Multiple periodic solutions of superlinear ordinary differential equations with a parameter

Nonlinear Analysis 74 (2011) 6442–6450

Contents lists available at ScienceDirect

Nonlinear Analysis journal homepage: www.elsevier.com/locate/na

Multiple periodic solutions of superlinear ordinary differential equations with a parameter✩ Jiabao Su ∗ , Ruiyi Zeng School of Mathematical Sciences, Capital Normal University, Beijing 100048, People’s Republic of China

article

abstract

info

We study the existence of multiple 2π -periodic solutions of ordinary differential equation −¨x = λx + f (t , x) with superlinear terms via homological linking and Morse theory.

Article history: Received 10 February 2011 Accepted 20 June 2011 Communicated by Ravi Agarwal

© 2011 Elsevier Ltd. All rights reserved.

MSC: 34C25 47J30 58E05 Keywords: Ordinary differential equation Homological linking Morse theory

1. Introduction In this paper we are concerned with the existence of 2π -periodic solutions of the ordinary differential equation



−¨x = λx + f (t , x), x(0) = x(2π ), x˙ (0) = x˙ (2π ),

( Pλ )

where λ ∈ R is a parameter and the nonlinear term f satisfies the following conditions: (f1 ) f ∈ C 1 (R × R, R) is 2π -periodic in t. (f2 ) f (t , 0) = 0, fx′ (t , 0) = 0. (f3 ) There exist r¯ > 0 and µ > 2 such that 0 ⩽ µF (t , x) := µ

x

∫

f (t , s) ds ⩽ f (t , x)x,

|x| ⩾ r¯ ,

t ∈ [0, 2π].

0

As f (t , 0) ≡ 0, (P )λ has a trivial solution x ≡ 0 for any parameter λ ∈ R. Our interest is in the multiplicity of nontrivial 2π -periodic solutions of (P )λ for a certain range of the parameter. We will prove that (P )λ has at least three nontrivial 2π -periodic solutions when the parameter λ is close to any a fixed eigenvalue of the linear periodic boundary value problem



−¨x = µx, x(0) = x(2π ),

x˙ (0) = x˙ (2π )

✩ Supported by NSFC10831005, KZ201010028027, SRFDP20070028004 and PHR201106118.

∗

Corresponding author. Tel.: +86 10 68902352 414; fax: +86 10 68903637. E-mail address: [email protected] (J. Su).

0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.06.027

(L0 )

J. Su, R. Zeng / Nonlinear Analysis 74 (2011) 6442–6450

6443

It is known that (L0 ) possesses all the eigenvalues µm = m2 (m = 0, 1, 2, . . .) with multiplicity 2 as m ⩾ 1. For the convenience of later use we denote by 0 = λ0 < λ1 < λ2 < · · · the distinct eigenvalues of (L0 ). We build the variational framework for (P )λ . The Sobolev space E := H 1 ([0, 2π ], R) =





x˙ ∈ L2 ([0, 2π ], R), x ∈ L2 ([0, 2π ], R)  x(0) = x(2π )



is a Hilbert space with inner product and norm given by

⟨x, y⟩ =

2π

∫

(˙xy˙ + xy) dt ,

‖x‖2 = ⟨x, x⟩,

x, y ∈ E .

0

On the basis of the compact embeddings E ↩→↩→ C ([0, 2π ], R),

E ↩→↩→ Lp ([0, 2π ], R)

for p ⩾ 1,

and the condition (f1 ), the 2π -periodic solutions of (P )λ are exactly the critical points of the functional

Φ (x) =

1 2

2π

∫

 2  |˙x| − λ|x|2 dt −

F (t , x)dt ,

x ∈ E.

(1.1)

0

0

Φ ∈ C 2 (E , R) with derivatives ∫ 2π ∫ ⟨Φ ′ (x), y⟩ = (˙xy˙ − λxy)dt − 0

⟨Φ ′′ (x)y, z ⟩ =

2π

∫

2π

f (t , x)ydt ,

x, y ∈ E ,

0

∫

2π

(˙yz˙ − λyz )dt −

0

2π

∫

fx′ (t , x)yzdt ,

x, y, z ∈ E .

0

We will emphasize by assuming λ ⩾ λ1 the situation that the trivial solution x = 0 acts as a local saddle point of the energy functional Φ of (P )λ . Define F − (t , x) = max{−F (t , x), 0},

M :=

sup

(t ,x)∈[0,2π ]×R

F − (t , x).

Our results are the following theorems. Theorem 1.1. Assume that f satisfies (f1 ) –(f3 ). Let k ∈ N be fixed. Then there is a δ > 0 such that when M ⩽ δ , for λ ∈ (λk+1 − δ, λk+1 ), (P )λ has at least three nontrivial 2π -periodic solutions. Theorem 1.2. Assume that f satisfies (f1 ) –(f3 ) and F ⩽ 0 for |x| ⩾ 0 small, t ∈ [0, 2π ]. Let k ∈ N be fixed. Then there is a δ > 0 such that when M ⩽ δ , for λ = λk+1 , (P )λ has at least three nontrivial 2π -periodic solutions. Theorem 1.3. Assume that f satisfies (f1 ) –(f3 ) and F (t , x) ⩾ 0. Let k ∈ N be fixed. Then there is a δ > 0 such that for λ ∈ (λk+1 − δ, λk+1 ), (P )λ has at least three nontrivial 2π -periodic solutions. Theorem 1.4. Assume that f satisfies (f1 ) –(f3 ). Let k ∈ N be fixed. Then there is a δ > 0 such that when 0 < M ⩽ δ , for λ ∈ (λk+1 , λk+1 + δ), (P )λ has at least one nontrivial 2π -periodic solutions. We give some comments on and explanations regarding the conditions and conclusions. One will see below that in the above theorems, the differences in the sign of F lead to different applications of an abstract critical point theorem in [1,2]. In Theorems 1.1 and 1.2, when F is negative somewhere, as λ is close to λk+1 from the left hand side or is λk+1 , we can only construct one homological linking starting from λk+2 . In Theorem 1.3, when F ⩾ 0 and λ is close to λk+1 from the left hand side, we can construct two homological linkings starting from λk+1 and λk+2 respectively. In Theorem 1.4, we reveal the fact that when λ is close to λk+1 from the right hand side, a linking starting from λk+2 can still be constructed even if F is negative somewhere. This is related to the situation considered in Rabinowitz [1,2] for an elliptic problem with a global sign condition fx′ (t , x) ⩾ 0 for all x ∈ R which implies F ⩾ 0. In Theorem 1.4, if F ⩽ 0 with 0 < M ⩽ δ is replaced by a local sign condition that f (t , x)x < 0 for |x| > 0 small, then (P)λ has at least three nontrivial 2π -periodic solutions. See [3]. The second-order Hamiltonian system is a typical modal in the applications of Morse theory and minimax methods. We refer the reader to the Refs. [1,2,4–10] for some historical progress. In this paper we consider the systems with one dimension. We apply Morse theory and homological linking to find multiple solutions. The basic idea used here is from [11] where a superlinear elliptic problem with a saddle structure near zero was studied by combining bifurcation methods, Morse theory and homological linking. Ref. [3] extends the ideas of [11] to the superlinear second-order Hamiltonian systems with indefinite linear part. The current paper is concerned with the one-dimensional case of [3]. On the basis of the ODE features, we apply Morse theory and homological linking to treat (P)λ under weaker local conditions on the signs of F . It should be pointed out that the bifurcation method cannot be applied here. Our results are new and extend the corresponding results for the one-dimensional case of [3]. The paper is organized as follows. In Section 2, we collect some basic abstract tools. In Section 3 we get solutions by linking arguments and give partial estimates of homological information. The proofs of Theorems 1.1–1.4 are given in Section 4.

6444

J. Su, R. Zeng / Nonlinear Analysis 74 (2011) 6442–6450

2. Preliminaries In this section we give some preliminaries that will be used to prove the main results of the paper. We first collect some results on Morse theory and homological linking. For our uses we just emphasize some basic results for a C 2 functional defined on a Hilbert space. Let E be a Hilbert space and suppose that Φ ∈ C 2 (E , R). Define K = {u ∈ E | Φ ′ (u) = 0}, Φ c = {u ∈ E | Φ (u) ⩽ c }, Kc = {u ∈ K | Φ (u) = c } for c ∈ R. We say that Φ satisfies the (PS)c condition at the level c ∈ R if any sequence {un } ⊂ E satisfying Φ (un ) → c , Φ ′ (un ) → 0 as n → ∞, has a convergent subsequence. Φ satisfies (PS) if Φ satisfies (PS)c at any c ∈ R. From now on we assume that Φ satisfies (PS) and #K < ∞. Suppose that u0 ∈ K with Φ (u0 ) = c ∈ R and let U be a neighborhood of u0 such that U ∩ K = {u0 }. The group Cq (Φ , u0 ) := Hq (Φ c ∩ U , Φ c ∩ U \ {u0 }),

q∈Z

is called the qth critical group of Φ at u0 , where H∗ (A, B) denotes the singular relative homology group of the pair (A, B) with coefficient field F (see [6,7]). Suppose that a < inf Φ (K ). The group Cq (Φ , ∞) := Hq (E , Φ a ),

q∈Z

is called the qth critical ∑ group of Φ at infinity (see [8]). a We call Mq := u∈K dim Cq (Φ , u) the qth Morse-type numbers of the pair (E , Φ ) and βq := dim Cq (Φ , ∞) the Betti numbers of the pair (E , Φ a ). The core of the Morse theory [6,7] are the following relations between Mq and βq : q q − − (−1)q−j Mj ⩾ (−1)q−j βj , j =0

for q ∈ Z,

j=0

∞ ∞ − − (−1)q βq . (−1)q Mq = q =0

q =0

If K = ∅ then βq = 0 for all q. Since Mq ⩾ βq for each q ∈ Z, it follows that if βq∗ ̸= 0 for some q∗ ∈ Z, then Φ must have a critical point u∗ with Cq∗ (Φ , u∗ ) ̸∼ = 0. If K = {u∗ } then Cq (Φ , ∞) ∼ = Cq (Φ , u∗ ) for all q. Thus if Cq (Φ , ∞) ̸∼ = Cq (Φ , u∗ ) for some q then Φ must have a new critical point. One can use critical groups to distinguish critical points obtained by other methods and use the Morse equality to find new critical points. For the critical groups of Φ at an isolated critical point, we have the following basic facts. Proposition 2.1. Assume that u is an isolated critical point of Φ ∈ C 2 (E , R) with finite Morse index µ(u) and nullity ν(u). Then (1) (2) (3) (4)

Cq (Φ , u) ∼ = δq,µ(u) F if ν(u) = 0; Cq (Φ , u) ∼ = 0 for q ̸∈ [µ(u), µ(u) + ν(u)] (Gromoll and Meyer [12]); if Cµ(u) (Φ , u) ̸∼ = 0 then Cq (Φ , u) ∼ = δq,µ(u) F; if Cµ(u)+ν(u) (Φ , u) ∼ ̸ 0 then Cq (Φ , u) ∼ = = δq,µ(u)+ν(u) F.

Proposition 2.2 ([13,14]). Let 0 be an isolated critical point of Φ ∈ C 2 (E , R) with finite Morse index µ0 and nullity ν0 . Assume that Φ has a local linking at 0 with respect to a direct sum decomposition E = E − ⊕ E + , κ = dim E − , i.e. there exists r > 0 small such that

Φ (u) > 0 for u ∈ E + ,

0 < ‖ u‖ ⩽ r ,

Φ (u) ⩽ 0 for u ∈ E − ,

‖ u‖ ⩽ r .

Then Cq (Φ , 0) ∼ = δq,κ Z for either κ = µ0 or κ = µ0 + ν0 . The concept of local linking was introduced in [15]. In [13] a partial result was given for a C 1 functional. The above result was obtained in [14]. Now we recall an abstract linking theorem which is from [2,7,13]. Proposition 2.3 ([2,7,13]). Let E be a real Banach space with E = X ⊕ Y and suppose that ℓ = dim X is finite. Suppose that Φ ∈ C 1 (E , R) satisfies (PS) and:

(81 ) there exist ρ > 0, α > 0 such that Φ (u) ⩾ α,

u ∈ Sρ = Y ∩ ∂ Bρ ,

(2.1)

where Bρ = {u ∈ E | ‖u‖ ⩽ ρ},

(82 ) there exist R > ρ > 0, and e ∈ Y with ‖e‖ = 1 such that Φ (u) < α,

u ∈ ∂Q ,

where Q = {u = v + se | ‖u‖ ⩽ R, v ∈ X , 0 ⩽ s ⩽ R}.

(2.2)

J. Su, R. Zeng / Nonlinear Analysis 74 (2011) 6442–6450

6445

Then Φ has a critical point u∗ with Φ (u∗ ) = c∗ ⩾ α and Cℓ+1 (Φ , u∗ ) ̸= 0.

(2.3)

We note here that under the framework of Proposition 2.3, Sρ and ∂ Q are homotopically linked with respect to the direct sum decomposition E = X ⊕ Y . Sρ and ∂ Q is also homologically linking. The conclusion (2.3) follows from Theorems 1.1′ and 1.5 of Chapter II in [7]. (See also [13].) 3. Linking solutions with homological information In this section we will apply Proposition 2.3 to find linking solutions of (P )λ and then to get partial homological information. We first verify (PS). Lemma 3.1. Assume that f satisfies (f1 ) and (f3 ); then for any fixed λ ∈ R, Φ satisfies the (PS) condition. Proof. Let {xn } ⊂ E be such that

|Φ (xn )| ⩽ C ,

Φ ′ (xn ) → 0,

n ∈ N,

n → ∞.

(3.1)

Take β ∈ ( µ , ); by (3.1), for n large, we have 1

1 2

C + β‖xn ‖ ⩾ Φ (xn ) − β⟨Φ ′ (xn ), xn ⟩.

(3.2)

By (f1 ) and (f3 ), for some C > 0, it holds that F (t , x) ⩾ C |x|µ − C ,

x ∈ R, t ∈ [0, 2π ].

(3.3)

For any given ϵ > 0, we have that

‖xn ‖22 ⩽ (2π )

µ−2 µ

‖xn ‖2µ ⩽ (2π )

2 µ − 2 2−µ 2 ϵ + ϵ ‖xn ‖µµ . µ µ

(3.4)

Take ϵ > 0 small enough; it follows from (3.2)–(3.4) and (f3 ) that C + β‖xn ‖ ⩾



2

 ⩾

1 2

 ⩾

1

1 2

−β

2π

∫

|˙xn |2 dt −

0

−β

2π

∫ 0



1 2

 ∫ − β λ‖xn ‖22 − (1 − βµ)

F (t , xn )dt − C |xn |⩾r¯

|˙xn |2 dt − C ε‖x‖µµ + (βµ − 1)C ‖x‖µµ − C



1

− β ‖xn ‖22 + (βµ − 1)C ‖xn ‖µµ − C 2

⩾ C ‖x n ‖ − C . 2

Therefore {xn } is bounded on E. Up to a subsequence if necessary, we can assume that there is an x ∈ E such that as n → ∞, xn ⇀ x in E and xn → x in C ([0, 2π ], R), and furthermore, xn → x in L2 ([0, 2π ], R). Thus

∫   

2π

0

 

f (t , xn )(xn − x)dt  ⩽ ‖f (t , xn )‖2 ‖xn − x‖2 → 0.

Since

⟨Φ ′ (xn ) − Φ ′ (x), xn − x⟩ =

2π

∫

|˙xn − x˙ |2 − λ|xn − x|2 dt −

0

∫

2π

(f (t , xn ) − f (t , x))(xn − x)dt → 0

0

it follows that 2π

∫

|x˙n − x˙ |2 dt → 0,

n→∞

0

and so xn → x in E.



From the proof of Lemma 3.1 and the embedding E ↩→ C ([0, 2π ], R), we see that the set of critical points of Φ is uniformly bounded in E. Therefore we may assume that F satisfies the growth condition

|F (t , x)| ⩽ C (1 + |x|p ), for any a fixed number p > µ.

x ∈ R, t ∈ [0, 2π ],

(F)

6446

J. Su, R. Zeng / Nonlinear Analysis 74 (2011) 6442–6450

Now we apply Proposition 2.3 to construct the linking. We need some notation. For j ∈ N, define d2



E (λj ) = ker −

dt 2

 − λj ,

j 

Ej =

E (λi ),

i=1

For each j ⩾ 1, νj := dim E (λj ) = 2, ℓj := dim Ej = 2π

∫

|˙x|2 + |x|2 dt ⩾ (λj+1 + 1)

∫

2π

∑j

i=0

νi = 2j + 1. We have the following variational inequalities:

|x|2 dt for x ∈ Ej⊥ .

(3.5)

0

0 2π

∫

E = Ej ⊕ Ej⊥ .

|˙x|2 + |x|2 dt ⩽ (λj + 1)

2π

∫

|x|2 dt for x ∈ Ej .

(3.6)

0

0

We construct a linking with respect to the decomposition E = Ej ⊕ Ej⊥ . We show some lemmas in verifying the conditions

(81 ) and (82 ).

Lemma 3.2. Let f satisfy (f1 ) –(f3 ). Then for any fixed λ < λj+1 , there are ρj > 0, αj > 0, such that

Φ (x) ⩾ αj for x ∈ Ej⊥ with ‖x‖ = ρj . Proof. By (f1 ), (f2 ) and (F ), for any ϵ > 0, there is a Cϵ > 0 such that 1

F (t , x) ⩽

2

ϵ|x|2 + Cϵ |x|p ,

(3.7)

For x ∈ Ej⊥ , by (3.5), we have

Φ (x) =

1

2π

∫

2

|˙x|2 − λ|x|2 dt −

0

2π

∫

F (t , x)dt

0

∫ 2π |˙x|2 + |x|2 − (λ + 1)|x|2 dt − F (t , x)dt 2 0 0   ∫ 2π 1 λ+1 ϵ 2 ⩾ ‖x‖2 − ‖x ‖2 − |x| + Cϵ |x|p dt 2 λj+1 + 1 2 0   1 λ+1 1 ⩾ 1− ‖x‖2 − ϵ‖x‖2 − Cˆ ‖x‖p 2 λj+1 + 1 2   1 λ j +1 − λ = − ϵ ‖x‖2 − Cˆ ‖x‖p . 2 λ j +1 + 1 =

1

2π

∫

Take

ϵ=

λj+1 − λ 2(λj+1 + 1)

,

η=

λ j +1 − λ > 0. λj+1 + 1

(3.8)

Then

Φ (x) ⩾

1 4

η‖x‖2 − Cˆ ‖x‖p .

(3.9)

Since p > 2, the function g (r ) = gmax =

p−2 2p

p

1 4

ηr 2 − Cˆ r p achieves its maximum 2

(2−1 η) p−2 (pCˆ ) 2−p := αj .

(3.10)

on (0, ∞) at

 rj =

η 2pCˆ

 p−1 2

.

(3.11)

Therefore

Φ (x) ⩾ αj ,

for x ∈ Ej⊥

with ‖x‖ = rj .

We note here that both αj and rj go decreasingly to zero as λ → λ− j +1 .

(3.12)

J. Su, R. Zeng / Nonlinear Analysis 74 (2011) 6442–6450

6447

Take an eigenfunction φj+1 with ‖φj+1 ‖ = 1 corresponding to the eigenvalue λj+1 of (L0 ). Now for fixed k ∈ N, we will apply Lemma 3.2 with j = k, k + 1. Accordingly, we set Sk = {x ∈ Ek⊥ | ‖x‖ = rk },

Sk+1 = {x ∈ Ek⊥+1 | ‖x‖ = rk+1 },

Vk := Ek ⊕ span{φk+1 },

(3.13)

Vk+1 := Ek+1 ⊕ span{φk+2 },

and keep in mind the corresponding numbers that appeared in Lemma 3.2.



Lemma 3.3. Let f satisfy (f1 ) and (f3 ). There exist δ > 0 and R > 0 independent of λ and δ , σ ∈ R such that when M ⩽ δ , for any fixed λ ∈ (λk+1 − δ, λk+1 + δ),

Φ (x) ⩽ σ < αk+1 ,

for x ∈ ∂ Qk+1 .

where Qk+1 = {x ∈ Vk+1 | ‖x‖ ⩽ R, x = y + sφk+2 , y ∈ Ek+1 , s ⩾ 0}. Moreover, if F (t , x) ⩾ 0, i.e. M = 0, then for any λ ∈ (λk , λk+1 ), there is an R > 0 independent of λ such that

Φ (x) ⩽ 0,

for x ∈ ∂ Qk ,

where Qk = {x ∈ Vk | ‖x‖ ⩽ R, x = y + sφk+1 , y ∈ Ek , s ⩾ 0}. Proof. For x ∈ Vk+1 , write x = y + z + w where y ∈ Ek , z ∈ E (λk+1 ), w ∈ span{φk+2 }. By (f3 ) and (3.3) we have

Φ (x) = ⩽ ⩽

1 2 1 2 1 2

2π

∫

|˙x|2 − λ|x|2 dt −

2π

∫

0

F (t , x)dt

0

1

1

2

2

(λk − λ)‖y‖22 + (λk+1 − λ)‖z ‖22 + (λk+2 − λ)‖w‖22 − C ‖x‖µµ + C (λk+2 − λk )‖x‖22 − C ‖x‖µµ + C .

Since µ > 2 and dim Vk+1 < ∞, there is an R > 0 independent of λ in a bounded interval such that

Φ (x) ⩽ 0,

x ∈ Vk+1 ,

‖x ‖ = R .

(3.14)

Fixing R > max{rk , rk+1 } > 0. For y ∈ Ek+1 with ‖y‖ ⩽ R, write y = w + z where w ∈ Ek , z ∈ E (λk+1 ). Then for λ ∈ (λk , λk+2 ), by (3.6), we have

Φ (y) ⩽ ⩽

1 2 1 2

1

(λk − λ)‖w‖ + (λk+1 − λ)‖z ‖ − 2 2

2 2

2

∫ {t : F (t ,y)⩽0}

F (t , y)dt

|λk+1 − λ|R2 + 2π M .

(3.15)

Now take

δ=

2σ 4π + R2

,

σ =

αk+1 2

.

Then

Φ (y) ⩽ σ < αk+1 for y ∈ Ek+1 ,

‖y‖ ⩽ R,

λ ∈ (λk+1 − δ, λk+1 + δ).

Notice that

∂ Qk+1 = {x = y + sφk+2 | ‖x‖ = R, y ∈ Ek+1 , s ⩾ 0} ∪ {y ∈ Ek+1 | ‖y‖ ⩽ R}, it follows from (3.14) and (3.16) that

Φ (x) ⩽ σ < αk+1 ,

∀ x ∈ ∂ Qk+1 ,

λ ∈ (λk+1 − δ, λk+1 + δ).

When λ ∈ (λk , λk+1 ), as Vk ⊂ Vk+1 , for R > 0 given above, we still have

Φ (x) ⩽ 0,

x ∈ Vk ,

‖x‖ = R.

(3.16)

6448

J. Su, R. Zeng / Nonlinear Analysis 74 (2011) 6442–6450

As F ⩾ 0, we have

Φ (y) =

1 2

(λk − λ)‖y‖22 −

2π

∫

F (t , y)dt ⩽ 0,

y ∈ Ek .

0

Thus

Φ (x) ⩽ 0,

x ∈ ∂ Qk .

The proof is complete.



Now we are ready to apply Proposition 2.3. We have the following existence results with partial homological information. Theorem 3.4. Let f satisfy (f1 ) –(f3 ) and suppose that k ⩾ 1. Then there is a δ > 0 such that when M ⩽ δ , for each λ ∈ (λk+1 − δ, λk+1 + δ), (P )λ has one nontrivial 2π -periodic solution x1 with critical group satisfying Cℓk+1 +1 (Φ , x1 ) ̸∼ = 0.

(3.17)

Moreover if F ⩾ 0, then for each λ ∈ (λk+1 − δ, λk+1 ), (P )λ has another nontrivial 2π -periodic solution x2 with critical group satisfying Cℓk +1 (Φ , x2 ) ̸∼ = 0.

(3.18)

Proof. By Lemma 3.1, Φ satisfies (PS). By Lemmas 3.2 and 3.3, for each fixed λ ∈ (λk+1 − δ, λk+1 + δ), Φ satisfies (81 ) and (82 ) in the sense that inf Φ (x) ⩾ αk+1 >

x∈Sk+1

αk+1 2

⩾ max Φ (y). y∈∂ Qk+1

Since R > rk+1 > 0, Sk+1 and ∂ Qk+1 homotopically link with respect to the decomposition E = Ek+1 ⊕ Ek⊥+1 . As dim Vk+1 = ℓk+1 + 1, it follows from Proposition 2.3 that Φ has a critical point x1 with Φ (x1 ) ⩾ αk+1 > 0 and satisfying (3.17). When λ ∈ (λk+1 − δ, λk+1 ) ⊂ (λk , λk+1 ), Φ satisfies (81 ) and (82 ) in the sense that inf Φ (x) ⩾ αk > 0 ⩾ max Φ (y).

x∈Sk

y∈∂ Qk

Since R > rk > 0, Sk and ∂ Qk homotopically link with respect to the decomposition E = Ek ⊕ Ek⊥ . As dim Vk = ℓk + 1, it follows from Proposition 2.3 that Φ has a critical point x2 with Φ (x2 ) ⩾ αk > 0 and satisfying (3.18). Since ℓk+1 + 1 − (ℓk + 1) = 2 and dim kerΦ ′′ (xi ) = dim{y ∈ E | −¨y = (λ + f ′ (t , xi ))y} ⩽ 2,

i = 1, 2,

by Proposition 2.1(2), we have that x1 ̸= x2 . The proof is complete.

(3.19)



We remark here that the global sign condition F ⩾ 0 is necessary in constructing the linking with respect to Ek ⊕ Ek⊥ when λ ∈ (λk+1 − δ, λk+1 ). On the other hand, if λ ∈ [λk , λk+1 ) and is bounded away from λk+1 , the linking with respect to Ek ⊕ Ek⊥ can be constructed provided the negative values of F are small. This phenomenon is exactly what appeared in the part for λ ∈ [λk+1 , λk+1 + δ) of Theorem 3.4. The idea of applying the kernel of the linearized equation in Morse theory was first adapted in [16]. 4. Proof of the main theorems In this section we give the proofs of the main theorems in this paper. We first compute the critical groups of Φ at both infinity and zero. Lemma 4.1 ([3,17]). Let f satisfy (f1 ) –(f3 ); then for any fixed λ ∈ R, Cq (Φ , ∞) ∼ = 0,

for all q ∈ Z.

Lemma 4.2. Let f satisfy (f1 ) and (f2 ). (1) (2) (3) (4)

For For For For

∼ δq,ℓ F. λ ∈ (λk , λk+1 ), Cq (Φ , 0) = k λ ∈ (λk+1 , λk+2 ), Cq (Φ , 0) ∼ = δq,ℓk+1 F. λ = λk+1 , if F (t , x) ⩽ 0 for |x| small, then Cq (Φ , 0) ∼ = δq,ℓk F. λ = λk+1 , if F (t , x) ⩾ 0 for |x| small, then Cq (Φ , 0) ∼ = δq,ℓk+1 F.

J. Su, R. Zeng / Nonlinear Analysis 74 (2011) 6442–6450

6449

Proof. By (f2 ), we have

⟨Φ ′′ (0)y, y⟩ =

2π

∫

(|˙y|2 − λ|y|2 )dt ,

y ∈ E.

0

(1) When λ ∈ (λk , λk+1 ), x = 0 is a nondegenerate critical point of Φ with Morse index µ0 = ℓk ; thus Cq (Φ , 0) ∼ = δq,ℓk F. (2) When λ ∈ (λk+1 , λk+2 ), x = 0 is a nondegenerate critical point of Φ with Morse index µ0 = ℓk+1 ; thus Cq (Φ , 0) ∼ = δq,ℓk+1 F. (3) When λ = λk+1 , x = 0 is a degenerate critical point of Φ with Morse index µ0 = ℓk and nullity ν0 = νk+1 = 2. When F (t , x) ⩽ 0 for |x| small, we can verify that Φ has a local linking structure at x = 0 with respect to E = Ek ⊕ Ek⊥ (see [18]). By Proposition 2.2, we have Cq (Φ , 0) ∼ = δq,ℓk F. (4) Similarly, when F (t , x) ⩾ 0 for |x| small, we can verify that Φ has a local linking structure at x = 0 with respect to E = Ek+1 ⊕ Ek⊥+1 (see [18]). By Proposition 2.2, we have Cq (Φ , 0) ∼ = δq,ℓk+1 F.  Finally we prove the theorems. Proof of Theorems 1.1 and 1.2. By Lemma 4.1 we have that Cq (Φ , ∞) ∼ = 0,

q ∈ Z.

(4.1)

By Lemma 4.2(1)(3), for λ ∈ (λk+1 − δ, λk+1 ], we have Cq (Φ , 0) ∼ = δq,ℓk F,

(4.2)

By the part for λ ∈ (λk+1 − δ, λk+1 ] of Theorem 3.4, (P )λ has a nontrivial 2π -periodic solution x1 satisfying Cℓk+1 +1 (Φ , x1 ) ∼ ̸ 0, =

(4.3)

By Proposition 2.1(2), we have Cq (Φ , x1 ) ∼ = 0,

for q ̸∈ [µ(x1 ), µ(x1 ) + ν(x1 )].

(4.4)

Since ν(x1 ) ⩽ 2, it follows from (4.3)–(4.4) and Proposition 2.1(3)(4) that Cq (Φ , x1 ) Cq (Φ , x1 ) Cq (Φ , x1 )



∼ = δq,ℓk+1 +1 F, if ν(x1 ) = 0, ∼ = δq,ℓk+1 +1 F, if ν(x1 ) = 2, ∼ = 0, ∀q ̸= ℓk+1 + 1, if ν(x1 ) = 1,

(4.5)

Assume that K = {0, x1 }. Then as ℓk+1 + 1 − ℓk = 3, the ℓk + 1th Morse inequality reads as −1 ⩾ 0. This is a contradiction. Therefore Φ has a nonzero critical point x∗ which satisfies Cq (Φ , x∗ ) ∼ = 0,

for q ̸∈ [µ(x∗ ), µ(x∗ ) + ν(x∗ )].

(4.6)

By Morse theory we have either Cℓk +1 (Φ , x∗ ) ̸∼ = 0,

(4.7)

Cℓk −1 (Φ , x∗ ) ∼ ̸ 0, =

(4.8)

or

By (4.3) and (4.7) or (4.8) and ℓk+1 − ℓk = 2, we see that x∗ ̸= x1 . Let (4.7) hold. Then we have by (4.6) and Proposition 2.1(3)(4) that Cq (Φ , x∗ ) Cq (Φ , x∗ ) Cq (Φ , x∗ )



∼ = δq,ℓk +1 F, if ν(x∗ ) = 0, ∼ = δq,ℓk +1 F, if ν(x∗ ) = 2, ∼ = 0, for q ̸= ℓk + 1, if ν(x∗ ) = 1.

(4.9)

Assume that K = {0, x1 , x∗ }. We divide the consideration into four cases. (a) ν(x1 ) = 0, 2 and ν(x∗ ) = 0, 2. It follows from (4.1), (4.2), (4.5) and (4.9) that the Morse equality reads as

(−1)ℓk+1 +1 = 0,

(4.10)

and this is a contradiction. (b) ν(x1 ) = 1 and ν(x∗ ) = 1. By the ℓk + 1th and ℓk + 2th Morse inequalities we have that rank Cℓk +1 (Φ , x∗ ) = 1,

(4.11)

Thus the Morse equality reads as rank Cℓk+1 +1 (Φ , x1 ) = 0. This contradicts (4.3).

(4.12)

6450

J. Su, R. Zeng / Nonlinear Analysis 74 (2011) 6442–6450

(c) ν(x∗ ) = 1 and ν(x1 ) = 0, 2. In this case we have (4.11) and then the Morse equality still reads as (4.10), contradicting (4.3). (d) ν(x1 ) = 1 and ν(x∗ ) = 0, 2. In this case the Morse equality reads as (4.10), contradicting (4.3). Therefore, (P )λ has at least three nontrivial 2π -periodic solutions. This completes the proof.  Proof of Theorems 1.3 and 1.4. Theorem 1.4 follows from the part for λ ∈ (λk+1 , λk+1 + δ) of Theorem 3.4. We still have (4.1) and (4.2). Since F ⩾ 0, by the part for λ ∈ (λk+1 −δ, λk+1 ) of Theorem 3.4, (P )λ has two nontrivial 2π -periodic solutions x1 and x2 with critical groups satisfying (4.3) and Cℓk +1 (Φ , x2 ) ∼ ̸ 0. = The remainder of the proof is similar and we omit the details. The proof is finished.

(4.13) 

We finally remark that when F ⩾ 0 for |x| small and 0 < M ⩽ δ , the conclusion of Theorem 1.4 is valid for λ = λk+1 . References [1] P.H. Rabinowitz, Some critical point theorems and applications to semilinear elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4) 5 (1978) 215–223. [2] P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, in: CBMS, in: Reg. Conf. Ser. Math., vol. 65, AMS, Providence, RI, 1986. [3] X.L. Li, J.B. Su, R.S. Tian, Multiple periodic solutions of the second order Hamiltonian systems with superlinear terms. J. Math. Anal. Appl. (in press). [4] P.H. Rabinowitz, Periodic solutions of Hamiltonian systems: a survey, SIAM J. Math. Anal. 13 (1982) 343–352. [5] S.J. Li, Periodic solutions of non-autonomous second order systems with superlinear terms, Differential and Integral Equations 5 (1992) 1419–1424. [6] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, Berlin, Heidelberg, 1989. [7] K.C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, in: Progress in Nonlinear Differential Equations and Their Applications, vol. 6, Birkhäuser, Boston, 1993. [8] T. Bartsch, S.J. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal. 28 (1997) 419–441. [9] S.J. Li, J.B. Su, Existence of multiple critical points for an asymtotically quadratic functional with applications, Abstr. Appl. Anal. 1 (1996) 277–289. [10] J.B. Su, Nontrivial critical points for asymptotically quadratic functional at resonance, in: Morse Theory, Minimax Theory and their Applications to Nonlinear Differential Equations, in: New Stud. Adv. Math., vol. 1, Int. Press, Somerville, MA, 2003, pp. 225–234. [11] P.H. Rabinowitz, J.B. Su, Z.-Q. Wang, Multiple solutions of superlinear elliptic equations, Rendiconti Lincei Matematicae Applicazioni 18 (2007) 97–108. [12] D. Gromoll, W. Meyer, On differential functions with isolated point, Topology 8 (1969) 361–369. [13] J.Q. Liu, A Morse index for a saddle point, Systems Sci. Math. Sci. 2 (1989) 32–39. [14] J.B. Su, Multiplicity results for asymptotically linear elliptic problems at resonance, J. Math. Anal. Appl. 278 (2003) 397–408. [15] S.J. Li, J.Q. Liu, Some existence theorems on multiple critical points and their applications, Kexue Tongbao 17 (1984) 1025–1027. [16] Z.-Q. Wang, Multiple solutions for indefinite functionals and applications to asymptotically linear problems, Acta Math. Sinica (N.S.) 5 (1989) 101–113. [17] Z.-Q. Wang, On a superlinear elliptic equation, Ann. Inst. H. Poincaré, Anal. Non Lin. 8 (1991) 43–57. [18] J.B. Su, L.G. Zhao, Multiple periodic solutions of ordinary differential equations with double resonance, Nonlinear Anal. 70 (2009) 1520–1527.