# Infinitely Many Periodic Solutions for Nonautonomous

Authors: Peng Zhang; Chun-Lei Tang
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

Two sequences of distinct periodic solutions for second-order Hamiltonian
systems with sublinear nonlinearity are obtained by using the minimax methods. One
sequence of solutions is local minimum points of functional, and the other is minimax type
critical points of functional. We do not assume any symmetry condition on nonlinearity.


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## Body

## 1. Introduction and Main Result

We are interested in the following second order Hamiltonian systems:(1.1)ü(t)+∇F(t,u(t))=0a.e.t∈[0,T],u(0)-u(T)=u̇(0)-u̇(T)=0,
where T>0 and F:[0,T]×RN→R satisfies the following assumption.(A)
F(t,x) is measurable in t for each x∈RN and continuously differentiable in x for a.e.t∈[0,T],F(t,0)=0 for a.e.t∈[0,T], and there exist a∈C(R+,R+),b∈L1(0,T;R+) such that(1.2)|F(t,x)|≤a(|x|)b(t),|∇F(t,x)|≤a(|x|)b(t)for allx∈RN and a.e.t∈[0,T].Then the corresponding functionalφ on HT1 given by (1.3)φ(u)=12∫0T|u̇(t)|2dt-∫0TF(t,u(t))dtis continuously differentiable and weakly lower semicontinuous onHT1, where (1.4)HT1={u:[0,T]→RN∣uisabsolutelycontinuous,u(0)=u(T),u̇∈L2(0,T;RN)}is a Hilbert space with the norm defined by(1.5)∥u∥=(∫0T|u(t)|2dt+∫0T‍|u̇(t)|2dt)1/2foru∈HT1 (see [1]). Moreover, (1.6)〈φ'(u),v〉=∫0T(u̇(t),v̇(t))dt-∫0T(∇F(t,u(t)),v(t))dtfor allu,v∈HT1. It is well known that the solutions of problem (1.1) correspond to the critical points of φ.There are large number of papers that deal with multiplicity results for this problem. Infinitely many solutions for problem (1.1) are obtained in [2–4], where the symmetry assumption on the nonlinearity F has played an important role. In recent years, many authors have paid much attention to weaken the symmetry condition, and some existence results on periodic and subharmonic solutions have been obtained without the symmetry condition (see [5–7]). Particularly, Ma and Zhang [6] got the existence of a sequence of distinct periodic solutions under some superquadratic and asymptotic quadratic cases. Faraci and Livrea [7] studied the existence of infinitely many periodic solutions under the assumption that F(t,x) is a suitable oscillating behaviour either at infinity or at zero.In this paper, we suppose that the nonlinearity∇F(t,x) is sublinear, that is, there exist f,g∈L1(0,T;R+) and α∈[0,1] such that(1.7)|∇F(t,x)|≤f(t)|x|α+g(t)
for all x∈RN and a.e.t∈[0,T]. We establish some multiplicity results for problem (1.1) under different assumptions on the potential F. Roughly speaking, we assume that F has a suitable oscillating behaviour at infinity. Two sequences of distinct periodic solutions are obtained by using the minimax methods. One sequence of solutions is local minimum points of functional, and the other is minimax type critical points of functional. In particular, we do not assume any symmetry condition at all.Our main result is the following theorem.Theorem 1.1.
Suppose thatF(t,x) satisfies assumptions (A) and (1.7). Assume that
(1.8)limsupr→+∞infx∈RN,|x|=r|x|-2α∫0TF(t,x)dt=+∞,(1.9)liminfR→+∞supx∈RN,|x|=R|x|-2α∫0TF(t,x)dt=-∞.
Then,(i)
there exists a sequence of periodic solutions{un} which are minimax type critical points of functional φ, and φ(un)→+∞, as n→∞;(ii)
there exists another sequence of periodic solutions{um*} which are local minimum points of functional φ, and φ(um*)→-∞, as m→∞.


## 2. Proof of Theorems

Foru∈HT1, let u¯=(1/T)∫0Tu(t)dt and ũ=u-u¯. Then one has (2.1)∥ũ∥∞2≤T12∫0T|u̇(t)|2dt(Sobolev'sinequality),∫0T|ũ(t)|2dt≤T24π2∫0T|u̇(t)|2dt(Wirtinger'sinequality).Lemma 2.1.
LetH̃T1 be the subspace of HT1 given by
(2.2)H̃T1={u∈HT1∣u¯=0}.
Suppose that (1.7) holds. Then
(2.3)φ(u)→+∞
as ∥u∥→∞ in H̃T1.Proof.
It follows from (1.7) and Sobolev's inequality that
(2.4)φ(u)=12∫0T|u̇(t)|2dt-∫0TF(t,u(t))dt≥12∫0T|u̇(t)|2dt-∫0Tf(t)|u(t)|α+1dt-∫0Tg(t)|u(t)|dt≥12∫0T|u̇(t)|2dt-∥u∥∞α+1∫0Tf(t)dt-∥u∥∞∫0Tg(t)dt≥12∫0T|u̇(t)|2dt-C1(∫0T|u̇(t)|2dt)(α+1)/2-C2(∫0T|u̇(t)|2dt)1/2
for all u in H̃T1. By Wirtinger's inequality, the norm
(2.5)∥u∥=(∫0T|u̇(t)|2dt)1/2
is an equivalent norm on H̃T1. Hence the lemma follows from the equivalence and the above inequality.Lemma 2.2.
Suppose that (1.7) and (1.8) hold. Then there exists positive real sequence {an} such that
(2.6)limn→∞an=+∞,limn→∞supu∈RN,|u|=anφ(u)=-∞.The proof of this lemma is similar to the following lemma.Lemma 2.3.
Suppose that (1.7) and (1.9) hold. Then there exists positive real sequence {bm} such that
(2.7)limm→∞bm=+∞,limm→∞infu∈Hbmφ(u)=+∞,
where Hbm={x∈RN:|u|=bm}⊕H̃T1.Proof.
For anyu∈Hbm, let u=u¯+ũ, where |u¯|=bm,ũ∈H̃T1. It follows from (1.7) and Sobolev's inequality that
(2.8)|∫0T[F(t,u(t))-F(t,u¯)]dt|=|∫0T∫01(∇F(t,u¯+sũ(t)),ũ(t))dsdt|≤∫0T∫01f(t)|u¯+sũ(t)|α|ũ(t)|dsdt+∫0T∫01g(t)|ũ(t)|dsdt≤2(|u¯|α+∥ũ∥∞α)∥ũ∥∞∫0Tf(t)dt+∥ũ∥∞∫0Tg(t)dt≤3T∥ũ∥∞2+T3|u¯|2α(∫0Tf(t)dt)2+2∥ũ∥∞α+1∫0Tf(t)dt+∥ũ∥∞∫0Tg(t)dt≤14∫0T|u̇(t)|2dt+C1|u¯|2α+C2(∫0T|u̇(t)|2dt)(α+1)/2+C3(∫0T|u̇(t)|2dt)1/2,
for all u∈HT1. Hence we have
(2.9)φ(u)=12∫0T|u̇(t)|2dt-∫0T[F(t,u(t))-F(t,u¯)]dt-∫0TF(t,u¯)dt≥14∫0T|u̇(t)|2dt-C2(∫0T|u̇(t)|2dt)(α+1)/2-C3(∫0T|u̇(t)|2dt)1/2-|u¯|2α(|u¯|-2α∫0TF(t,u¯)dt+C1),
for all u∈HT1. As (|u¯|2+∥u̇(t)∥L2)1/2→∞ if and only if ∥u∥→∞, then the lemma follows from (1.9) and the above inequality.Now we give the proof Theorem1.1.Proof of Theorem1.1.
LetBan be a ball in RN with radius an. Define
(2.10)Sn={γ∈C(Ban,HT1),γ|∂Ban=id|∂Ban},cn=infγ∈Sn[maxx∈Banφ(γ(x))].
We claim that each γ intersects the hyperplane H̃T1. In fact, let π:HT1→RN be the projection of HT1 onto RN, defined by
(2.11)π(u)=1T∫0Tu(t)dt.
For t∈[0,1],u∈RN, define
(2.12)γt(u)=tπ(γ(u))+(1-t)u.
Then γt∈C(RN;RN) is a homotopy of γ0=id with γ1=π∘γ. Moreover, γt|∂Ban=id for all t∈[0,1]. By homotopy invariance and normalization of the degree, we have
(2.13)deg(π∘γ,Ban,0)=deg(id,Ban,0)=1,
which means that 0∈π(γ(Ban)). Thus γ(Ban) intersects the hyperplane H̃T1.
By Lemma2.1, the functional φ is coercive on H̃T1. There is a constant M such that
(2.14)maxx∈Banφ(γ(x))≥infu∈H̃T1φ(u)≥M.
Hence cn≥M. In view of Lemma 2.2, for all large values of n,
(2.15)cn>maxx∈RN,|x|=anφ(x).
For such n, there exists a sequence {γk} in Sn such that
(2.16)maxx∈Banφ(γk(x))→cn,k→∞.
Applying Theorem  4.3 and Corollary  4.3 in [1], there exists a sequence {vk} in HT1 such that
(2.17)φ(vk)→cn,dist(vk,γk(Ban))→0,φ'(vk)→0,
as k→∞.
Now, let us prove that the sequence{vk} is bounded in HT1. For any large enough k,
(2.18)cn≤maxx∈Banφ(γk(x))≤cn+1,
we can find wk∈γk(Ban), such that
(2.19)∥vk-wk∥≤1.
For fixed n, by Lemma 2.3, we can choose m such that bm>an and γk(Ban) cannot intersect the hyperplane Hbm. Let wk=w¯k+w̃k, where w¯k∈RN and w̃k∈H̃T1. Then we have
(2.20)|w¯k|=|1T∫0Twk(t)dt|≤1T∫0T|wk(t)|dt≤∥wk∥∞=maxt∈[0,T]|wk(t)|<bm,
for bm large enough. Besides, by Sobolev's inequality and (1.7), it is obvious that
(2.21)cn+1≥φ(wk)=12∫0T|ẇk(t)|2dt-∫0TF(t,wk(t))dt≥12∫0T|ẇk(t)|2dt-∫0Tf(t)|wk(t)|1+αdt-∫0Tg(t)|wk(t)|dt≥12∫0T|ẇk(t)|2dt-4∫0Tf(t)(|w̅k(t)|1+α+|w̃k(t)|1+α)dt-∫0Tg(t)(|w¯k(t)|+|w̃k(t)|)dt=12∫0T|ẇk(t)|2dt-C1(∫0T|ẇk(t)|2dt)(1+α)/2-C2(∫0T|ẇk(t)|2dt)1/2-C3.
As (|u¯|2+∥u̇(t)∥L2)1/2 is an equivalent norm in HT1, it follows that w̃k(t) is bounded. Hence, wk is bounded. Also {vk} is bounded in HT1.
We assume that(2.22)vk⇀unweaklyinHT1,vk→ununiformlyinC([0,T];RN),
hence
(2.23)〈φ'(vk)-φ'(un),vk-un〉→0,∫0T(∇F(t,vk)-∇F(t,un),vk-un)dt→0
as k→∞. Moreover, an easy computation shows that
(2.24)〈φ'(vk)-φ'(un),vk-un〉=∥v̇k-u̇n∥L22-∫0T(∇F(t,vk)-∇F(t,un),vk-un)dt
so ∥v̇k-u̇n∥L2→0 as n→∞. Then, it is not difficult to obtain ∥vk-un∥→0, as k→∞.
Now we have(2.25)φ'(un)=limk→∞φ'(vk)=0,φ(un)=limk→∞φ(vk)=cn.
Thus, un is critical point and cn is critical value of functional φ. For any γ∈Sn, if an>bm, γ(Ban) intersects the hyperplane Hbm={x∈RN,|u|=bm}⊕H̃T1. It follows that
(2.26)maxx∈Banφ(γ(x))≥infu∈Hbmφ(u).
This inequality and Lemma 2.3 deduce that
(2.27)limn→∞cn=+∞.
The first result of Theorem 1.1 is obtained.
For fixedm, define the subset Pm of HT1 by
(2.28)Pm={u∈HT1:u=u¯+ũ,|u¯|≤bm,ũ∈H̃T1}.
For u∈Pm, we have(2.29)φ(u)=12∫0T|u̇(t)|2dt-∫0TF(t,u(t))dt≥12∫0T|u̇(t)|2dt-∫0Tf(t)|u(t)|1+αdt-∫0Tg(t)|u(t)|dt≥12∫0T|u̇(t)|2dt-4∫0Tf(t)(|w¯k(t)|1+α+|ũ(t)|1+α)dt-∫0Tg(t)(|u¯(t)|+|ũ(t)|)dt=12∫0T|u̇(t)|2dt-C4(∫0T|u̇(t)|2dt)(1+α)/2-C5(∫0T|u̇(t)|2dt)1/2-C6.
It follows that φ is bounded below on Pm.Define
(2.30)μm=infu∈Pmφ(u),
and let {uk} be a minimizing sequence in Pm, that is,
(2.31)φ(uk)→μmask→∞.
From (2.29), {uk} is bounded in HT1. Then, there is a subsequence, we also denoted by {uk}, such that
(2.32)uk⇀um*weaklyinHT1.
The case that Pm is a convex closed subset of HT1 implies that um*∈Pm.As φ is weakly lower semicontinuous, we have
(2.33)μm=limk→∞φ(uk)≥φ(um*).
Since um*∈Pm,
(2.34)μm=φ(um*).
Suppose that um* is in the interior of Pm, then um* is a local minimum of functional φ. In fact, let um*=u¯m*+ũm*. For large m, from Lemmas 2.2 and 2.3, we have |u¯m*|≠bm, which means that um* is not on the boundary of Pm.
Finally, asum* is a minimum of φ in Pm,
(2.35)φ(um*)≤sup|x|=bmφ(x).
It follows from Lemma 2.2 that
(2.36)φ(um*)=-∞.
Therefore Theorem 1.1 is proved.



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