An existence result on periodic solutions of an ordinary p-Laplacian system
Xingyong Zhang
∗, Peixiang Zhou
School of Mathematical Science and Computing Technology, Central South University, Changsha, Hunan 410083, P.R.China
Abstract: A solvability condition of periodic solutions is obtained for an ordinary p-Laplacian system by the Linking Theorem in critical point theory.
Key words: Periodic solution; p-Laplacian system; Critical point; Linking Theorem 2000 Mathematics Subject Classification: 37J45, 34C25, 58E50.
1. Introduction
This paper is concerned with the existence of periodic solutions for the following p- Laplacian system:
−(|u0(t)|p−2u0(t))0 =∇F(t, u(t)), a.e. t ∈[0, T], (1.1) where p ≥ 2, T > 0 and F : R×RN → R with N ≥ 2 is T-periodic in its first variable and satisfies the following assumption:
(A) F(t, x) is measurable in t for all x ∈ RN and continuously differentiable in x for a.e.t ∈[0, T] and there exist a∈C(R+,R+) and b∈L1(0, T;R+) such that
|F(t, x)| ≤a(|x|)b(t), |∇F(t, x)| ≤a(|x|)b(t) for all x∈RN and a.e.t ∈[0, T].
As p = 2, problem (1.1) becomes the second order Hamiltonian system. By using the variational methods, many existence results have been obtained (see [1-5] and the references therein). For the general casep≥2, some papers calling it vector p-Laplacian, there are not so many results using the dual least action principle in variational method.
In [7] an existence result is obtained, which generalizes theorem 3.5 in [6]; in [8] two
∗E-mail: [email protected]
existence results are obtained by the least action principle and Mountain-pass Lemma in nonlinear boundary conditions; in [9] some existence results are got by the Schauder’s fixed point theorem; In paper [10], the authors discuss a general vector valued operator and get some existence results by the topological methods. Besides, in [11,12] problem (1.1) is generalized to differential inclusion systems and some existence results are obtained by the nonsmooth critical point theory.
Recently, for p= 2, Tao and Tang [1] have obtained the following Theorem:
Theorem A Assume that F satisfies (A) and the following conditions:
F(t, x)≥0, ∀(t, x)∈[0, T]×RN,
|x|→0lim
F(t, x)
|x|2 < ω2
2 uniformly for a.e. t ∈[0, T], lim inf
|x|→∞
F(t, x)
|x|2 > ω2
2 uniformly for a.e. t∈[0, T], where ω= 2πT . There exist constantsr >2 and µ > r−2such that
lim sup
|x|→∞
F(t, x)
|x|r <∞ uniformly for a.e. t∈[0, T], lim inf
|x|→∞
¡∇F(t, x), x¢
−2F(t, x)
|x|µ >0 uniformly for a.e. t∈[0, T].
Then there exists a non-constant T-periodic solution of system (1.1).
In this paper, the corresponding conditions in Theorem A are generalized and it is proved that under these conditions, the corresponding energy functional also satisfies (C) condition. Then an existence result for problem (1.1) is obtained by Linking Theorem.
2. Main results
The Sobolev’s Space WT1,p is defined by
WT1,p = {u: [0, T]→RN| u is absolutely continuous, u(0) =u(T) and u0 ∈Lp(0, T;RN)}
and is endowed with the norm kuk=
µZ T
0
|u(t)|pdt+ Z T
0
|u0(t)|pdt
¶1p .
It follows from [6] that WT1,p is reflexive and uniformly convex Banach space. From [13], one can know that a locally uniformly convex Banach space has the Kadec−Klee
property, that is for any sequence {un} satisfying un* u weakly in Banach spaceE and kunk → kuk, one has un →u strongly inE. This property will be used later.
Define functional ϕ onWT1,p by ϕ(u) = 1
p Z T
0
|u0(t)|pdt− Z T
0
F(t, u(t))dt
for all u ∈ WT1,p. It follows from assumption (A) that the functional ϕ is continuously differentiable on WT1,p. Moreover, one has
(ϕ0(u), v) = Z T
0
£¡|u0(t)|p−2u0(t), v0(t)¢
−(∇F(t, u(t)), v(t))¤ dt
for all u, v ∈ WT1,p, where (·,·) and | · | are usual inner product and norm of RN. It is well known that the solutions of problem (1.1) correspond to the critical points of the functionalϕ. By proposition 1.1 in [6], there is a constantC0 >0 such that
kuk∞:= max
t∈[0,T]|u(t)| ≤C0kuk, for every u∈WT1,p. (2.1) Let ˜WT1,p ={u∈WT1,p|RT
0 u(t)dt = 0}.It is easy to know that ˜WT1,p is a subset ofWT1,p and WT1,p=RN ⊕W˜T1,p. It follows from the proof of proposition 1.1 in [6] that
Z T
0
|u(t)|pdt ≤ Tp Z T
0
|u0(t)|pdt, for every u∈W˜T1,p. (Wirtinger’s inequality)(2.2) Hence,
kukp ≤(Tp+ 1) Z T
0
|u0(t)|pdt, for every u∈W˜T1,p. (2.3) The main result of this paper is the following theorem:
Theorem 1 Assume that F satisfies (A) and the following conditions:
F(t, x)≥0, ∀ (t, x)∈[0, T]×RN, (2.4)
|x|→0lim
F(t, x)
|x|p < 1
pTp uniformly for a.e. t∈[0, T], (2.5) lim inf
|x|→∞
F(t, x)
|x|p > ωp
p uniformly for a.e. t∈[0, T], (2.6) where ω= 2πT .There exist constants r > p and µ > r−psuch that
lim sup
|x|→∞
F(t, x)
|x|r <∞ uniformly for a.e. t ∈[0, T], (2.7) lim inf
|x|→∞
¡∇F(t, x), x¢
−pF(t, x)
|x|µ >0 uniformly for a.e. t∈[0, T]. (2.8) Then there exists a non-constant T-periodic solution of system (1.1).
3. Proof of theorem
Lemma 1 Assume that condition (A),(2.7) and (2.8) hold, then the functional ϕ satisfies condition (C), that is{un}has a convergent subsequence inWT1,p, wheneverϕ(un) is bounded andkϕ0(un)k ×(1 +kunk)→0 as n→ ∞.
Proof Let {un} be a sequence in WT1,p such that ϕ(un) is bounded and kϕ0(un)k × (1 +kunk)→0 as n→ ∞. Then there exists a constantM such that
|ϕ(un)| ≤M,kϕ0(un)k(1 +kunk)≤M (3.1) for every n∈N. On one hand, by (2.7), there are constants C1 >0 andδ1 >0 such that F(t, x)≤C1|x|r, for all |x|> δ1 and a.e. t∈[0, T]. (3.2) It follows from (A) that
F(t, x)≤ max
s∈[0,δ1]a(s)b(t), for all |x| ≤δ1 and a.e. t∈[0, T].
So for all x∈RN and a.e. t∈[0, T], one has F(t, x)≤ max
s∈[0,δ1]a(s)b(t) +C1|x|r. (3.3) It follows from (3.1),(3.3) and H¨older’s inequality that
1
pkunkp = ϕ(un) + Z T
0
F(t, un)dt+1 p
Z T
0
|un|pdt
≤ M +C2+C1 Z T
0
|un|rdt+ 1 p
Z T
0
|un|pdt
≤ M +C2+C1 Z T
0
|un|rdt+ 1 pTr−pr
µZ T
0
|un|rdt
¶pr
, (3.4)
where C2 = maxs∈[0,δ1]a(s)RT
0 b(t)dt. On the other hand, by (2.8), there are constants C3 >0 and δ2 >0 such that
(∇F(t, x), x)−pF(t, x)≥C3|x|µ>0, for all |x|> δ2 and a.e.t ∈[0, T].
By (A), one has
¯¯
¯(∇F(t, x), x)−pF(t, x)
¯¯
¯≤C4b(t), for every |x| ≤δ2 and a.e.t∈[0, T], where C4 = (p+δ2) maxs∈[0,δ2]a(s). Hence one can obtain that
(∇F(t, x), x)−pF(t, x)≥C3|x|µ−C3δ2µ−C4b(t)
for all x∈RN and a.e. t ∈[0, T]. Then one has (p+ 1)M ≥ pϕ(un)−(ϕ0(un), un)
= Z T
0
|u0n|pdt−p Z T
0
F(t, un(t))dt− Z T
0
¡|u0n(t)|p−2u0n(t), u0n(t)¢ dt +
Z T
0
¡∇F(t, un(t)), un(t)¢ dt
= Z T
0
¡∇F(t, un(t)), un(t)¢ dt−p
Z T
0
F(t, un(t))dt
= Z T
0
h¡∇F(t, un(t)), un(t)¢
−pF(t, un(t)) i
dt
≥ C3
Z T
0
|un(t)|µdt−T C3δ2µ−C4
Z T
0
b(t)dt.
SoRT
0 |un(t)|µdt is bounded. If µ≥r, by (3.4) and H¨older’s inequality Z T
0
|un(t)|rdt≤Tµ−rµ
³ Z T 0
|un(t)|µdt
´r
µ, one has kunk is bounded. Ifµ≤r, by (2.1), one has
Z T
0
|un(t)|rdt = Z T
0
|un(t)|r−µ|un(t)|µdt
≤ kunkr−µ∞ Z T
0
|un(t)|µdt
≤ C0r−µkunkr−µ Z T
0
|un(t)|µdt.
Thus, by (3.4) and r−µ < p, one can know that kunk is bounded, too. Hence kunk is bounded in WT1,p. Since for the set in a reflexive Banach space, boundedness and weak compactness are equivalent, then there is a subsequence of {un}, again denoted by{un}, such that
un* u weakly in WT1,p. (3.5)
Furthermore, by proposition 1.2 in [6], one has
un→u strongly in C([0, T],RN). (3.6) Note that
(ϕ0(un), u−un) = Z T
0
¡|u0n|p−2u0n, u0−u0n¢ dt−
Z T
0
¡∇F(t, un), u−un¢ dt.
Since {kunk} is bounded and ϕ0(un)→0,
|(ϕ0(un), u−un)| ≤ kϕ0(un)k ku−unk →0, n→ ∞.
From (3.6), {un} is bounded in C([0, T],RN). Then by assumption (A), one has
¯¯
¯ Z T
0
¡∇F(t, un), u−un¢ dt
¯¯
¯ ≤ Z T
0
|∇F(t, un)||u−un|dt
≤ C5 Z T
0
b(t)|u−un|dt
≤ C5ku−unk∞ Z T
0
b(t)dt for some positive constant C5. Thus, it follows from (3.6) that
Z T
0
¡∇F(t, un), u−un¢
dt →0, as n→ ∞.
Hence one has Z T
0
(|u0n|p−2u0n, u0−u0n)dt→0, asn → ∞.
Besides, it is easy to derive from (3.6) that Z T
0
(|un|p−2un, u−un)dt→0, asn → ∞.
Let
ψ(u) = 1
pkukp = 1 p
µZ T
0
|u|pdt+ Z T
0
|u0|pdt
¶ .
Then one has
(ψ0(un), u−un) = Z T
0
(|un|p−2un, u−un)dt+ Z T
0
(|u0n|p−2u0n, u0−u0n)dt and
(ψ0(un), u−un)→0, as n→ ∞. (3.7) Using the H¨older’s inequality, one has
0≤(kunkp−1− kukp−1)(kunk − kuk)≤¡
ψ0(u)−ψ0(un), u−un¢ ,
which together with (3.7) yields kunk → kuk. Because of the uniform convexity of WT1,p and (3.5), it follows that un→u strongly in WT1,p from theKadec−Klee property. The proof is completed.
The Linking Theorem introduced in [14] by Rabinowitz will be used to obtain the critical point ofϕ.
Linking Theorem Let E = E1 ⊕ E2 be a Banach space, where E1 is a finite dimensional subspace of E and E2 = E1⊥. Suppose that ϕ(·) ∈ C1(E,R) satisfies the Palais-Smale condition and the following conditions:
(i) there are constantsρ >0and α such thatϕ|∂Bρ∩E2 ≥α,where Bρ ={u∈E :kukE <
ρ},
(ii) there is a constant d < α and e ∈ E2, kekE = 1, s1 >0, s2 > ρ such that ϕ|∂Q ≤ d where Q={u∈E|u=z+λe, z ∈E1,kzk ≤s1, λ∈[0, s2]}.
Then ϕ possesses a critical value.
Proof of theorem 1 As shown in [15], a deformation lemma can be proved with the weaker condition (C) replacing the usual Palais-Smale condition, and it turns out that the Linking Theorem holds under the condition (C). Let E1 = RN, E2 = ˜WT1,p = {u ∈ WT1,p|RT
0 u(t)dt = 0}. Then, by Lemma 1, one only needs to prove (i) and (ii) in Linking Theorem hold.
By (2.5), there is ε0 >0 such that
|x|→0lim
F(t, x)
|x|p ≤ 1
pTp −2ε0. Thus, there is a constantδ0 ∈(0, δ1) such that
F(t, x)≤¡ 1 pTp −ε0
¢|x|p, for all |x| ≤δ0 and a.e. t∈[0, T]. (3.8)
It follows from (A) that
|F(t, x)| ≤ max
s∈[δ0,δ1]a(s)b(t), for all δ0 ≤ |x| ≤δ1 and a.e. t∈[0, T]. (3.9) Then, by (3.2),(3.8) and (3.9), for all x∈RNand a.e. t ∈[0, T], one has
F(t, x)≤ µ 1
pTp −ε0
¶
|x|p+
³
s∈[δmax0,δ1]a(s)b(t)δ0−r+C1
´
|x|r. (3.10) So based on (2.2),(2.3),(3.10) and (2.1), for everyu∈W˜T1,p, one has
ϕ(u) = 1 p
Z T
0
|u0|pdt− Z T
0
F(t, u)dt
≥ 1 p
Z T
0
|u0|pdt− µ 1
pTp −ε0
¶ Z T
0
|u|pdt− max
s∈[δ0,δ1]a(s) Z T
0
b(t)dt δ0−rkukr∞
−C1kukr−p∞ Z T
0
|u|pdt
≥ 1 p
Z T
0
|u0|pdt− µ 1
pTp −ε0
¶ Tp
Z T
0
|u0|pdt− max
s∈[δ0,δ1]a(s) Z T
0
b(t)dt δ0−rkukr∞
−C1kukr−p∞ Z T
0
|u|pdt
= ε0Tp Z T
0
|u0|pdt− max
s∈[δ0,δ1]a(s) Z T
0
b(t)dt δ0−rkukr∞−C1kukr−p∞ Z T
0
|u|pdt
≥ ε0Tp(Tp + 1)−1kukp−(C1C0r−p+C2δ0−rC0r)kukr.
Hence, there exist constants α >0 and ρ∈(0,1), such that ϕ(u)≥α, for every u∈W˜T1,p and kuk=ρ.
which shows that (i) holds.
Next it will be shown that (ii) also holds . By (2.6), for ε1 = essinf
t∈[0,T]lim inf
|x|→∞
F(t, x)
|x|p −ωp p >0, there exists δ3 > ρT−1p such that
F(t, x)≥ µωp
p +ε1
¶
|x|p, for all |x| ≥δ3 and a.e. t∈[0, T].
Hence, for all x∈RN and a.e. t ∈[0, T], one has F(t, x)≥
µωp p +ε1
¶
|x|p− µωp
p +ε1
¶
δp3. (3.11)
Let e= sinωt
(T+ωpT)1pe1+ cosωt
(T+ωpT)p1e2, e1 = (1,0,0, . . . ,0)∈ RN, e2 = (0,1,0, . . . ,0)∈ RN. By calculation, it is easy to know that |e|= 1
(T+ωpT)1p, |e|˙ = ω
(T+ωpT)1p, kek= 1 ande∈W˜T1,p. Let Q = {x ∈ RN| |x| ≤
³ωp pε1 + 1
´1
pδ3} ⊕ {se| 0 ≤ s ≤
³ωp pε1 + 1
´1
p δ3(T +T ωp)1p} and C6 =
³ωp p +ε1
´
T δp3. It follows from H¨older’s inequality that Z T
0
|x+se|2dt ≤
·Z T
0
¡|x+se|2¢p
2 dt
¸2p
·T1−2p.
Thus one has ωp
Z T
0
|x+se|pdt ≥ ωpT1−p2 µZ T
0
|x+se|2dt
¶p2
= ωpT1−p2 µZ T
0
|x|2dt+ Z T
0
|se|2dt
¶p2
(3.12)
≥ ωpT1−p2 µZ T
0
|se|2dt
¶p2
= T spωp T +ωpT
= Z T
0
|se|˙pdt.
Then for every x+se∈Q, by (3.11) and (3.12), one has ϕ(x+se) = 1
p Z T
0
|se|˙pdt− Z T
0
F(t, x+se)dt
≤ ωp p
Z T
0
|x+se|pdt−(ωp p +ε1)
Z T
0
|x+se|pdt+C6
= −ε1
Z T
0
|x+se|pdt+C6
≤ −ε1T1−p2 ÃZ T
0
|x|2dt+ Z T
0
s2
(T +ωpT)2pdt
!p
2
+C6. (3.13)
For every x+se ∈Q, where|x|=
³ωp pε1 + 1
´1
p δ3, one has
ϕ(x+se) ≤ −ε1T1−p2 µZ T
0
|x|2dt
¶p2
+C6 = 0. (3.14)
For every x+se ∈Q, wheres=
³ωp pε1 + 1
´1
pδ3(T +T ωp)1p, one has
ϕ(x+se)≤ −ε1T1−p2 ÃZ T
0
s2
(T +ωpT)2pdt
!p
2
+C6 = 0. (3.15)
If s= 0, for all x∈RN, by (2.4), one has ϕ(x) =−
Z T
0
F(t, x)dt≤0. (3.16)
Therefore, by (3.14)-(3.16), one has ϕ|∂Q ≤ 0. Let d = 0, s1 =
³ωp pε1 + 1
´1
p δ3 and s2 =
³ωp pε1 + 1
´1
pδ3(T +T ωp)1p > ρ. Thus (ii) is proved. Hence, by Linking Theorem, there exists a non-constant T-periodic solution of system (1.1).
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