (V 6) imply that for each ε > 0, there is where p > 2 is the parameter in (V 6).Then for all x ∈ M, we have Lemma 2.6 Assume that (V 2 ) and (V 4) hold.Then M is bounded away from 0 and closed.Moreover, there exists ρ > 0 such that c = inf M ϕ ≥ inf S ρ ϕ > 0, where S ρ = {x ∈ E : Proof By (V 2 ), for any = 1 For any x ∈ E, without loss of generality, we can assume that x = 1.Then there exists a constant ρ such that ρ = δ M ∞ > 0 and |ρx| ≤ δ .Then Proof Assume on the contrary that ϕ is not coercive on M, i.e., there exists ( for all t ∈ U, and meas(U) ≥ δ for some δ > 0, where meas(•) denotes the Lebesgue measure.By (V 5) and Fatou's lemma we have which is a contradiction.
(b) Let v = 0. We can write x n = τ v n , where τ := x n .It follows from the dominated convergence theorem that T 0 V (sv n ) dt → 0 for every s ∈ R + .Hence, for any s > 0, we have which is a contradiction when we choose s > √ 2d.Hence ϕ is coercive on M. Lemma 2.8 If (V 4) and (V 5) hold, and U ⊂ E \ {0} is a compact set, then there exists R > 0 such that ϕ ≤ 0 on E(x) \ B R (0) for every x ∈ U.
Proof Without loss of generality, we may assume that x = 1 for every x ∈ U. Suppose, on the contrary, that there exist (x n ) ⊂ U and (s n ) ⊂ R + such that y n := s n x n ∈ Ê(x n ), ϕ(y n ) > 0 for all n, and s n → +∞ as n → ∞.Passing to a subsequence, we may assume that Arguing similarly as for (2.6), we have which contracts to (2.8).
Lemma 2.9 Suppose that all assumptions of Theorem 1.1 hold, Then the map E \ {0} → M, x → m(x), is continuous.
Proof For a sequence (x n ) ⊂ E \ {0} such that x n → x, we show that m(x n ) → m(x) for a subsequence.
Without loss of generality, we may assume that x 2 = R 2 for every n.
Hence by the coercivity of ϕ, m(x n ) is bounded.Passing to a subsequence, we may assume that t n := m(x n ) → t, and by Lemma 2.5 we have t ≥ α 0 > 0. Since M is closed and m(x n ) → tx, we have tx ∈ M. Hence tx = m(x) and m(x n ) → m(x).
Next, we consider the unit sphere S := {x ∈ E : x = 1}.Note that the restriction of the map m to S is a homeomorphism with inverse given by We will also consider the functionals ˆ : E \ {0} → R and : S → R defined by Arguing similarly as in Proposition 9 and Corollary 10 in [30], we have the following conclusions.Since the proofs are basically the same, we omit them.

Lemma 2.10
Suppose that all assumptions of Theorem 1.
A sequence (x n ) is called a Palais-Smale sequence (PS-sequence for short) for is bounded and there exist ˆ (x n ) → 0 as n → ∞.We say that ˆ satisfies the PS-condition if every PS-sequence for ˆ contains a convergent subsequence.
Lemma 2.11 (see [30]) Suppose that all assumptions of Theorem 1.1 hold.Then (a) ∈ C 1 (S, R), and Moreover, the corresponding values of and ϕ coincide, and inf S = inf M ϕ.(d) if ϕ is even, then so is .Lemma 2.12 Assume that all assumptions of Theorem 1.1 hold.Then ϕ satisfies PScondition on M, and so does .
Next, assume that (y n ) is a PS-sequence for .According to Lemma 2. 11, (m(y n )) ⊂ M is a PS-sequence for ϕ.Since ϕ satisfies PS-condition, passing to a subsequence, m(y n ) → z.

Thus y n → m(z). Hence satisfies PS-condition.
Proof of Theorem 1.1 According to Lemmas 2.6 and 2.11, inf x∈S (x) = inf x∈M ϕ(x) = c.Let (y n ) be a minimizing sequence for restricted to S. By Ekeland's variational principle [11] we may assume that (y n ) → 0 as n → ∞.It is clear that ( (y n )) is bounded.Then (y n ) is a PS-sequence for .Since satisfies PS-condition, (y n ) contains a subsequence converging to some limit y.Thus y is a critical point of .According to Lemma 2.11 again, It remains to show that ϕ(x) = c.Obviously, ϕ(x) ≥ c.By (V 9) and Fatou's lemma where x n := m(y n ).Hence ϕ(x) ≤ c.So ϕ(x) = c, and x is a nonconstant T-periodic solution for system (1.1).Finally, we will show that x has T as its minimal period.Suppose that x has a minimal period T/k, where k ≥ 2 is an integer.Denote w(t) = x(t/k).Obviously, w ∈ E, and there exists r > 0 such that rw ∈ M. Hence which is a contradiction.Hence x has T as its minimal period.
Let X be a Banach space such that the unit sphere S in X is a submanifold of class (at least) C 1 , and let ψ ∈ C 1 (S, R).We have the following result.Lemma 2.13 [30] If X is infinite-dimensional and ψ ∈ C 1 (S, R) is bounded below and satisfies PS-condition, then ψ has infinitely many pairs of critical points.
Proof of Theorem 1.2 Since V is even, so do ϕ and .By Lemmas 2.6, 2.11, and 2.12, inf x∈S (x) = c > 0, and satisfies PS-condition.Then Lemma 2.13 yields that has infinitely many pairs of critical points.Applying Proposition 2.11 again, ϕ has infinitely many pairs of critical points.Hence system (1.1) has infinitely many pairs of T-periodic solutions.

The noneven case
Recall that The operator A is a self-adjoint operator with a sequence of eigenvalues (counted with multiplicity) Denote by |A| the absolute value of A, and let |A| 1/2 be the square root of |A| with domain D(|A 1/2 ).Decompose the space H 1 as follows: where H 0 and H + are the null space and the positive eigenvalue space of A. Obviously, H 0 = R N .For any x ∈ H 1 , x = x 0 + x + , where x 0 ∈ H 0 and x + ∈ H + .Define a new inner product and the associated norm by Then • 1 and • 2 are equivalent norms on H 1 .We can rewrite ϕ on H 1 by Moreover, for all y = y 0 + y Obviously, H(x) is an (N + 1)-dimensional space.According to the Sobolev embedding theorem, there exist C i > 0 such that where • L i and • ∞ are the usual norms in L i (S T , R N ) and C(S T , R N ), respectively.Define the generalized Nehari manifold Obviously, N contains all nonconstant solutions of ϕ.Moreover, our assumptions on V imply that solutions of (1.1) are critical points of the functional (3.1).
Lemma 3.1 Suppose that (V 1 ), (V 5), (V 8), (V 9), and (V 10) hold.Let x ∈ H 1 \ H 0 , y ∈ H 0 , and s ≥ 0 with x = sx + y.Then Proof Let x, y, and s be as in the statement.Obviously, for such x, y, and s, x = sx + y implies s = 1 or y = 0. Then we need to show that We next claim that g(s, y) < 0. Obviously, g(1, 0) = 0, but in this case, x = sx + y, which contradicts to the assumption.Using (V 1 ) and (V 9), we have Using (V 9) again, we have ( If M is large enough, then the quadratic form (in s and y) above is positive definite, and V (sx + y) -M|sx + y| 2 is bounded below.Then we have g(s, y) → -∞ as s + |y| → ∞.Therefore g(s, y) attains its maximum on the set {(s, y)|s ≥ 0, y ∈ R N }.Suppose that g attains its maximum at some point (s, y) with s > 0. Then , and thus (V (x), sx + y) 2 ≤ (V (x), x) 2 .Since (V (x), y) = 0, we have s 2 ≤ 1.However, if V (x) = V (sx + y) and s = 1, then by (V10) we get y = 0, which contradicts the assumption x = sx + y.Thus s < 1. Hence Hence the claim holds, and the conclusion follows.
Proof Since x ∈ N , by the definition of N , ϕ (x), s 2 -1 2 x + sy 2 = 0. Hence by Lemma 3.1 we have ϕ(x) > ϕ(sx + y).Therefore the maximum point of ϕ is unique, and the lemma holds.
Obviously, Lemma 3.2 implies that if x ∈ N , then x is a unique maximum of ϕ| Ĥ(x) .Proof (i) Arguing similarly as in the proof of Lemma 2.6, we can show this conclusion.
(ii) Denote c = inf N ϕ.For all x ∈ N , by (i) we have Hence x + 2 ≥ √ 2c.Clearly, ϕ| H 0 ≤ 0. Then N is bounded away from H 0 .We can take a sequence (y n ) ⊂ N and prove that its limit y ∈ N .Then N is closed.
Proof If not, then there exists (x n ) ⊂ N such that ϕ(x n ) ≤ d for some d > 0 as x n 2 → ∞.Let s n := x n 2 , v n := x n / x n 2 .Passing to a subsequence, (v n ) converges weakly to some point v.
(a) Let v = 0. Arguing similarly as for (2.6), we have 0, then there exist some α > 0 and N such that v + n 2 ≥ α for all n ≥ N .It follows from the dominated convergence theorem that T 0 V (sv n ) dt → 0 for every s ∈ (0, ∞).Hence, for any s > 0 and n ≥ N , we have which is a contradiction when we choose s > √ 2d/α.Hence ϕ is coercive on N .
Proof Without loss of generality, we may assume that y 2 = 1 for every y ∈ U. Suppose, on the contrary, that there exist (y n ) ⊂ U and z n ∈ Ĥ(y n ) such that ϕ(z n ) > 0 for all n and z n 2 → ∞ as n → ∞.Passing to a subsequence, we may assume that which is a contradiction.This finishes the proof of the lemma.

Lemma 3.6
Assume that all assumptions of Theorem 1.4 hold.Then for each x ∈ H 1 \ H 0 , the set N ∩ Ĥ(x) consists of precisely one point n(x), which is the unique global maximum of ϕ| Ĥ(x) .
Arguing similarly as in [29], we can prove the following two lemmas.

Lemma 3.7
Assume that all assumptions of Theorem 1.4 hold.Then the map H + \ {0} → N , x → n(x), is continuous.
Define the following maps: where S + := {x ∈ H + : x 2 = 1} in H + .Then ˆ is continuous by Lemma 3.7.Moreover, n is a homeomorphism between S + and N .Lemma 3.9 Assume that all assumptions of Theorem 1.4 hold.Then ϕ satisfies PScondition on N , and so does .
Proof Let (x n ) ⊂ N be a PS-sequence of ϕ.Then (ϕ(x n )) is bounded.By the coercivity of ϕ, (x n ) is bounded.Set x n = x 0 n + x + n for all n, where x 0 n ∈ H 0 and x + n ∈ H + .So both (x 0 n ) and (x + n ) are bounded.Since φ : H 1 → (H 1 ) * is compact and ϕ (x n ) = x + nφ (x n ) → 0, (x + n ) has a convergent subsequence.As dim H 0 = N , (x 0 n ) has a convergent subsequence.Hence (x n ) has a convergent subsequence.Thus ϕ satisfies PS-condition.Following the same way as in the proof of Lemma 2.12, we can attain that also satisfies PS-condition.Let (w n ) be a minimizing sequence for on S + .Then (w n ) → c as n → ∞.By Ekeland's variational principle we have (w n ) → 0 as n → ∞.Put x n := n(w n ) ∈ N .Then ϕ(x n ) → c and ϕ (x n ) → 0 as n → ∞.Hence (x n ) is a PS-sequence of ϕ.Since ϕ satisfies PS-condition, (x n ) contains a converging subsequence; denote its limit by x.Since N is closed, x ∈ N , and x is a critical point of ϕ.Clearly, ϕ(x) ≥ c.Arguing similarly as for (2.10), we can show that ϕ(x) ≤ c.Consequently, ϕ(x) = c, and x is a nonconstant T-period for system (1.1).

Lemma 3 . 8
Assume that all assumptions of Theorem 1.4 hold.Then (a)∈ C 1 (S + , R), and(w), z 2 = n(w) + 2 ϕ n(w) , z 2 , z ∈ T w S + := v ∈ H + : w, v 2 = 0 ; (b) (w n ) n is a PS-sequence for if and only if ( n(x n )) n is a PS-sequence for ϕ; (c) We have inf S + = inf N ϕ = c.Moreover, x ∈ S + is a critical point of if and only if n(x) ∈ N is a critical point of ϕ, and the corresponding critical values coincide.