On periodic solutions of nonautonomous second order Hamiltonian systems with ( q, p ) -Laplacian

On periodic solutions of nonautonomous second order Hamiltonian systems with ( q, p ) -Laplacian

The first author dedicates this paper to Professor George Dinc˘a on the occasion of his 75th birthday with deep esteem and respect

Daniel Pas

ca

and Zhiyong Wang

1Department of Mathematics and Informatics, University of Oradea University Street 1, 410087 Oradea, Romania

2Department of Mathematics, Nanjing University of Information Science & Technology Nanjing 210044, Jiangsu, People’s Republic of China

Received 7 May 2016, appeared 23 November 2016 Communicated by Petru Jebelean

Abstract. A new existence result is obtained for nonautonomous second order Hamil- tonian systems with(q,p)-Laplacian by using the minimax methods.

Keywords: periodic solutions, Hamiltonian systems with(q,p)-Laplacian, Cerami con- dition; saddle point theorem.

2010 Mathematics Subject Classification: 34C25, 58E50.

1 Introduction

Consider the second-order Hamiltonian systems with(q,p)_-Laplacian



















−^d

dt |u˙₁(t)|^q−2u˙₁(t) =∇_u1F(t,u₁(t),u₂(t)), a.e. t ∈[0,T],

−^d

dt |u˙₂(t)|^p−2u˙₂(t) =∇_u2F(t,u₁(t)_,u₂(t))_{, a.e.} t ∈[_0,T]_, u₁(0)−u₁(T) =u˙₁(0)−u˙₁(T) =0,

u2(0)−u2(T) =u˙2(0)−u˙2(T) =0,

(1.1)

where 1< p,q< +_∞,T>0, and F:[0,T]×_R^N×_R^N →_Rsatisfy the following assumption (A):

• Fis measurable int for each(x₁,x₂)∈ _R^N×_R^N;

• Fis continuously differentiable in(x₁,x₂)for a.e.t ∈[0,T];

BCorresponding author. Email: dpasca@uoradea.ro

• there exista₁,a₂∈C(_R⁺,R⁺)andb∈ L¹(0,T;R⁺)such that

|F(t,x₁,x₂)|, |∇_x1F(t,x₁,x₂)|, |∇_x2F(t,x₁,x₂)| ≤a₁(|x₁|) +a₂(|x₂|)b(t) for all(x₁,x₂)∈_R^N×_R^N and a.e. t∈ [0,T].

When p = q and F(t,x₁,x₂) = F₁(t,x₁), problem (1.1) reduces to the following second order Hamiltonian system:







−^d

dt |u˙(t)|^p−2u˙(t)= ∇F₁(t,u(t)) a.e. t ∈[0,T], u(0)−u(T) =u˙(0)−u˙(T) =0.

(1.2) In the past decades, there are many papers concerning the existence of periodic solutions for problem (1.2) with p = 2 or more general with p > 1 via critical point theory, we refer the reader to [2,4,12–17] and the references therein. Specially, in [14], Tang and Wu estab- lished the existence of periodic solutions for problem (1.2) with p = 2 when potential F was subquadratic. Concretely speaking, they obtained the following theorems.

Theorem 1.1 (Tang and Wu [14]). Suppose that F₁ satisfies assumption (A) and the following conditions:

(_S1) There exists0<µ<_2,R>_{0such that}

(∇F₁(t,x)_,x)≤µF₁(t,x) for all|x| ≥R and a.e. t∈[0,T];

(S₂) F₁(t,x)→+_∞ as|x| →+_∞uniformly for a.e. t∈[0,T]. Then problem(1.2)with p=2has at least one solution.

Theorem 1.2(Tang and Wu [14]). Suppose that F₁satisfies assumption(A),(S₁)and the following conditions:

(S₃) RT

0 F₁(t,x)dt→+_∞as|x| →+_∞.

(S₄) F₁(t,·)is(β,γ)-subconvex withγ>0for a.e. t∈ [0,T], that is, F₁(t,β(x+y))≤ γ(F₁(t,x) +F₁(t,y)) for all x,y∈_R^N.

Then problem(1.1)with p=2has at least one solution.

Inspired by some of our early papers in [6–10], the aim of this paper is to obtain new exis- tence result for system (1.1) by imposing a more general growth conditions on the potentialF.

For the sake of convenience, in the sequel,Hwill denote the space of continuous function space such that, for anyθ ∈ H, there exists constantM >0 such that

(i) θ(t)>0 ∀t ∈_R⁺, (ii) Rt

M 1

sθ(s)ds→+_{∞ as} t→+_∞.

The main result is the following theorem.

Theorem 1.3. Suppose that F satisfies assumption(A)and the following conditions:

(H₁) there existθ(|(x₁,x2)|)∈ Hwith0< ¹

θ(|(x1,x2)|) <r, r:=min(q,p), M₁ >0such that (∇_(x1,x2)F(t,x₁,x₂),(x₁,x₂))≤

r− ¹

θ(|(x₁,x₂)|)

F(t,x₁,x₂) for all|(x₁,x₂)| ≥ M₁ and a.e. t∈[0,T];

(H2) F(t,x₁,x2)≥0as|(x₁,x2)| →+_∞uniformly for a.e. t∈[0,T]; (H₃) RT

0

F(t,x₁,x₂)

θ(|(x1,x2)|)dt→+_∞as|(x₁,x₂)| →+_∞. Then problem(1.1)has at least one solution.

Remark 1.4. Let inf_{|(x1,x2)|≥M θ(|(x}¹

1,x2)|) :=k, wherekis a constant. We point out that

(a) It is clear that the set of hypotheses assumed in Theorem1.3is weaker than Theorem1.1 even if p = q = 2,F(t,x₁,x₂) = F₁(t,x₁). Therefore, Theorem 1.3 generalizes Theorem 1.1completely.

(b) Theorem 1.3 also can be viewed as a partial extension of Theorem 1.2. In fact, on one hand, condition (S₄)in Theorem 1.2 is completely removed, on the other hand, under assumption(H₂), we can easily see that(H₁),(H₃)withp=q=2,F(t,x₁,x₂) =F₁(t,x₁) are equivalent to(_S1)_,(_S3)respectively whenk>0, however,(_H1)is much weaker than (S₁)whenk=0.

(c) There are functions F satisfying our Theorem 1.3 and not satisfying the results in [14].

For example, let

F(t,x₁,x₂) =d(t) ²+|x₁|^q+|x|^p

ln(₂+|x₁|²+|x₂|²)^, ∀(t,x)∈_R×_R^N, where

d(t)_:=

(sin^2πt_T , t ∈[0,T/2], 0, t ∈[T/2,T].

Settingθ(|(x₁,x2)|) =ln(2+|x₁|²+|x2|²), a straightforward computation shows that F satisfies the conditions(H₁)–(H₃)of Theorem1.3, but it does not satisfy the correspond- ing conditions of Theorem1.1, Theorem1.2.

The rest of this paper is organized as follows. In Section 2, we recall some important nota- tions and present some preliminary results which will be used for the proofs of Theorem1.3.

In Section 3, we prove our main result.

2 Preliminaries

For the sake of convenience, in the following we will denote various positive constants as c_i, i = 1, 2, 3, . . . Firstly, we introduce some functional spaces. Let T > 0, 1 < q,p < +_∞ and use | · | to denote the Euclidean norm in R^N. We denote by W_T^1,p the Sobolev space of

functionsu ∈ L^p(0,T;R^N) having a weak derivative ˙u ∈ L^p(0,T;R^N). The norm inW_T^1,p is defined by

kuk

W_T^1,p := _{Z T}

0

(|u(t)|^p+|u˙(t)|^p)dt ¹_p

. Furthermore, we use the spaceW defined by

W :=W_T^1,q×W_T^1,p with the normk(u₁,u₂)k_W :=ku₁k

W_T^1,q+ku₂k

W_T^1,p. It is clear thatWis a reflexive Banach space.

For(u₁,u₂)∈W, let (u¯₁, ¯u₂):= ¹

T _{Z T}

0 u₁(t)dt, Z _T

0 u₂(t)dt

and (u˜₁(t), ˜u₂(t)):= (u₁(t),u₂(t))−(u¯₁, ¯u₂), then one has

ku˜₁k_∞ ≤c₁ku˙₁k_q, ku˜₂k_∞ ≤c₁ku˙₂k_p, (Sobolev’s inequality) ku˜₁k_q≤c₂ku˙₁k_q, ku˜₂k_p ≤c₂ku˙₂k_p (Wirtinger’s inequality) for each (u₁,u₂) ∈ W, where ku₁k_q := RT

0 |u₁(t)|^qdt¹_q

, ku₂k_p := RT

0 |u₂(t)|^pdt¹_p and ku˜_ik_∞:=_max0≤t≤T|u˜_i(_t)|fori=1, 2. Since the embedding ofWintoC(_0,T;R^N)×C(_0,T;R^N) is compact, there exists a constantd>0 such that

k(u₁,u₂)k_∞ ≤dk(u₁,u₂)k_W (2.1) for all(u₁,u₂)∈W.

It follows from assumption (A) that functional ϕonW given by ϕ(u₁,u₂) = ¹

q Z _T

0

|u˙₁(t)|^qdt+ ¹ p

Z _T

0

|u˙₂(t)|^pdt−

Z _T

0 F(t,u₁(t),u₂(t))dt

is continuously differentiable and weakly lower semicontinuous onW (see [7]). Moreover, one has

(ϕ⁰(u₁,u₂),(v₁,v₂)) =

Z _T

0

(|u˙₁|^q−2u˙₁, ˙v₁)dt+

Z _T

0

(|u˙₂|^p−2u˙₂, ˙v₂)dt

−

Z _T

0

(∇_(u1,u2)F(t,u₁,u₂),(v₁,v₂))dt

for all u_i ∈ W_T^1,q,v_i ∈ W_T^1,p,i = 1, 2. It is well known that the solutions of problem (1.1) correspond to the critical points of the functional ϕ.

To prove our main theorem, we need the following auxiliary result.

Proposition 2.1. Suppose that F(t,x₁,x₂)satisfies assumption(A),(H₁)and(H₂), then we have F(t,x₁,x2)≤ ^h(t)

M^r |(x₁,x2)|^rG(|(x₁,x2)|) +h(t) for all x∈_R^N and a.e. t∈ [0,T], where

h(t):= max

|(x₁,x2)|≤M

a₁(|x₁|) +a2(|x2|)b(t),

G(|(x₁,x₂)|):=exp

−

Z _|(x1,x2)|

M

1 tθ(t)^dt

.

Proof. Take f(s):= F(t,sx₁,sx₂). By(H₂), we know that there exists M₂>0 such that f(s)≥0 for all s≥ ^M2

|(x₁,x₂)|^{. (2.2)}

In light of (H₁), one may prove that f⁰(s) = ¹

s

∇_(x1,x2)F(t,s(x₁,x₂)),s(x₁,x₂)

≤ ¹ s

r− ¹

θ(s|(x₁,x₂)|)

F(t,s(x₁,x₂))

= ¹ s

r− ¹

θ(s|(x₁,x₂)|)

f(s) (2.3)

for all s≥ _|(x^M1

1,x₂)|. Then, by (2.2), integrating the inequality (2.3), we derive f(s)≤ ^f

M

|(x₁,x₂)|

|(x₁,x₂)|^r

M^r s^rG(s|(x₁,x₂)|) for all s≥ _|(x^M

1,x₂)|, whereM :=max{M₁,M₂}. Therefore, for|(x₁,x₂)| ≥ M, we obtain F(t,x₁,x₂) = f(1)≤ ^F

t,_|(x^M

1,x2)|(x₁,x₂)|(x₁,x₂)|^r

M^r G(|(x₁,x₂)|). (2.4) Furthermore, by assumption (A), we also have

F

t, M

|(x₁,x₂)|(x₁,x₂)

≤h(t) (2.5)

for all (x₁,x₂)∈_R^N×_R^N and a.e. t∈[0,T]. From (2.4), (2.5) and assumption(A), we obtain F(t,x₁,x2)≤ ^h(_t)

M^r |(x₁,x2)|^rG(|(x₁,x2)|) +h(t) for all x∈_R^N and a.e.t∈ [0,T].

Remark 2.2. Making use of property (ii) ofθ, we know thatG(|(x₁,x₂)|)→ 0 as|(x₁,x₂)| → +_∞. It should be noted that function t^rG(t)is increasing ont. This fact follows easily from the range of ¹_θ, and(t^rG(t))⁰ =t^r−1G(t) r− ¹

θ(t)

>0.

3 Proof of the main result

We start with a compactness condition, which plays a crucial role in establishing our result.

Recall that a sequence {(u_1n,u_2n)} ⊂ W is said to be a (C) sequence of ϕ if ϕ(u_1n,u_2n) _is bounded and (1+k(u_1n,u_2n)k)kϕ⁰(u_1n,u_2n)k → 0 as n → ∞. The functional ϕ satisfies condition(C)if every(C)sequence of ϕhas a convergent subsequence. This condition is due to G. Cerami [3].

Lemma 3.1. Assume that(A),(H₁)and(H₃)hold, then the functionalϕsatisfies condition(C). Proof. Suppose that {(u_1n,u_2n)} ⊂ W is a(C)sequence of ϕ, that is, ϕ(u_1n,u_2n)is bounded and(1+k(u1n,u2n)k)kϕ⁰(u1n,u2n)k →0 asn →+∞. Then there exists a constant L>0 such that

|ϕ(u_1n,u_2n)| ≤L, (₁+k(u_1n,u_2n)k)kϕ⁰(u_1n,u_2n)k ≤L (3.1) for all n ∈ N. In a similar way to the proof of Lemma 8 in [8], we only need to prove {(u_1n,u_2n)}is bounded.

Combining assumption(A)_with(H₁)_{, we have}

−h^˜(t) +∇_(x1,x2)F(t,x₁,x₂),(x₁,x₂)≤

r− ¹

θ(|(x₁,x₂)|)

F(t,x₁,x₂) (3.2) for all (x₁,x₂) ∈ _R^N×_R^N and a.e. t ∈ [0,T], where ˜h(t) = (r+M)h(t) ≥ 0. Taking into account of (3.2) and assumption(A), we conclude from (3.1) that

(r+1)L≥(1+k(u1n,u2n)k)kϕ⁰(u1n,u2n)k −rϕ(u1n,u2n)

≥(ϕ⁰(u_1n,u_2n),(u_1n,u_2n))−rϕ(u_1n,u_2n)

=

1− ^r q

_{Z T}

0

|u˙_1n(t)|^qdt+

1− ^r p

_{Z T}

0

|u˙_2n(t)|^pdt

−

Z _T

0

(∇_(u1n,u2n)F(t,u_1n(t),u2n(t)),(u_1n(t),u2n(t)))dt +r

Z _T

0 F(t,u_1n(t),u_2n(t))dt

≥

1− ^r q

^Z T 0

|u˙_1n(t)|^qdt+1− ^r p

_{Z T}

0

|u˙_2n(t)|^pdt +

Z _T

0

F(t,u_1n(t),u2n(t)) θ(|(u_1n(t),u_2n(t))|)^dt−

Z _T

0

h˜(t)dt

≥

Z _T

0

F(t,u_1n(t),u_2n(t)) θ(|(u_1n(t),u2n(t))|)^dt−

Z _T

0

h˜(t)dt, (3.3)

for alln∈N, taking into account the fact thatr=min(q,p). Hence, we get Z _T

0

F(t,u1n(t),u2n(t))

θ(|(u_1n(t),u_2n(t))|)^dt≤c3 (3.4) for all n ∈ N. In addition, by using the relation (3.1), (2.1), Proposition 2.1, Remark2.2 and Wirtinger’s inequality, one has

L≥ ϕ(u_1n,u_2n) = ¹ q

Z _T

0

|u˙_1n(t)|^qdt+ ¹ p

Z _T

0

|u˙_2n(t)|^pdt−

Z _T

0 F(t,u_1n(t),u_2n(t))dt

≥c₄(ku˙_1nk^r_Lq+ku˙_2nk^r_Lp)−

Z _T

0

h(t)

M^r |(u_1n(t),u_2n(t))|^rG(|(u_1n(t),u_2n(t))|) +h(t)

≥c₅

ku˜_1nk^r

W_T^1,q +ku˜_2nk^r

W_T^1,p

−c₆k(u_1n,u_2n)k^r_∞G(k(u_1n,u_2n)k_∞)−c₇

≥c₈ ku˜_1nk

W_T^1,q+ku˜_2nk

W_T^1,p

r

−c₉k(u_1n,u_2n)k^r_WG(dk(u_1n,u_2n)k_W)−c₇

=c₈k(u˜_1n, ˜u_2n)k_W^r −c₉k(u_1n,u_2n)k^r_WG(dk(u_1n,u_2n)k_W)−c₇ (3.5)

for all n∈_N.

Finally, we claim(u_1n,u_2n)is bounded, otherwise, going if necessary to a subsequence, we can assume thatk(u_1n,u_2n)k →+_∞asn→+_{∞. Put}

(v_1n,v_2n) = (u_1n,u_2n)

k(u_1n,u_2n)k_W = (u¯_1n, ¯u_2n)

k(u_1n,u_2n)k_W + (u˜_1n, ˜u_2n) k(u_1n,u_2n)k_W

= (v¯1n, ¯v2n) + (v˜1n, ˜v2n). (3.6) Then, {(v_1n,v_2n)} is bounded inW and by the compactness of the embedding W = W_T^1,q× W_T^1,p ⊂C(0,T;R^N)×C(0,T;R^N), there is a subsequence, again denoted by{(v_1n,v_2n)}, such that

(v_1n,v_2n)*(v₁,v₂) weakly inW, (3.7) (v_1n,v_2n)→(v₁,v₂) strongly inC(_0,T;R^N)×C(_0,T;R^N)_{. (3.8)} Dividing both sides of (3.5) byk(u_1n,u_2n)k^r_W, by Remark2.2and (3.6), we find that

k(v˜_1n, ˜v_2n)k_W →0 asn→+_∞. (3.9) Moreover, it follows from (3.8) and (3.9) that

(v_1n,v_2n)→(v¯₁, ¯v₂) asn→+_∞, which implies that

(v₁,v₂) = (v¯₁, ¯v₂) _and |(v¯₁, ¯v₂)|^r ≥ |v¯₁|^r+|v¯₂|^r≥c₁₀k(v¯₁, ¯v₂)k^r_W =c₁₀. Consequently,|(u_1n(t),u_2n(t))| →+_∞uniformly for a.e. t∈[0,T]. From(H₃), we get

|(u1n(t),ulim2n(t))|→+_∞ Z _T

0

F(t,u1n(t),u2n(t))

θ(|(u_1n(t),u_2n(t))|)^dt→+_∞,

which contradicts (3.4). Therefore,{(u_1n,u_2n)}is bounded inW, thenϕsatisfies condition(C).

Now, we are ready to prove our main result.

Proof of Theorem1.3. LetWe =We_T^1,q×We_T^1,pbe the subspace ofWgiven byWe :={(u₁,u₂)∈W | (u¯₁, ¯u₂) = (_{0, 0})}_{. Then}W =We ^L(_R^N×_R^N). From Lemma3.1, we obtain thatϕ∈C¹(W,R) satisfies condition (C). As shown in [1], a deformation lemma can be proved with the weaker condition (C) replacing the usual Palais–Smale condition, and it turns out that the saddle point theorem holds true under condition (C). By saddle point theorem (see Theorem 4.6 in [11]), we have only to verify the assertion:

(ϕ₁) ϕ(_u1,u2)→+_{∞ as}k(u1,u2)k →+_∞inWe and (ϕ₂) ϕ(u₁,u₂)→ −_{∞ as}|(u₁,u₂)k →+_∞inR^N×_R^N_.

We first prove(ϕ₁). For(u₁,u₂)∈W, by Propositione 2.1, Remark2.2, (2.1) and Wirtinger’s inequality, we obtain

ϕ(u₁,u2) = ¹ q

Z _T

0

|u˙₁(t)|^qdt+ ¹ p

Z _T

0

|u˙2(t)|^pdt−

Z _T

0 F(t,u₁(t),u2(t))dt

≥c₁₁

ku₁k^q

W_T^1,q+ku₁k^p

W_T^1,p

−

Z _T

0

h(t)

M^r |(u₁,u₂)|^rG(|(u₁,u₂)|) +h(t)

dt

≥c₁₂k(u₁,u₂)k^r_W−c₁₃k(u₁,u₂)k^r_∞G(k(u₁,u₂)k_∞)−c₁₄

≥[c₁₂−c₁₃G(dk(u₁,u2)k_W)]k(u₁,u2)k^r_W−c₁₄. (3.10) Hence,(ϕ₁)holds.

On the other hand, since 0< ¹

θ(|(x1,x2)|) <r, by (H₂)and(H₃), one then arrives at ϕ(u₁,u₂) =−

Z _T

0 F(t,u₁(t),u₂(t))dt

≤ −¹ r

Z _T

0

F(t,u₁(t),u₂(t)) θ(|(u₁(t),u₂(t))|)^dt

→ −_∞ as|(u₁(t),u2(t))| →+_∞inR^N×_R^N, (3.11) which implies(ϕ₂). It follows from the saddle point theorem that Theorem1.3holds.

Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version. The second author is supported by Natural Science Foundation of China (No. 11571176).

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