Subharmonic solutions of first-order Hamiltonian systems

Definition 2.1

([25, Definition 5.4.3]) For γ ∈ P τ ( 2 n ) , there is a Maslov-type index theory that assign to the path a pair of integers

where i 1 ( γ ) is the index part or rotation number of γ and ν 1 ( γ ) ≔ dim ker ( γ ( τ ) − I 2 n ) is the nullity.

Definition 2.2

(cf. [25, page 177]) Given a T > 0 , a positive integer k and a path γ ∈ P T ( 2 n ) , the k th iteration γ k of γ is defined by γ ˜ ∣ [ 0 , k T ] with

Proposition 2.3

(cf. [25, Theorem 10.1.3]) For any γ ∈ P T ( 2 n ) and k ∈ N ,

Proposition 2.4

(cf. [14, Theorem 3.2]) Let γ ∈ P T ( 2 n ) . If the integers k , n k ∈ N , and n 1 ∈ Z satisfy n k ≥ n 1 , n k ≥ n , and the conditions

then k ≤ n k − n 1 + 1 .

Proposition 2.5

(cf. [10, Theorem 11.1]) Let γ ∈ P T ( 2 ) and k ∈ N . Suppose

Then k = 1 .

Lemma 2.6

Assume B ∈ C ( S T , L s ( R 2 n ) ) satisfies (B1), then for any integer j, we have

Proof

By assumption, B ∈ C ( S j T , L s ( R 2 n ) ) for any integer j . Let γ B be the associated symplectic path of B . By [4, Proposition 3.2.11] for semi-positive path,

where for M ∈ Sp ( 2 n ) , Gr ( M ) ≔ { ( x , M x ) ; x ∈ C 2 n } ⊂ C 2 n × C 2 n and the Maslov-type index Mas { ⋅ , ⋅ } was defined in loc. cit.□

Lemma 2.7

(cf. [22, Lemma 4.1]) If x j is a jT-periodic solution of (1.2), then i ( l * x j ) = i ( x j ) and ν ( l * x j ) = ν ( x j ) for l ∈ Z .

Definition 2.8

Given a Banach space X and let f ∈ C 1 ( X , R ) . We call f satisfies the Palais-Smale condition if every sequence { x k } ⊂ X such that

possesses a convergent subsequence in X .

Definition 2.9

(cf. [1, Definition 4.1.5] and [7, Definition 2.1.2]) Let Q be a closed q -dimensional closed ball topologically embedded into a Banach space X and given a closed subset A of X such that ∂ Q ∩ A = ∅ . We call that ∂ Q and A homologically link if ∂ Q is the support of a nonvanishing homology class in homology group H q − 1 ( X \ A ) .

Theorem 2.10

Let X be a real Hilbert space with direct sum decomposition X = Z ⊕ Y , where dim Z < + ∞ . Assume the function f ∈ C 2 ( X , R ) satisfies ( PS ) condition and

1. There are two positive numbers ρ and α > 0 such that f ( x ) ≥ α for any x ∈ ∂ B ρ ( 0 ) ∩ Y .

2. There exist e ∈ ∂ B 1 ( 0 ) ∩ Y and R > ρ > 0 such that

   f ( x ) ≤ 0 , ∀ x ∈ ∂ Q ,

   where Q = ( B R ( 0 ) ∩ Z ) ⊕ { r e ; 0 ≤ r ≤ R } and B r ( 0 ) = { x ∈ X ; ‖ x ‖ ≤ r } .

1. f possesses a critical value c ≥ α , which is given by the minimax

   c = inf ξ ∈ Γ sup x ∈ ∣ ξ ∣ f ( x ) .

2. If f ″ ( x ) is Fredholm for x ∈ K c ( f ) ≔ { x ∈ X ; f ′ ( x ) = 0 , f ( x ) = c } , then there is an element x 0 ∈ K c ( f ) such that the Morse index m − ( f , x 0 ) and the nullity m 0 ( f , x 0 ) of f at x 0 satisfy the following inequality:

   m − ( f , x 0 ) ≤ dim Z + 1 ≤ m − ( f , x 0 ) + m 0 ( f , x 0 ) .

Remark 2.11

In the literature, there are critical theorems that show the multiple solutions of the corresponding functional. Papageorgiou et al. [26] used the Nehari method to give multiple solutions with sign information for the double-phase functional with the ( p − 1 ) -sublinear reaction function; while in the study by Papageorgiou et al. [27], the nonlinear eigenvalue problems with the ( p − 1 ) -superlinear reaction were studied.

Theorem 3.1

Let H ( t , x ) = 1 2 B ˜ ( t ) x ⋅ x + H ˜ ( t , x ) where B ˜ ∈ C ( S T , ℒ s ( R 2 n ) ) satisfies (B1) and H ˜ satisfies (H1)–(H2), and (SH1)–(SH4). Then for any j ∈ N and 1 ≤ j < 2 π T ‖ B ˜ ‖ C 0 , the system (1.2) possesses a nonconstant jT-periodic solution x j satisfying

Definition 3.2

[25, Definition 4.1.2] Let Γ = { P m ∈ ℒ ( E T ) ; m ∈ N ∪ { 0 } } be a sequence of orthogonal projections. Γ is called a Galerkin approximation scheme with respect to A if the following three conditions hold:

1. P m E T is finite dimensional for all m ∈ N ∪ { 0 } .

2. For any x ∈ E T , lim m → + ∞ ‖ P m − x ‖ = 0 .

3. P m A − A P m → 0 in the operator norm of ℒ ( E T ) as m → + ∞ .

Theorem 3.3

[15, Theorem 2.1] Let B ∈ C ( S T , ℒ s ( R 2 n ) ) . For any positive number d satisfying d ≤ 1 4 ‖ ( A − B ) ♯ ‖ − 1 , there exists a positive integer m * such that for all m ≥ m * , we have

Proof of Theorem 3.1

Step 1 We will modify the function H ˜ to obtain a C 2 functional on E T , whose critical point is just a periodic solution of (1.2) for j = 1 . The Sobolev embedding theorem (cf. [18, Appendix A.3]) tells us that for each 1 ≤ s < + ∞ , E T is compactly embedded in L s ( S T , R 2 n ) , and there exists a positive number C s such that

Following [28], for the fixed r ¯ in (SH3), we choose K > r ¯ , where K will be determined later; and select χ ∈ C ∞ ( R , R ) such that χ ( y ) = 1 for y ≤ K , χ ( y ) = 0 for y ≥ K + 1 , and χ ′ ( y ) < 0 for y ∈ ( K , K + 1 ) . Set

with the positive constant R K satisfying R K ≥ max K ≤ ∣ x ∣ ≤ K + 1 0 ≤ t ≤ T H ˜ ( t , x ) ∣ x ∣ 4 .

Let g ( x ) = ∫ 0 T H K ( t , x ( t ) ) d t for any x ∈ E T . Since there are constants a 1 , a 2 > 0 such that

for all x ∈ R 2 n , g ∈ C 2 ( E T , R ) , whose derivative is a compact operator (cf. [29]).

Then H ˜ K also satisfies (H1)–(H2) and (SH1)–(SH4), where μ is replaced by μ ¯ = min { μ , 4 } , α is replaced by α ¯ = max { α , 1 } , and R 1 is replaced by R ¯ 1 = max { R 1 , 1 } . By integrating the inequality in (SH3), we obtain

for all t ∈ R , x ∈ R 2 n , where both a 3 > 0 and a 4 are constants independent of K . In virtue of (SH2), for any ε > 0 , there exists a δ > 0 such that H ˜ K ( t , x ) ≤ ε ∣ x ∣ 2 for ∣ x ∣ ≤ δ and t ∈ R . Since H ˜ K ( t , x ) ∣ x ∣ − 4 is uniformly bounded as ∣ x ∣ → + ∞ , there exists a C = C ( ε , K ) such that H ˜ K ( t , x ) ≤ C ∣ x ∣ 4 for ∣ x ∣ ≥ δ and t ∈ R . Hence,

which will be used later. For x ∈ E T , we set

Then f K ∈ C 2 ( E T , R ) . We will show that there is a nonconstant critical point x K of f K on E T . By the similar method in the study by Rabinowitz [29, Theorem 6.10], x K ( t ) is also a nonconstant T -periodic classical solution of the modified Hamiltonian system

Similar estimates in the study by Rabinowitz [28] then prove that there is a K 0 > 0 such that for any K ≥ max { K 0 , r ¯ } , ‖ x K ‖ L ∞ ≤ K 0 . Thus for such sufficient large K , H K ( t , x K ) = H ( t , x K ) for t ∈ R , and therefore, x K satisfies (1.2) for j = 1 .