Infinitely Many Periodic Solutions for Nonautonomous Sublinear Second-Order Hamiltonian Systems

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.

1. Introduction and Main Result

We are interested in the following second order Hamiltonian systems:̈̇̇𝑢(𝑡)+∇𝐹(𝑡,𝑢(𝑡))=0a.e.𝑡∈[0,𝑇],𝑢(0)−𝑢(𝑇)=𝑢(0)−𝑢(𝑇)=0, (1.1) where 𝑇>0 and 𝐹∶[0,𝑇]×𝑅^𝑁→𝑅 satisfies the following assumption.(A)𝐹(𝑡,𝑥) is measurable in 𝑡 for each 𝑥∈𝑅^𝑁 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 |𝐹(𝑡,𝑥)|≤𝑎(|𝑥|)𝑏(𝑡),|∇𝐹(𝑡,𝑥)|≤𝑎(|𝑥|)𝑏(𝑡) (1.2)

for all 𝑥∈𝑅^𝑁 and a.e.𝑡∈[0,𝑇].

Then the corresponding functional 𝜑 on 𝐻¹_𝑇 given by 1𝜑(𝑢)=2^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡−^𝑇₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡 (1.3)

is continuously differentiable and weakly lower semicontinuous on 𝐻¹_𝑇, where 𝐻¹_𝑇={𝑢∶[0,𝑇]→𝑅^𝑁∣𝑢isabsolutelycontinuous,𝑢(0)=𝑢(𝑇),𝑢̇∈𝐿²(0,𝑇;𝑅^𝑁)} (1.4)

is a Hilbert space with the norm defined by ‖𝑢‖=(^𝑇₀|𝑢(𝑡)|²𝑑𝑡+^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡)^1/2 (1.5)

for 𝑢∈𝐻¹_𝑇 (see [1]). Moreover, ⟨𝜑^′(𝑢),𝑣⟩=^𝑇₀(𝑢̇(𝑡),𝑣(̇𝑡))𝑑𝑡−^𝑇₀(∇𝐹(𝑡,𝑢(𝑡)),𝑣(𝑡))𝑑𝑡 (1.6)

for all 𝑢,𝑣∈𝐻¹_𝑇. 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|∇𝐹(𝑡,𝑥)|≤𝑓(𝑡)|𝑥|^𝛼+𝑔(𝑡) (1.7) for all 𝑥∈𝑅^𝑁 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 that 𝐹(𝑡,𝑥) satisfies assumptions (A) and (1.7). Assume that limsup_𝑟→+∞inf_{𝑥∈𝑅𝑁,|𝑥|=𝑟}|𝑥|^−2𝛼𝑇₀𝐹(𝑡,𝑥)𝑑𝑡=+∞,liminf_𝑅→+∞sup_{𝑥∈𝑅𝑁,|𝑥|=𝑅}|𝑥|^−2𝛼𝑇₀𝐹(𝑡,𝑥)𝑑𝑡=−∞. (1.8)(1.9) Then,(i)there exists a sequence of periodic solutions {𝑢_𝑛} which are minimax type critical points of functional 𝜑, and 𝜑(𝑢_𝑛)→+∞, as 𝑛→∞;(ii)there exists another sequence of periodic solutions {𝑢^∗_𝑚} which are local minimum points of functional 𝜑, and 𝜑(𝑢^∗_𝑚)→−∞, as 𝑚→∞.

2. Proof of Theorems

For 𝑢∈𝐻¹_𝑇, let 𝑢=(1/𝑇)∫^𝑇₀𝑢(𝑡)𝑑𝑡 and 𝑢=𝑢−𝑢. Then one has ‖𝑢‖²_∞≤𝑇12^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡(Sobolev'sinequality),_𝑇0|𝑢(𝑡)|²𝑇𝑑𝑡≤²4𝜋^2𝑇₀|𝑢̇(𝑡)|²𝑑𝑡(Wirtinger'sinequality). (2.1)

Lemma 2.1. Let 𝐻¹_𝑇 be the subspace of 𝐻¹_𝑇 given by 𝐻¹_𝑇={𝑢∈𝐻¹_𝑇∣𝑢=0}. (2.2) Suppose that (1.7) holds. Then 𝜑(𝑢)→+∞ (2.3) as ‖𝑢‖→∞ in 𝐻¹_𝑇.

Proof. It follows from (1.7) and Sobolev's inequality that 1𝜑(𝑢)=2^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡−^𝑇₀≥1𝐹(𝑡,𝑢(𝑡))𝑑𝑡2^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡−^𝑇₀𝑓(𝑡)|𝑢(𝑡)|^𝛼+1𝑑𝑡−^𝑇₀≥1𝑔(𝑡)|𝑢(𝑡)|𝑑𝑡2^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡−‖𝑢‖_∞^𝛼+1𝑇₀𝑓(𝑡)𝑑𝑡−‖𝑢‖_∞^𝑇₀≥1𝑔(𝑡)𝑑𝑡2^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡−𝐶₁(^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡)^(𝛼+1)/2−𝐶₂(^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡)^1/2 (2.4) for all 𝑢 in 𝐻¹_𝑇. By Wirtinger's inequality, the norm ‖𝑢‖=(^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡)^1/2 (2.5) is an equivalent norm on 𝐻¹_𝑇. 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 {𝑎_𝑛} such that lim_𝑛→∞𝑎_𝑛=+∞,lim_𝑛→∞sup_{𝑢∈𝑅𝑁,|𝑢|=𝑎𝑛}𝜑(𝑢)=−∞. (2.6)

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 {𝑏_𝑚} such that lim_𝑚→∞𝑏_𝑚=+∞,lim_𝑚→∞inf_{𝑢∈𝐻𝑏𝑚}𝜑(𝑢)=+∞, (2.7) where 𝐻_𝑏𝑚={𝑥∈𝑅^𝑁∶|𝑢|=𝑏_𝑚}⊕𝐻¹_𝑇.

Proof. For any 𝑢∈𝐻_𝑏𝑚, let 𝑢=𝑢+𝑢, where |𝑢|=𝑏_𝑚,𝑢∈𝐻¹_𝑇. It follows from (1.7) and Sobolev's inequality that |^𝑇₀[𝐹(𝑡,𝑢(𝑡))−𝐹(𝑡,𝑢)]𝑑𝑡|=|^𝑇₀¹₀(∇𝐹(𝑡,≤𝑢+𝑠𝑢(𝑡)),𝑢(𝑡))𝑑𝑠𝑑𝑡|_𝑇010𝑓(𝑡)|𝑢+𝑠𝑢(𝑡)|^𝛼|𝑢(𝑡)|𝑑𝑠𝑑𝑡+^𝑇₀¹₀𝑔(𝑡)|𝑢(𝑡)|𝑑𝑠𝑑𝑡≤2(|𝑢|^𝛼+‖𝑢‖^𝛼_∞)‖𝑢‖_∞^𝑇₀𝑓(𝑡)𝑑𝑡+‖𝑢‖_∞^𝑇₀≤3𝑔(𝑡)𝑑𝑡𝑇‖𝑢‖²_∞+𝑇3|𝑢|^2𝛼(^𝑇₀𝑓(𝑡)𝑑𝑡)²+2‖𝑢‖_∞^𝛼+1𝑇₀𝑓(𝑡)𝑑𝑡+‖𝑢‖_∞^𝑇₀≤1𝑔(𝑡)𝑑𝑡4^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡+𝐶₁|𝑢|^2𝛼+𝐶₂(^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡)^(𝛼+1)/2+𝐶₃(^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡)^1/2, (2.8) for all 𝑢∈𝐻¹_𝑇. Hence we have 1𝜑(𝑢)=2^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡−^𝑇₀[𝐹(𝑡,𝑢(𝑡))−𝐹(𝑡,𝑢)]𝑑𝑡−^𝑇₀𝐹(𝑡,≥1𝑢)𝑑𝑡4^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡−𝐶₂(^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡)^(𝛼+1)/2−𝐶₃(^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡)^1/2−|𝑢|^2𝛼(|𝑢|^−2𝛼𝑇₀𝐹(𝑡,𝑢)𝑑𝑡+𝐶₁), (2.9) for all 𝑢∈𝐻¹_𝑇. 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 Theorem 1.1.

Proof of Theorem 1.1. Let 𝐵_𝑎𝑛 be a ball in 𝑅^𝑁 with radius 𝑎_𝑛. Define 𝑆_𝑛={𝛾∈𝐶(𝐵_𝑎𝑛,𝐻¹_𝑇),𝛾|_{𝜕𝐵𝑎𝑛}=id|_{𝜕𝐵𝑎𝑛}},𝑐_𝑛=inf_{𝛾∈𝑆𝑛}[max_{𝑥∈𝐵𝑎𝑛}𝜑(𝛾(𝑥))]. (2.10) We claim that each 𝛾 intersects the hyperplane 𝐻¹_𝑇. In fact, let 𝜋∶𝐻¹_𝑇→𝑅^𝑁 be the projection of 𝐻¹_𝑇 onto 𝑅^𝑁, defined by 1𝜋(𝑢)=𝑇^𝑇₀𝑢(𝑡)𝑑𝑡. (2.11) For 𝑡∈[0,1],𝑢∈𝑅^𝑁, define 𝛾_𝑡(𝑢)=𝑡𝜋(𝛾(𝑢))+(1−𝑡)𝑢. (2.12) Then 𝛾_𝑡∈𝐶(𝑅^𝑁;𝑅^𝑁) is a homotopy of 𝛾₀=id with 𝛾₁=𝜋∘𝛾. Moreover, 𝛾_𝑡|_{𝜕𝐵𝑎𝑛}=id for all 𝑡∈[0,1]. By homotopy invariance and normalization of the degree, we have deg(𝜋∘𝛾,𝐵_𝑎𝑛,0)=deg(id,𝐵_𝑎𝑛,0)=1, (2.13) which means that 0∈𝜋(𝛾(𝐵_𝑎𝑛)). Thus 𝛾(𝐵_𝑎𝑛) intersects the hyperplane 𝐻¹_𝑇.
By Lemma 2.1, the functional 𝜑 is coercive on 𝐻¹_𝑇. There is a constant 𝑀 such that max_{𝑥∈𝐵𝑎𝑛}𝜑(𝛾(𝑥))≥inf_𝑢∈𝐻¹_𝑇𝜑(𝑢)≥𝑀. (2.14) Hence 𝑐_𝑛≥𝑀. In view of Lemma 2.2, for all large values of 𝑛, 𝑐_𝑛>max_{𝑥∈𝑅𝑁,|𝑥|=𝑎𝑛}𝜑(𝑥). (2.15) For such 𝑛, there exists a sequence {𝛾_𝑘} in 𝑆_𝑛 such that max_{𝑥∈𝐵𝑎𝑛}𝜑(𝛾_𝑘(𝑥))→𝑐_𝑛,𝑘→∞. (2.16) Applying Theorem  4.3 and Corollary  4.3 in [1], there exists a sequence {𝑣_𝑘} in 𝐻¹_𝑇 such that 𝜑(𝑣_𝑘)→𝑐_𝑛,dist(𝑣_𝑘,𝛾_𝑘(𝐵_𝑎𝑛𝜑))→0,_′(𝑣_𝑘)→0, (2.17) as 𝑘→∞.
Now, let us prove that the sequence {𝑣_𝑘} is bounded in 𝐻¹_𝑇. For any large enough 𝑘, 𝑐_𝑛≤max_{𝑥∈𝐵𝑎𝑛}𝜑(𝛾_𝑘(𝑥))≤𝑐_𝑛+1, (2.18) we can find 𝑤_𝑘∈𝛾_𝑘(𝐵_𝑎𝑛), such that ‖𝑣_𝑘−𝑤_𝑘‖≤1. (2.19) For fixed 𝑛, by Lemma 2.3, we can choose 𝑚 such that 𝑏_𝑚>𝑎_𝑛 and 𝛾_𝑘(𝐵_𝑎𝑛) cannot intersect the hyperplane 𝐻_𝑏𝑚. Let 𝑤_𝑘=𝑤_𝑘+𝑤_𝑘, where 𝑤_𝑘∈𝑅^𝑁 and 𝑤_𝑘∈𝐻¹_𝑇. Then we have |𝑤_𝑘1|=|𝑇^𝑇₀𝑤_𝑘1(𝑡)𝑑𝑡|≤𝑇^𝑇₀|𝑤_𝑘(𝑡)|𝑑𝑡≤‖𝑤_𝑘‖_∞=max_{𝑡∈[0,𝑇]}|𝑤_𝑘(𝑡)|<𝑏_𝑚, (2.20) for 𝑏_𝑚 large enough. Besides, by Sobolev's inequality and (1.7), it is obvious that 𝑐_𝑛+1≥𝜑(𝑤_𝑘1)=2^𝑇₀|𝑤̇_𝑘(𝑡)|²𝑑𝑡−^𝑇₀𝐹(𝑡,𝑤_𝑘≥1(𝑡))𝑑𝑡2^𝑇₀|𝑤̇_𝑘(𝑡)|²𝑑𝑡−^𝑇₀𝑓(𝑡)|𝑤_𝑘(𝑡)|^1+𝛼𝑑𝑡−^𝑇₀𝑔(𝑡)|𝑤_𝑘≥1(𝑡)|𝑑𝑡2^𝑇₀|𝑤̇_𝑘(𝑡)|²𝑑𝑡−4^𝑇₀𝑓(𝑡)(|𝑤_𝑘(𝑡)|^1+𝛼+|𝑤_𝑘(𝑡)|^1+𝛼−)𝑑𝑡_𝑇0𝑔(𝑡)(|𝑤_𝑘(𝑡)|+|𝑤_k=1(𝑡)|)𝑑𝑡2^𝑇₀|𝑤̇_𝑘(𝑡)|²𝑑𝑡−𝐶₁(^𝑇₀|𝑤̇_𝑘(𝑡)|²𝑑𝑡)^(1+𝛼)/2−𝐶₂(^𝑇₀|𝑤̇_𝑘(𝑡)|²𝑑𝑡)^1/2−𝐶₃. (2.21) As (|𝑢|²+‖𝑢̇(𝑡)‖_𝐿²)^1/2 is an equivalent norm in 𝐻¹_𝑇, it follows that 𝑤_𝑘(𝑡) is bounded. Hence, 𝑤_𝑘 is bounded. Also {𝑣_𝑘} is bounded in 𝐻¹_𝑇.
We assume that 𝑣_𝑘⇀𝑢_𝑛weaklyin𝐻¹_𝑇,𝑣_𝑘→𝑢_𝑛uniformlyin𝐶([0,𝑇];𝑅^𝑁), (2.22) hence ⟨𝜑^′(𝑣_𝑘)−𝜑^′(𝑢_𝑛),𝑣_𝑘−𝑢_𝑛∫⟩→0,_𝑇0(∇𝐹(𝑡,𝑣_𝑘)−∇𝐹(𝑡,𝑢_𝑛),𝑣_𝑘−𝑢_𝑛)𝑑𝑡→0 (2.23) as 𝑘→∞. Moreover, an easy computation shows that ⟨𝜑^′(𝑣_𝑘)−𝜑^′(𝑢_𝑛),𝑣_𝑘−𝑢_𝑛⟩=‖𝑣̇_𝑘−𝑢̇_𝑛‖²_𝐿2−^𝑇₀(∇𝐹(𝑡,𝑣_𝑘)−∇𝐹(𝑡,𝑢_𝑛),𝑣_𝑘−𝑢_𝑛)𝑑𝑡 (2.24) so ‖𝑣̇_𝑘−𝑢̇_𝑛‖_𝐿2→0 as 𝑛→∞. Then, it is not difficult to obtain ‖𝑣_𝑘−𝑢_𝑛‖→0, as 𝑘→∞.
Now we have 𝜑^′(𝑢_𝑛)=lim_𝑘→∞𝜑^′(𝑣_𝑘)=0,𝜑(𝑢_𝑛)=lim_𝑘→∞𝜑(𝑣_𝑘)=𝑐_𝑛. (2.25) Thus, 𝑢_𝑛 is critical point and 𝑐_𝑛 is critical value of functional 𝜑. For any 𝛾∈𝑆_𝑛, if 𝑎_𝑛>𝑏_𝑚, 𝛾(𝐵_𝑎𝑛) intersects the hyperplane 𝐻_𝑏𝑚={𝑥∈𝑅^𝑁,|𝑢|=𝑏_𝑚}⊕𝐻¹_𝑇. It follows that max_{𝑥∈𝐵𝑎𝑛}𝜑(𝛾(𝑥))≥inf_{𝑢∈𝐻𝑏𝑚}𝜑(𝑢). (2.26) This inequality and Lemma 2.3 deduce that lim_𝑛→∞𝑐_𝑛=+∞. (2.27) The first result of Theorem 1.1 is obtained.
For fixed 𝑚, define the subset 𝑃_𝑚 of 𝐻¹_𝑇 by 𝑃_𝑚={𝑢∈𝐻¹_𝑇∶𝑢=𝑢+𝑢,|𝑢|≤𝑏_𝑚,𝑢∈𝐻¹_𝑇}. (2.28) For 𝑢∈𝑃_𝑚, we have1𝜑(𝑢)=2^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡−^𝑇₀≥1𝐹(𝑡,𝑢(𝑡))𝑑𝑡2^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡−^𝑇₀𝑓(𝑡)|𝑢(𝑡)|^1+𝛼𝑑𝑡−^𝑇₀≥1𝑔(𝑡)|𝑢(𝑡)|𝑑𝑡2^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡−4^𝑇₀𝑓(𝑡)(|𝑤_𝑘(𝑡)|^1+𝛼+|𝑢(𝑡)|^1+𝛼−)𝑑𝑡_𝑇0𝑔(𝑡)(|=1𝑢(𝑡)|+|𝑢(𝑡)|)𝑑𝑡2^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡−𝐶₄(^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡)^(1+𝛼)/2−𝐶₅(^𝑇₀|𝑢̇(𝑡)|²𝑑𝑡)^1/2−𝐶₆. (2.29) It follows that 𝜑 is bounded below on 𝑃_𝑚.Define 𝜇_𝑚=inf_{𝑢∈𝑃𝑚}𝜑(𝑢), (2.30) and let {𝑢_𝑘} be a minimizing sequence in 𝑃_𝑚, that is, 𝜑(𝑢_𝑘)→𝜇_𝑚as𝑘→∞. (2.31) From (2.29), {𝑢_𝑘} is bounded in 𝐻¹_𝑇. Then, there is a subsequence, we also denoted by {𝑢_𝑘}, such that 𝑢_𝑘⇀𝑢^∗_𝑚weaklyin𝐻¹_𝑇. (2.32) The case that 𝑃_𝑚 is a convex closed subset of 𝐻¹_𝑇 implies that 𝑢^∗_𝑚∈𝑃_𝑚.As 𝜑 is weakly lower semicontinuous, we have 𝜇_𝑚=lim_𝑘→∞𝜑(𝑢_𝑘)≥𝜑(𝑢^∗_𝑚). (2.33) Since 𝑢^∗_𝑚∈𝑃_𝑚, 𝜇_𝑚=𝜑(𝑢^∗_𝑚). (2.34) Suppose that 𝑢^∗_𝑚 is in the interior of 𝑃_𝑚, then 𝑢^∗_𝑚 is a local minimum of functional 𝜑. In fact, let 𝑢^∗_𝑚=𝑢^∗_𝑚+𝑢^∗_𝑚. For large 𝑚, from Lemmas 2.2 and 2.3, we have |𝑢^∗_𝑚|≠𝑏_𝑚, which means that 𝑢^∗_𝑚 is not on the boundary of 𝑃_𝑚.
Finally, as 𝑢^∗_𝑚 is a minimum of 𝜑 in 𝑃_𝑚, 𝜑(𝑢^∗_𝑚)≤sup_{|𝑥|=𝑏𝑚}𝜑(𝑥). (2.35) It follows from Lemma 2.2 that 𝜑(𝑢^∗_𝑚)=−∞. (2.36) Therefore Theorem 1.1 is proved.

Acknowledgments

This work is supported by NNSF of China under Grant 10771173 and Natural Science Foundation of Education of Guizhou Province under Grant 2008067.