Periodic solutions of second order Hamiltonian systems with nonlinearity of general linear growth

Periodic solutions of second order Hamiltonian systems with nonlinearity of general linear growth

Guanggang Liu

School of Mathematical Sciences, Liaocheng University, Liaocheng 252000, P. R. China Received 31 October 2020, appeared 2 April 2021

Communicated by Gabriele Bonanno

Abstract.In this paper we consider a class of second order Hamiltonian system with the nonlinearity of linear growth. Compared with the existing results, we do not assume an asymptotic of the nonlinearity at infinity to exist. Moreover, we allow the system to be resonant at zero. Under some general conditions, we will establish the existence and multiplicity of nontrivial periodic solutions by using the Morse theory and two critical point theorems.

Keywords:second order Hamiltonian systems, periodic solutions, Morse theory, critical groups.

2020 Mathematics Subject Classification: 34C25, 37B30, 37J45.

1 Introduction

Consider the following second order Hamiltonian systems

−x¨ =Vx(t,x), (1.1)

where V ∈ C²(_R×_R^N,R) with V(t+T,x) = V(t,x) for some T > 0. During the past forty years, the existence and multiplicity of periodic solutions for second order Hamiltonian systems have been extensively studied by variational methods. There has been a lot of results under various suitable solvability conditions, such as the sublinear conditions (see [14,18,22, 23,27,28] and references therein), the superlinear conditions (see [3,8,9,16,17,21,24,29] and references therein), and the asymptotically linear conditions (see [2,6,10,15,19,20,30] and references therein).

In this paper, we shall study the existence and multiplicity of nontrivial periodic solutions for (1.1) when the nonlinearityVx(t,x)has linear growth. Compared with the existing results, we do not make any assumptions at infinity on the asymptotic behaviors of the nonlinearity V_x(t,x). Specifically, we do not require the system to be asymptotically linear at infinity.

Instead, we assume that there exists a T-periodic symmetric matrix function A_∞(t)such that for someK>0,

V_xx(t,x)≥ A_∞(t) (orV_xx(t,x)≤ A_∞(t)), ∀t ∈[0,T], |x| ≥K,

BEmail: lgg112@163.com

where for two symmetric matricesAand B, A ≤ Bmeans that B−A is semi-positively defi- nite. Under this general linear growth condition, we will construct a sequence of approximate systems and use the Morse theory and two critical point theorems to establish the existence and multiplicity of nontrivial periodic solutions for the system. The idea of our proof is closely related to the work of Liu, Su and Wang [13], where they dealt with the existence of nontrivial solutions of elliptic problems. Note that in [13] the authors assumed that the elliptic problem was nonresonant at zero. By contrast, here we allow system (1.1) to be resonant at zero. On the other hand, system (1.1) with periodic boundary condition is rather different from the elliptic problems with Dirichlet boundary condition. These lead us to need some new technique.

Now let us say some words about the idea of the proof. We first construct a sequence of approximate systems which are asymptotically linear and non-resonant at infinity. Then in a crucial step we establish theL^∞ bound to the solutions of the approximate systems whose Morse index is controlled by the Morse index at infinity. Finally, we use the Morse theory and two critical point theorems to obtain the nontrivial periodic solutions with the controlled Morse index for the approximate systems, therefore using the previous L^∞ estimate they are also the nontrivial periodic solutions of the original system.

We make the following assumptions:

(H₁) V(t,x)∈C²(_R×_R^N,R)withV(t, 0) =0 andV(t+T,x) =V(t,x); (H₂) There existC₁>_{0 and}C₂>0 such that

|V_x(t,x)| ≤C₁(1+|x|), |V_xx(t,x)| ≤C₂, t∈ [0,T], x∈_R^N;

(H3) Vx(t,x) = A0(t)x+ (G0)_x(t,x), where A0(t)is aT-periodic continuous symmetric ma- trix function and(G₀)_x(t,x) =o(|x|)as|x| →0;

(H₄^±) There existsδ >0 such that

±G₀(t,x)>0, ∀t ∈[0,T], 0<|x|<δ;

(H₅^±) There exists a T-periodic continuous symmetric matrix function A_∞(t) such that for someK>0,

±V_xx(t,x)≥ ±A_∞(t), ∀t∈[0,T], |x| ≥K.

Let E := H¹_T(_R,R^N), the Hilbert space of T-periodic functions on R with values in R^N under the inner product

hx,yi=

Z _T

0

(x˙·y˙+x·y)dt, ∀x,y ∈E, and normkxk=hx,xi¹². We define the functional I on Eby

I(x) = ¹ 2

Z _T

0

|x˙(t)|²dt−

Z _T

0 V(t,x)dt. (1.2)

By (H₁) and (H₂),I ∈ C²(E,R)and the critical points ofI inEareT-periodic solutions of (1.1).

Clearly, the setσ ={(^2kπ_T )² |k∈_Z⁺}is the set of the eigenvalues of

−x¨ =λx (1.3)

with T-periodic boundary condition. Consider the eigenvalue problem of the following sys- tem

−x¨−A_∞x= λx (1.4)

with T-periodic boundary condition. Without loss of generality, in (H₅^±) by considering A_∞(t)∓eI_N instead of A_∞(t)foresmall if necessary we may assume that 0 is not the eigen- value of (1.4). Letλ₁< λ₂<· · ·<λ_l <0<λ_l+1 <λ_l+2< · · ·be distinct eigenvalues of (1.4).

Clearly,λ_i →_∞asi→_{∞. Let} E(λ_i)be the eigenspace of (1.4) corresponding toλ_i,i∈_Z⁺. We define the linear operator ˜Lon Eby

hLx,^˜ yi:=

Z _T

0 x˙·ydt,˙ ∀x,y∈ E.

Then ˜Lis a bounded self-adjoint operator. Define the linear operatorsB₀andB_∞ on Eby hB₀x,yi:=

Z _T

0 A₀(t)x·ydt, ∀x,y∈E and

hB_∞x,yi:=

Z _T

0 A_∞(t)x·ydt, ∀x,y∈ E.

Then B₀ and B_∞ are bounded self-adjoint compact operators on E. Let L₀ := L^˜ −B₀ and L_∞ := L^˜ −B_∞. Since 0 is not an eigenvalue of (1.4), we have that L_∞ is a non-degenerate operator on E. Denote by E₀⁺, E₀⁻,E_∞⁺andE_∞⁻the positive and negative spectral subspaces of L₀ and L_∞ respectively, and let E⁰₀ = kerL₀. Then there exists a constantc₀ > 0 such that for anyx ∈E₀⁺andy∈E₀⁻,

hL₀x,xi ≥c₀kxk², hL₀y,yi ≤ −c₀kyk². (1.5) Clearly,

E⁻_∞=

l

M

i=1

E(λ_i), E_∞⁺=

M∞ i=l+1

E(λ_i), E=E₀^+ME⁰₀^ME₀⁻= E⁺_∞^ME_∞⁻. Set

i⁰₀=dimE⁰₀, i⁻₀ =dimE⁻₀, i⁻_∞=dimE⁻_∞.

By (H₃), we see that x = 0 is a periodic solutions of (1.1) which is called trivial periodic solution. Our aim is to find nontrivial periodic solutions of (1.1). Now we give our main results as follows.

Theorem 1.1. Assume that (H₁), (H₂), (H₃) hold. Then (1.1) has at least one nontrivial periodic solution in each of the following cases:

(1) (H₄⁺), (H₅⁺) and i⁻₀ +i⁰₀<i⁻_∞−1;

(2) (H₄⁻), (H₅⁺) and i⁻₀ <i⁻_∞−1;

(3) (H₄⁺), (H₅⁻) and i⁻₀ +i⁰₀>i⁻_∞+1;

(4) (H₄⁻), (H₅⁻) and i⁻₀ >i⁻_∞+1.

Theorem 1.2. Assume that (H₁), (H₂), (H₃) hold, and V(t,−x) =V(t,x)for any(t,x)∈_R×_R^N. (1) If (H₄⁺), (H₅⁺) hold and i⁻₀ +i⁰₀ < i⁻_∞−1, then (1.1) has at least i⁻_∞−i⁻₀ −i⁰₀−1 pairs of

nontrivial periodic solutions;

(2) If (H₄⁻), (H₅⁺) hold and i⁻₀ < i⁻_∞−1, then (1.1) has at least i⁻_∞−i⁻₀ −1 pairs of nontrivial periodic solutions;

(3) If (H₄⁺), (H₅⁻) hold and i⁻₀ +i⁰₀ > i⁻_∞+1, then (1.1) has at least i⁻₀ +i⁰₀−i_∞⁻−1 pairs of nontrivial periodic solutions;

(4) If (H₄⁻), (H₅⁻) hold and i⁻₀ > i⁻_∞+1, then (1.1) has at least i⁻₀ −i⁻_∞−1 pairs of nontrivial periodic solutions.

Remark 1.3. In what follows, we assume thatx = 0 is an isolated critical point of I in E. In fact, if x = 0 is not an isolated critical point of I, then I has infinitely many critical points and therefore (1.1) has infinitely many periodic solutions. Therefore Theorem1.1and1.2hold naturally.

The paper is organized as follows. In Section 2, we construct a sequence of approximate systems and establish the L^∞ bound to the solutions of these approximate systems with ap- propriate Morse indexes. In Section 3, we will give the proof of Theorem1.1by using Morse theory and previous estimate. In Section 4, we will prove Theorem1.2 by using two critical point theorems for even functional and previous estimate.

2 Preliminaries

In this section we give some important preliminary lemmas. Let H be a real Hilbert space and J ∈ C²(H,R). Denote K(J) = {u ∈ H | J⁰(u) = 0}. For u ∈ K(J), we denote the Morse index ofu by m⁻(J⁰⁰(u)) which is the dimension of the negative spectral subspace of J⁰⁰(u)_. The augmented Morse index ofuis defined by

m^∗(J⁰⁰(u)) =m⁻(J⁰⁰(u)) +dim ker(J⁰⁰(u)), where ker(J⁰⁰(u))is the kernel of J⁰⁰(u).

To construct a sequence of approximate systems of (1.1), we first construct a sequence of approximate functionsV_k(t,x). The following result is from [13].

Lemma 2.1. Assume that (H₁), (H₂) and (H₅⁺) (resp. (H₅⁻)) hold. Then there exists a sequence functions V_k(t,x)∈ C²(_R×_R^N,R)satisfying the following properties:

(a) V_k(t+T,x) = V_k(t,x) and there exists an increasing sequence of real numbers R_k → _∞ (k→_∞)such that

V_k(t,x) =V(t,x), ∀|x| ≤R_k,t∈ [0,T]; (b) there exist C₁⁰ >0and C₂⁰ >0independent of k such that

|(V_k)_x(t,x)| ≤C⁰₁(1+|x|), |(V_k)_xx(t,x)| ≤C⁰₂;

(c) for each k∈_Z⁺,(V_k)_xx(t,x)≥ A_∞(t)(resp.(V_k)_xx(t,x)≤ A_∞(t)) for all t∈ [_0,T]_,|x| ≥K;

(d) there isγ > 0independent of k such that ^2pπ_T 2

< γ< ^2(p+_T^1)π2 for some p∈ _Z⁺, and for each k ∈_Z⁺fixed,

V_k(t,x) = ^γ

2|x|²+o(|x|²)_, (V_k)_x(t,x) =γx+o(|x|)_, (V_k)_xx(t,x) =γI_N+o(₁) as|x| →_∞;

(e) if V(t,−x) = V(t,x), ∀t ∈ [0,T],x ∈ _R^N, then for every k ∈ _Z⁺, V_k(t,−x) = V_k(t,x),

∀t ∈[0,T],x ∈_R^N. Let

I_k(x):= ¹ 2

Z _T

0

|x˙|²dt−ψ_k(x), x∈ E, (2.1) where

ψ_k(x):=

Z _T

0 V_k(t,x)dt.

Clearly, I_k(x) ∈ C²(E,R) and the critical points of I_k correspond to the periodic solutions of the following system

−x¨= (V_k)_x(t,x)_{. (2.2)}

By Lemma2.1(a) and Remark1.3,x=0 is also an isolated critical point of I_kfor everyk∈_Z⁺. Define the linear operator B_γ :E→Eby

hBγx,yi:=

Z _T

0 γx·ydt, ∀x,y∈E.

Let Lγ := eL−Bγ, then by Lemma 2.1, Lγ is a non-degenerate bounded linear self-adjoint operator on E. We have the decomposition E= E_γ⁻⊕E_γ⁺, whereE_γ⁻ andE⁺_γ are the negative and positive spectral subspaces of Lγ. Then there exists a constant cγ > 0 such that for any x∈ E⁺_γ andy∈ E⁻_γ,

hL_γx,xi ≥c_γkx²k, hL_γy,yi ≤ −c_γky²k. (2.3) Denote

j_∞⁻=dimE⁻_γ.

By Lemma2.1(c), (d), if (H₅⁺) holds, then γIN ≥ A_∞(t), which implies that

E_∞⁻⊂E_γ⁻ and j_∞⁻≥i⁻_∞. (2.4) If (H₅⁻) holds, then γI_N ≤ A_∞(t), which implies that

E_γ⁻⊂E_∞⁻ and j_∞⁻≤i⁻_∞. (2.5) Let

G_k(t,x) =V_k(t,x)−^γ

2|x|², G_0k(t,x) =V_k(t,x)− ¹

2A₀(t)x·x and

ϕ_k(x) =

Z _T

0 G_k(t,x)dt, ϕ_0k(x) =

Z _T

0 G_0k(t,x)dt.

By (H₃), Lemma 2.1 (a), (d), we see that (G_k)_x(t,x) = o(|x|) as |x| → _∞ and (G_0k)_x(t,x) = o(|x|)as|x| →0. Then we have

ϕ⁰_k(x) =o(kxk) _askxk →_{∞ and} ϕ⁰_0k(x) =o(kxk) _askxk →_{0. (2.6)} And we can rewrite the functional I_k by

I_k(x) = ¹

2hL_γx,xi −ϕ_k(x) = ¹

2hL₀x,xi −ϕ_0k(x), x∈E. (2.7)

Lemma 2.2. Assume that (H₁), (H₂), (H₃) and (H₅⁺) (resp. (H₅⁻)) hold. For every k ∈ _Z⁺, if x_k is a critical point of I_k with m⁻(I_k⁰⁰(x_k)) ≤ i⁻_∞−1 (resp. m^∗(I_k⁰⁰(x_k)) ≥ i⁻_∞+1), then there exists a constantβ>0independent of k such thatkx_kk_L∞ ≤ β.

Proof. We use an indirect argument. Assume that kx_kk_L∞ → _∞ as k → ∞. By the Sobolev inequalitykxk_L∞([0,T])≤Ckxk, we have thatkx_kk →_∞ask→_∞.

Let

¯

x_k = ^xk kx_kk^. Then ¯x_k satisfies

−x¨¯_k = (_Vk)_x(_t,xk)

kx_kk ^{. (2.8)}

Up to a subsequence, we have that for some ¯x ∈ E, ¯x_k * x¯ in E, ¯x_k → x¯ in L²([0,T]). And it follows from Proposition 1.2 in [20] that ¯x_k converges uniformly to ¯x on [0,T]. By (H₂), (H₃) and Lemma2.1, there existsC⁰₁>0 such that|(V_k)_x(t,x_k)| ≤C₁⁰|x_k|. Thus for everyk,

(V_k)_x(t,x_k) kx_kk

≤C₁⁰|x¯_k|. (2.9)

Multiplying (2.8) by ¯x_k, one has

1= kx¯_kk²≤(C₁⁰ +1)kx¯_kk²_L2([0,T]). Lettingk →_{∞, we get}

kx¯k²_L2([0,T])≥ ¹

C₁⁰ +1 >0. (2.10)

Now we show that up to a subsequence ˙¯x_k converges uniformly to ˙¯x on [0,T]. For any t∈ [0,T], by (2.8), (2.9) and Hölder inequality, we have

|x˙¯_k(0)|=

˙¯

x_k(t) +

Z _t

0

(V_k)_x(s,x_k) kx_kk ^ds

≤

˙¯

x_k(t)|+

Z _t

0 C₁⁰|x¯_k(s)

ds

≤ |x˙¯_k(t)|+C⁰₁√

Tkx¯_kk_L2

≤ |x˙¯_k(t)|+C⁰₁√ T, thus

Z _T

0

|x˙¯_k(0)|dt≤

Z _T

0

|x˙¯_k(t)|dt+

Z _T

0 C₁⁰√ Tdt

≤√

Tkx˙¯_kk_L2 +C₁⁰√ TT

≤√

T+C⁰₁√ TT.

Hence

|x˙¯_k(₀)| ≤C₂,

whereC₂ =

√T

T +C₁⁰√

T. Then for anyt ∈[0,T],

|x˙¯_k(t)|=

˙¯

x_k(0) +

Z _t

0

−(V_k)_x(s,x_k) kx_kk ^ds

≤ |x˙¯_k(0)|+

Z _t

0 C₁⁰|x¯_k(s)|ds

≤C₂+C₁⁰√

Tkx¯_kk_L2

≤C₂+C₁⁰√ T, which implies that for everyk ∈_Z⁺,

kx˙¯_k(t)k_C0 ≤C₂+C⁰₁√

T. (2.11)

For any∆t∈R, by (2.8) and (2.9) we have

|x˙¯_k(t+_∆t)−x˙¯_k(t)|=

Z _t+∆t

t x¨¯_k(s)ds

=

Z _t+_∆t t

−(V_k)_x(t,x_k) kx_kk ^ds

≤

Z _t+_∆t

t C₁⁰|x¯_k|ds

≤C₁⁰|_∆t|¹²kx¯_kk_L2 ≤ C₁⁰|_∆t|¹². (2.12) Thus by (2.11) and (2.12), we have

kx˙¯_k(t)k

C¹² ≤C.

Then by the Arzelà–Ascoli theorem, ˙¯x_k converges uniformly to ˙¯x on[_0,T]_.

We claim that ¯x(t)6= 0 a.e. in[0,T]. In fact, conversely, if ¯x(t) = 0 in a positive measure subset of [0,T], then there exists a pointt₀ ∈ [0,T]such that ¯x(t₀) =0 and ˙¯x(t₀) = 0. Recall that ¯x_k and ˙¯x_k converge uniformly to ¯xand ˙¯x respectively on[_0,T]_{, we have}

¯

x_k(t₀)→0 and x˙¯_k(t₀)→0 (2.13) ask →_{∞. Let ¯}y_k := x˙¯_k, then(x¯_k, ¯y_k)satisfies the following system

(x˙¯_k = y¯_k,

˙¯

y_k =−^(Vk_k^)x(t,xk)

xkk . (2.14)

For anyt ∈[0,T],

|(x¯_k(t), ¯y_k(t))|=

(x¯_k(t0), ¯y_k(t0)) +

Z _t

t0

¯

y_k(s),−(V_k)_x(s,x_k) kx_kk

ds

≤ |(x¯_k(t₀), ¯y_k(t₀))|+

Z _t

t0

¯

y_k(s),−(V_k)_x(s,x_k) kx_kk

ds

≤ |(x¯_k(t₀), ¯y_k(t₀))|+

Z _t

t₀

q

1+C₁⁰²|(x¯_k(s), ¯y_k(s))|ds . Thus by Gronwall’s inequality, we have

|(x¯_k(t)_{, ¯}y_k(t))| ≤ |(x¯_k(t₀)_{, ¯}y_k(t₀))|e^|

Rt t0

√1+C₁⁰²ds|

≤C|(x¯_k(t₀)_{, ¯}y_k(t₀))|_{, (2.15)}

where C = e

√

1+C⁰₁²T. Then letting k → _∞ in (2.15), we get ¯x(t) = 0 and ¯y(t) = 0 for any t∈ [0,T], which is contrary to (2.10). Hence the claim is proved. Note thatkx_kk →_{∞, then by} this claim one has

|x_k| →_∞ a.e. in[0,T] (2.16)

ask→_∞.

If (H₅⁺) holds, then by (2.16), Lemma 2.1 (b), (c) and Fatou’s Lemma, for any fixed x ∈ E⁻_∞\ {0},

lim sup

k→_∞

hI_k⁰⁰(x_k)x,xi= hLx,^˜ xi −lim inf

k→_∞ Z _T

0

(V_k)_xx(t,x_k)x·xdt

≤ hLx,^˜ xi −

Z _T

0 lim inf

k→_∞ (V_k)_xx(t,x_k)x·xdt

≤ hLx,^˜ xi −

Z _T

0 A_∞(t)x·xdt

= hL_∞x,xi<0,

which implies that there existsk(x)∈_Z⁺such thathI_k⁰⁰(x_k)x,xi<0 whenk ≥k(x). Note that E⁻_∞ is finite dimensional, there must existk₀∈ _Z⁺ independent ofx∈ E⁻_∞\ {0}such that

hI_k⁰⁰(x_k)x,xi<0

for allx ∈E_∞⁻\ {₀}_andk≥ k₀. This means thatm⁻(I_k⁰⁰(x_k))≥i⁻_∞ fork≥k₀, which leads to a contradiction.

If (H₅⁻) holds, sinceE⁺_∞is infinite dimensional, the above argument cannot be used directly.

To overcome this difficulty, we will splitE⁺_∞ into two parts. Let M= max

t∈[0,T]|A_∞(t)|.

Sinceλ_i →_∞ asi→ _∞, then there existsi₀ ∈ _Z⁺such that λ_i0 ≥2(M+C₂⁰)where C⁰₂is the constant as in Lemma2.1(b). Let

E₁ =

i0−1

M

i=l+1

E(λ_i), E2 =

M∞ i=i0

E(λ_i).

ThenE⁺_∞ =E₁⊕E₂andE₁is finite dimensional. For any y₁ ∈E₂\ {0}, note that Z _T

0

(|y˙₁|²−A_∞y₁·y₁)dt≥λ_i0 Z _T

0

|y₁|²dt, then

hI_k⁰⁰(x_k)y₁,y₁i=

Z _T

0

|y˙₁|²dt−

Z _T

0

(V_k)_xx(t,x_k)y₁·y₁dt

≥λ_i0 Z _T

0

|y₁|²dt+

Z _T

0 A_∞y₁·y₁dt−

Z _T

0

(V_k)_xx(t,x_k)y₁·y₁dt

≥λ_i0 Z _T

0

|y₁|²dt−

Z _T

0 M|y₁|²dt−

Z _T

0 C⁰₂|y₁|²dt

≥ ^λi0 2

Z _T

0

|y₁|²dt>_{0. (2.17)}

For anyy₂∈ E₁\ {0}, by (2.16), Lemma2.1(b), (c) and Fatou’s Lemma, lim inf

k→_∞ hI_k⁰⁰(x_k)y₂,y₂i=

Z _T

0

|y˙₂|²dt−lim sup

k→_∞ Z _T

0

(V_k)_xx(t,x_k)y₂·y₂dt

≥

Z _T

0

|y˙₂|²dt−

Z _T

0 lim sup

k→_∞

(V_k)_xx(t,x_k)y₂·y₂dt

≥

Z _T

0

|y˙₂|²dt−

Z _T

0 A_∞(t)y₂·y₂dt

=hL_∞y₂,y₂i>0,

which implies that there exists k(y₂) ∈ _Z⁺ such that hI_k⁰⁰(x_k)y₂,y₂i > 0 for k ≥ k(y₂). Note that E₁is finite dimensional, there must exist k₁ ∈_Z⁺independent ofy₂ ∈E₁\ {0}such that hI_k⁰⁰(x_k)y₂,y₂i>0 (2.18) for all y₂ ∈ E₁\ {0}andk ≥ k₁. Hence by (2.17) and (2.18), for any y ∈ E_∞⁺\ {0} and every k≥k₁,

hI_k⁰⁰(x_k)y,yi>0.

This implies that m^∗(I_k⁰⁰(x_k))≤i⁻_∞fork ≥k₁, which leads to a contradiction.

Therefore the lemma is proved.

3 Proof of Theorem 1.1

In this section, we will use Morse theory to prove the existence of nontrivial periodic solution for (1.1). Let H be a real Hilbert space and J ∈ C²(H,R) be a functional satisfying the (PS) condition, i.e., any sequence{u_n} ⊂ Hfor which J(u_n)is bounded and J⁰(u_n)→0 asn→_∞ possesses a convergent subsequence. Denote by Hq(A,B)theq-th singular relative homology group of the topological pair(A,B)with coefficients in a field F. Letube an isolated critical point of J with J(u) =c. The groups

C_q(J,u):=H_q(J^c,J^c\ {u}), q∈_Z

are called the critical groups of J atu, where J^c = {u ∈ H | J(u) ≤ c}. Denote K = K(J) = {u ∈ H| J⁰(u) =₀}. Suppose that J(K)is bounded from below bya ∈R. The critical groups of J at infinity are defined by

Cq(J,∞):=Hq(H,J^a), q∈_Z.

We say thatJhas a local linking structure at 0 with respect to the direct sum decomposition H= H⁻⊕H⁺if there existsr>0 such that

J(u)>0 foru∈ H⁺with 0<kuk ≤r, J(u)≤0 foru∈ H⁻ withkuk ≤r.

The following results can be found in [1], [26] and [4].

Proposition 3.1(See [1]). Suppose J satisfies (PS) condition. If K = _{∅, then Cq}(J,∞)∼= 0,q∈ _Z.

If K={u₀}, then C_q(J,∞)∼= C_q(J,u₀)_,q∈_Z.

Proposition 3.2 (See [26]). Let 0 be an isolated critical point of J ∈ C²(H,R) with Morse index µ₀ and nullity ν₀. Assume that J has a local linking structure at 0 with respect to the direct sum decomposition H= H⁻⊕H⁺and k=dimH⁻ <_{∞. If k}=µ₀or k=µ₀+ν₀, then

C_q(J,u) =δ_q,kF_, q∈ _Z.

Let A be a nondegenerate bounded self-adjoint operator defined on H. According to its spectral decomposition,H= H⁺⊕H⁻, whereH⁺,H⁻are invariant subspaces corresponding to the positive and negative spectrum ofArespectively. Let

J(x) = ¹

2hAx,xi+g(x), and the following assumptions are given:

(A₁) A±:= A|_H± has a bounded inverse on H^±; (A₂) κ:=dimH⁻<_∞;

(A₃) g∈C¹(H,R¹)has a compact derivativeg⁰(x)andkg⁰(x)k=o(kxk)askxk →_∞.

Proposition 3.3(See [4]). Under the assumptions(A₁),(A₂)and(A₃), we have that J satisfies (PS) condition and C_q(J,∞) =δ_q,κF.

Proposition 3.4(See [4]). Suppose that J ∈C²(H,R)satisfies (PS) condition, and K= {u₁, . . . ,u_k},

then ∞

q

∑

Mqt^q =

∑

βqt^q+ (1+t)Q(t),

where Q(_t) is a formal series with nonnegative coefficients, Mq = _∑^k_i=0rankCq(_J,uk) _and β_q = rankCq(J,∞), q=0, 1, 2, . . .

Now we compute the critical groups of I_k at zero and at infinity.

Lemma 3.5. Assume that (H₁)–(H₃) hold. Then for every k∈_Z⁺, (1) if (H₄⁺) holds,

C_q(I_k, 0) =δ_q,i⁻

0+i⁰₀F_, q∈ _Z.

(2) if (H₄⁻) holds,

Cq(I_k, 0) =δ_q,i⁻

0 F, q∈_Z.

Proof. (1) We first show that I_k has a local linking structure at 0 with respect to E=E⁻⊕E⁺, whereE⁻ =E₀⁻⊕E₀⁰andE⁺=E₀⁺. For x∈ E⁺₀, by (1.5) and (2.6) we have

I_k(x) = ¹

2hL₀x,xi −ϕ_0k(x)

≥ ^c0

2kxk²−o(kxk²) (3.1)

askxk →0. This means that there exists smallr>0 such that

I_k(x)>_{0, for}x ∈E⁺₀ with 0<kxk ≤r. (3.2)

Forx ∈E₀⁻⊕E⁰₀, we writex =x⁻+x⁰ withx⁻∈ E⁻₀ andx⁰ ∈E⁰₀. Then I_k(x) = ¹

2hL₀x⁻,x⁻i −

Z _T

0 G_0k(t,x)dt

≤ −^c0

2kx⁻k²−

Z _T

0 G_0k(t,x)dt. (3.3)

By (H₄⁺) and Lemma2.1(a),

Z

|x|≤δ

G_0k(t,x)dt≥0. (3.4)

If|x|>δ, sinceE₀⁰is finite dimensional, we have

|x⁻| ≥ |x| − |x⁰| ≥ |x| − kx⁰k_L∞ ≥ |x| −Ckx⁰k ≥ |x| −Ckxk, thus let 0<r < _3C^δ , forkxk ≤r, we have

|x⁻| ≥ |x| −^δ

3 ≥ |x| −¹

3|x|= ²

3|x|. (3.5)

By Lemma2.1(b), (d), there existsC_δ >0 such that for |x|>δ,

|G_0k(t,x)| ≤C_δ|x|³. (3.6) Hence, by (3.3)–(3.6), forx ∈E₀⁻⊕E⁰₀with kxk ≤r, we have

I_k(x)≤ −^c0

2kx⁻k²−

Z _T

0 G_0k(t,x)dt

≤ −^c0

2kx⁻k²−

Z

|x|≤δ

G_0k(t,x)dt−

Z

|x|>δ

G_0k(t,x)dt

≤ −^c0

2kx⁻k²+

Z

|x|>δ

C_δ|x|³dt

≤ −^c0

2kx⁻k²+C_δ Z

|x|>δ

3 2

3

|x⁻|³dt

≤ −^c0

2kx⁻k²+C⁰_δkx⁻k³. (3.7)

This implies that there existsr >0 small enough such that

I_k(x)<0, forx∈ E₀⁻⊕E⁰₀ withkxk ≤r andkx⁻k>0. (3.8) On the other hand, for x⁰ ∈ E₀⁰, we can chooser>0 small enough such that

0<kx⁰k_L∞ <δ, when 0< kx⁰k ≤r.

Then forx⁰ ∈E⁰₀with 0<kx⁰k ≤r, sincex⁰ ∈C²([0,T],R^N), there must exist 0<t₁<t₂ <T such that

0< |x⁰(t)|<δ, ∀t∈[t₁,t₂]. Then by (H₄⁺) and Lemma2.1(a), forx⁰∈ E⁰₀ with 0<kx⁰k ≤r,

I_k(x⁰) =−

Z _T

0 G_0k(t,x⁰)dt= −

Z _T

0 G0(t,x⁰)dt≤ −

Z _t2

t1

G0(t,x⁰)dt<0. (3.9)

Hence, by (3.8) and (3.9), there existsr >0 such that

I_k(x)<0, for x∈E₀⁻⊕E⁰₀with 0< kxk ≤r. (3.10) Therefore, it follows from (3.2) and (3.10) that I_k has a local linking structure at 0 with respect toE= E⁻⊕E⁺, whereE⁻= E⁻₀ ⊕E₀⁰andE⁺ =E₀⁺. Then by Proposition3.2, we have

C_q(I_k, 0) =δ_q,i⁻

0+i⁰₀F, q∈_Z.

(2) By a similar argument as (1), we can prove that I_k has a local linking structure at 0 with respect toE= E⁻⊕E⁺, whereE⁻= E⁻₀ andE⁺= E₀⁺⊕E⁰₀. Then by Proposition3.2, we have

Cq(I_k, 0) =δ_q,i⁻

0F, q∈_Z.

Lemma 3.6. Assume that (H₁)-(H₃), (H₅⁺)(or (H₅⁻)) hold. Then for every k ∈ _Z⁺, I_k satisfies (PS) condition and the critical groups of I_k at infinity are

C_q(I_k,∞) =δ_q,j⁻_∞F, q∈_Z.

Proof. Note that

I_k(x) = ¹

2hL_γx,xi −ϕ_k(x) SinceLγ is a nondegenerate operator on E, then Lγ |_E^±

γ has a bounded inverse on E^±_γ. Recall that dimE⁻_γ = j_∞⁻ < ∞. Thus the assumptions (A1) and (A₂) in Proposition3.3 are satisfied.

On the other hand, note that ϕ_k(x) ∈ C²(E,R) has compact derivative ϕ⁰_k(x) and ϕ⁰_k(x) = o(kxk)as kxk → ∞, then the assumption (A3) in Proposition 3.3 is also satisfied. Hence, by Proposition3.3, we have

Cq(I_k,∞) =δ_q,j⁻_∞F, q∈_Z.

Remark 3.7. Since I_k⁰(_x) = _Lγx+ϕ⁰_k(_x) =_Lγx+_o(kxk)askxk →_∞andLγ is invertible, it is easy to see that the critical point setK(I_k)is bounded for everyk∈ _Z⁺. Then sinceI_k satisfies (PS) condition by Lemma3.6, we conclude thatK(I_k)is a compact set for everyk∈_Z⁺. Proof of Theorem1.1. We only prove the result for the case (1), the proofs for the cases (2), (3) and (4) are similar.

For everyk∈_Z⁺, sincex =0 is an isolated critical point of I_k, there existsσ >0 such that I_k(x)has no nontrivial critical points inBσ(0) := {x | kxk ≤ σ}. Since i⁻₀ +i⁰₀ < i⁻_∞−1, then by (2.4), Lemma3.5(1) and Lemma3.6we have

Cq(I_k,∞)6= Cq(I_k, 0)

for some q∈ _Z. So by Proposition 3.1and Remark 3.7, the setK(I_k)\ {0}is not empty and compact. DenoteK_k = K(I_k)\ {0}.

Now we show that for every k ∈ _Z⁺ there exists a nontrivial critical point x_k ∈ K_k such that

m⁻(I_k⁰⁰(x_k))≤i⁻_∞−1. (3.11) We use an indirect argument. Suppose that for anyx_k ∈ K_k,

m⁻(I_k⁰⁰(x_k))>i⁻_∞−1. (3.12) For A⊂Eanda>0, set

N_a(A):={x ∈E|dist(x,A)< a}.

Using the Marino–Prodi perturbation technique from [25], for any e > 0 and 0 < τ <

min{^σ_{3, 1}}, we can obtain aC²functional J_k such that:

(i) kI_k−J_kk_C2 <e;

(ii) I_k(x) = J_k(x),x∈ E\N_2τ(K_k);

(iii) I_k⁰⁰(x) = J_k⁰⁰(x)for any x ∈ N_τ(K(I_k)), K(J_k)\ {0} ⊂ N_τ(K_k), and the nontrivial critical points of J_k are all non-degenerate.

By (iii), J_k⁰⁰(0) = I_k⁰⁰(0), thus by Proposition3.2and Lemma 3.5, we have Cq(J_k, 0) =Cq(I_k, 0) =δ_q,i⁻

0+i⁰₀F. (3.13)

By (ii), I_k(x) = J_k(x) forx ∈ E\N_2τ(K_k), then by Lemma 3.6, J_k also satisfies (PS) condition and

C_q(J_k,∞) =C_q(I_k,∞) =δ_q,j⁻_∞F. (3.14) Since K(J_k) ⊂ N_τ(K(I_k)) and K(I_k) is compact, K(J_k) is also a compact set. Moreover, note that the notrivial critical points of J_k are all non-degenerate, we have thatK(J_k)is a finite set.

Suppose that

K(J_k)\ {0}={x_k1,x_k2,x_k3, . . . ,x_kn}.

By (iii) and (3.12), we can chooseτsmall enough such that for all 1≤i≤n,

m⁻(J_k⁰⁰(x_ki))>i⁻_∞−1. (3.15) By (3.13), (3.14), and Proposition3.4 we have

t^i−0+i00+

∑

t^{m−(Jk00(xki))} =t^j−∞+ (1+t)Q(t). (3.16) Note that i⁻₀ +i⁰₀ < i⁻_∞−1 and i⁻_∞ ≤ j⁻_∞, it follows from (3.16) that (1+t)Q(t)has a nonzero term with exponent i₀⁻+i⁰₀. Then this means that the left hand side of (3.16) has a nonzero term with exponent i₀⁻+i⁰₀−1 ori⁻₀ +i⁰₀+1. Thus there exists a 1≤i≤nsuch that

m⁻(J_k⁰⁰(x_ki)) =i⁻₀ +i⁰₀−1 or m⁻(J_k⁰⁰(x_ki)) =i⁻₀ +i⁰₀+1.

Sincei⁻₀ +i⁰₀ <i⁻_∞−1, we have thatm⁻(J_k⁰⁰(x_ki))≤i⁻_∞−1 for some 1≤i≤ n. This is contrary to (3.15), thus (3.11) is proved.

By Lemma2.2and (3.11), for everyk ∈_Z⁺the functional I_k has a nontrivial critical point x_k such that kx_kk_L∞ ≤ β. By Lemma 2.1, for k large enough such that R_k > β, x_k is also a nontrivial critical point of I, and thusx_k is a nontrivial periodic solution of (1.1).

4 Proof of Theorem 1.2

We introduce two critical point theorems which will be used in proving Theorem1.2. LetHbe a Hilbert space. Assume that J ∈ C²(H,R)is an even functional, satisfies the (PS) condition, J(0) = 0 and K(J)is a compact set. Let Ba = {y ∈ H | kyk ≤ a}and Sa = ∂Ba = {y ∈ H | kyk = a}. The following two critical point theorems follow from Ghoussoub [7] and Chang [4] (see also [13] ).