# Existence of Subharmonic Solutions for a Class of Second-Orderp-Laplacian Systems with Impulsive Effects

Authors: Wen-Zhen Gong; Qiongfen Zhang; X. H. Tang
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

By using minimax methods in critical point theory, a new existence theorem of infinitely many periodic solutions is obtained for a class of second-orderp-Laplacian systems with impulsive effects. Our result generalizes many known works in the literature.


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## Body

## 1. Introduction

Consider the followingp-Laplacian system with impulsive effects: (1.1)ddt(|u̇(t)|p-2u̇(t))-L(t)|u(t)|p-2u(t)+∇F(t,u(t))=0,a.e.t∈R,u(0)-u(T)=u̇(0)-u̇(T)=0,Δ(|u̇(tj)|p-2u̇(tj))=|u̇(tj+)|p-2u̇(tj+)-|u̇(tj-)|p-2u̇(tj-)=∇Ij(u(tj)),j=1,2,…,m,
where p>1, T>0, 0=t0<t1<t2<⋯<tm<tm+1=T, and ∇Ij:ℝN→ℝN(j=1,2,…,m) are continuous and F:ℝ×ℝN→ℝ is T-periodic in t for all u∈ℝN, ∇F(t,u) is the gradient of F(t,u) with respect to u. L∈C(ℝ,ℝN×N) is a T-periodic positive definite symmetric matrix.Throughout this paper, we always assume the following condition holds.(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+), b∈L1([0,T];ℝ+) such that
(1.2)|F(t,x)|≤a(|x|)b(t),|∇F(t,x)|≤a(|x|)b(t)
for all x∈ℝN and a.e. t∈[0,T].For the sake of convenience, in the sequel, we defineB={1,2,…,m}.Whenp=2, ∇Ij≡0, j∈B, problem (1.1) becomes the following second-order Hamiltonian system: (1.3)ü(t)-L(t)u(t)+∇F(t,u(t))=0,a.e.t∈R.There are many papers concerning the existence of periodic solutions or homoclinic solutions for problem (1.3) by minimax methods. Here for identifying a few, we only mention [1–8].For∇Ij≠0, j∈B, problem (1.1) involves impulsive effects. Impulsive differential equations are suitable for the mathematical simulation of evolutionary processes in which the parameters undergo relatively long periods of smooth variation followed by a short-term rapid change (that is jumps) in their values. Since these processes are subject to short-term perturbations whose duration is negligible in comparison with the duration of the processes, it is natural to suppose that these perturbations act instantaneously, that is, in the form of impulse. Processes of this type are often investigated in various fields of science and technology, for example, many biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics and frequency modulated systems, and so on. For more details of impulsive differential equations, we refer the readers to the books [9, 10].There are many methods for finding periodic solutions of impulsive differential equations, such as the monotone-iterative technique, a numerical-analytical method, the method of upper and lower solutions, and the method of bilateral approximations. For more information about periodic solutions of impulsive differential equations, one can refer to the papers [11–18]. However, there are few papers [19–25] concerning periodic solutions of impulsive differential equations by variational methods. So it is a novel method to employ variational methods to investigate the existence of periodic solutions for impulsive differential equations.Motivated by the above papers, we study the existence of subharmonic solutions for problem (1.1) by applying minimax methods in critical point theory. Our result is new, which seems not to be found in the literature.Throughout this paper, letq∈(1,+∞) satisfy 1/p+1/q=1.


## 2. Preliminaries

In this section, we recall some basic facts which will be used in the proofs of our main results. In order to apply the critical point theory, we construct a variational structure. With this variational structure, we can reduce the problem of finding solutions of problem (1.1) to that of seeking the critical points of the corresponding functional.Letk be a positive integer and WkT1,p the Sobolev space defined by(2.1)WkT1,p={u:R→RN∣uis  absolutely  continuous,u(t)=u(t+kT),u̇∈Lp([0,kT];RN)}
with the norm (2.2)‖u‖=(∫0kT|u(t)|pdt+∫0kT|u̇(t)|pdt)1/p.Takev∈WkT1,p and multiply the two sides of the equality(2.3)ddt(|u̇(t)|p-2u̇(t))-L(t)|u(t)|p-2u(t)+∇F(t,u(t))=0
by v and integrate from 0 to kT; we have(2.4)∫0kT((|u̇(t)|p-2u̇(t))′,v(t))dt=∫0kT(L(t)|u(t)|p-2u(t),v(t))dt-∫0kT(∇F(t,u(t)),v(t))dt.
Moreover, by u̇(0)=u̇(T), one has(2.5)∫0kT((|u̇(t)|p-2u̇(t))′,v(t))dt=k∫0T((|u̇(t)|p-2u̇(t))′,v(t))dt=k∑j=0m∫tjtj+1((|u̇(t)|p-2u̇(t))′,v(t))dt=k∑j=0m[|u̇(tj+1-)|p-2u̇(tj+1-)v(tj+1-)-|u̇(tj+)|p-2u̇(tj+)v(tj+)-∫tjtj+1(|u̇(t)|p-2u̇(t),v̇(t))dt]=k∑j=0m(|u̇(tj+1-)|p-2u̇(tj+1-)v(tj+1-)-|u̇(tj+)|p-2u̇(tj+)v(tj+))-∫0kT(|u̇(t)|p-2u̇(t),v̇(t))dt=k|u̇(T)|p-2u̇(T)v(T)-k|u̇(0)|p-2u̇(0)v(0)-k∑j=1m∇Ij(u(tj))v(tj)-∫0kT(|u̇(t)|p-2u̇(t),v̇(t))dt=-k∑j=1m∇Ij(u(tj))v(tj)-∫0kT(|u̇(t)|p-2u̇(t),v̇(t))dt.
Together with (2.4), we get (2.6)∫0kT(|u̇(t)|p-2u̇(t),v̇(t))dt+k∑j=1m∇Ij(u(tj))v(tj)+∫0kT(L(t)|u(t)|p-2u(t),v(t))dt=∫0kT(∇F(t,u(t)),v(t))dt.Definition 2.1.
We say that a functionu∈WkT1,p is a weak solution of problem (1.1) if the identity
(2.7)∫0kT(|u̇(t)|p-2u̇(t),v̇(t))dt+k∑j=1m∇Ij(u(tj))v(tj)+∫0kT(L(t)|u(t)|p-2u(t),v(t))dt=∫0kT(∇F(t,u(t)),v(t))dt
holds for any v∈WkT1,p.Define the functionalϕk on WkT1,p by (2.8)ϕk(u)=1p∫0kT[|u̇(t)|p+(L(t)|u(t)|p-2u(t),u(t))]dt-∫0kTF(t,u(t))dt+k∑j=1mIj(u(tj))=φk(u)+ψk(u),u∈WkT1,p,
where (2.9)φk(u)=1p∫0kT[|u̇(t)|p+(L(t)|u(t)|p-2u(t),u(t))]dt-∫0kTF(t,u(t))dt,ψk(u)=k∑j=1mIj(u(tj)).It follows from assumption (A) that the functionalφk is continuously differentiable on WkT1,p and(2.10)〈φk′(u),v〉=∫0kT[(|u̇(t)|p-2u̇(t),v̇(t))+(L(t)|u(t)|p-2u(t),v(t))-(∇F(t,u(t)),v(t))]dt
for u,v∈WkT1,p. By the continuity of ∇Ij, j∈B, one has that ψk∈(WkT1,p,ℝ). Hence, ϕk(u)∈(WkT1,p,ℝ). For any v∈WkT1,p, we have (2.11)〈ϕk′(u),v〉=∫0kT[(|u̇(t)|p-2u̇(t),v̇(t))+(L(t)|u(t)|p-2u(t),v(t))-(∇F(t,u(t)),v(t))]dt+k∑j=1m∇Ij(u(tj))v(tj).By Definition2.1, the weak solutions of problem (1.1) correspond to the critical points of the functional ϕk.Foru∈WkT1,p, let u̅=(1/kT)∫0kTu(t)dt and ũ(t)=u(t)-u̅; then it follows from Proposition  1.1 in [26] that (2.12)‖u‖∞≔max⁡t∈[0,kT]⁡|u(t)|≤((kT)-1/p+(kT)1/q)‖u‖=dk‖u‖,
where dk=(kT)-1/p+(kT)1/q, and if (1/kT)∫0kTu(t)dt=0, then (2.13)‖ũ‖∞≔max⁡t∈[0,kT]⁡|ũ(t)|≤(kT)1/q‖u̇‖Lp,(2.14)‖ũ‖Lpp≤(kT)p‖u̇‖Lpp,
where 1/p+1/q=1. Let W̃kT1,p={u∈WkT1,p∣u̅=0}; then WkT1,p=W̃kT1,p⊕ℝN. We will use the following lemma to prove our main results.Lemma 2.2 (see [27]).
LetE be a real Banach space with  E=X1⊕X2, where  X1 is finite dimensional. Suppose that  φ∈C1(E,ℝ) satisfies the (PS) condition, and (a)
there exist constantsρ,  α>0 such that  φ|∂Bρ⋂X2≥α, where  Bρ:={u∈E∣∥u∥≤ρ}, and ∂Bρ denotes the boundary of Bρ;(b)
there exists ane∈∂B1⋂X2 and  L>ρ such that if  Q≡(B̅L⋂X1)⊕{re∣0≤r≤L},  then  φ|∂Q≤0.
Thenφ possesses a critical value  c≥α which can be characterized as  c=infh∈Γmaxu∈Qφ(h(u)), where  Γ={h∈C(Q̅,E)∣h=idon∂Q}.It is well known that a deformation lemma can be proved with the weaker condition (C) replacing the usual (PS) condition. So Lemma2.2 holds true under condition (C).


## 3. Main Result and Proof

Theorem 3.1.
Assume that  (A)  holds andF,Ij satisfy the following conditions: (I1)
there existscj>0 such that
(3.1)0≤Ij(x)≤cjk|x|p,j∈B,∀x∈RN;(I2)
for anyj∈B,
(3.2)∇Ij(x)x≤pIj(x),∀x∈RN;(H1)
∫0TF(t,x)dt≥0, for all x∈ℝN;(H2)
lim|x|→0(F(t,x)/|x|p)=0 uniformly for a.e.  t∈[0,T];(H3)
lim|x|→∞(F(t,x)/|x|p)=+∞ uniformly for a.e.  t∈[0,T];(H4)
there exists a positive constantM such that  limsup|x|→∞(F(t,x)/|x|r)≤M uniformly for  a.e. t∈[0,T];(H5)
there existsM1>0 such that  liminf|x|→∞((∇F(t,x),x)-pF(t,x))/|x|μ≥M1 uniformly for a.e.  t∈[0,T],
wherer>p and  μ>r-p. Then problem (1.1) has a sequence of distinct periodic solutions with period  kjT satisfying  kj∈ℕ and  kj→∞ as  j→∞.Remark 3.2.
As far as we know, there is no paper considering subharmonic solutions of impulsive differential equations. Our result is new.Proof.
The proof is divided into three steps. In the following,Ci(i=1,…) denote different positive constants.Step 1.
The functionalϕk satisfies condition (C). Let {un}⊂WkT1,p satisfying (1+∥un∥)∥ϕk′(un)∥→0 as n→∞ and ϕk(un) is bounded; then, there exists a constant C1 such that
(3.3)|ϕk(un)|≤C1,(1+‖un‖)‖ϕk′(un)‖≤C1.
From (H4), there exists M2>0 such that
(3.4)F(t,x)≤M|x|r∀|x|≥M2,a.e.t∈[0,T].
By assumption (A), for |x|≤M2, there exists C2=max|x|≤M2a(|x|)>0 such that
(3.5)|F(t,x)|≤C2b(t),
which together with (3.4) implies that
(3.6)F(t,x)≤M|x|r+C2b(t),∀x∈RN,a.e.t∈[0,T].
By (3.3) and (3.6), we have
(3.7)ϕk(un)+∫0kTF(t,un)dt≤C1+∫0kT(M|un(t)|r+C2b(t))dt=C1+C2k‖b‖L1+M∫0kT|un(t)|rdt=C3+M∫0kT|un(t)|rdt.
Since L(t) is continuous T-periodic positive definite symmetric matrix on [0,T], there exist constants c1,c2>0 such that
(3.8)c1|x|p≤(L(t)|x|p-2x,x)≤c2|x|p,∀x∈RN.
It follows from (3.8) and (I1) that
(3.9)ϕk(un)+∫0kTF(t,un)dt=1p∫0kT[|u̇n(t)|p+(L(t)|un(t)|p-2un(t),un(t))]dt+k∑j=1mIj(u(tj))≥1p∫0kT[|u̇n(t)|p+c1|un(t)|p]dt≥min⁡⁡{1p,c1p}‖un‖p=C4‖un‖p.
By (3.7) and (3.9), we get
(3.10)C4‖un‖p≤C3+M∫0kT|un(t)|rdt.
From (H5), there exists M3>0 such that
(3.11)(∇F(t,x),x)-pF(t,x)≥M1|x|μfor|x|≥M3,a.e.t∈[0,T].
By assumption (A), for |x|≤M3, there exists C5=max|x|≤M3a(|x|)>0 such that
(3.12)|(∇F(t,x),x)-pF(t,x)|≤C5(p+M3)b(t).
Thus, from (3.11) and (3.12), we have
(3.13)(∇F(t,x),x)-pF(t,x)≥M1|x|μ-M1M3μ-C5(p+M3)b(t)forx∈RN,a.e.t∈[0,T],
which together with (3.3) and (I2) implies that
(3.14)(p+1)C1≥pϕk(un)-〈ϕk′(un),un〉=∫0kT[(∇F(t,un),un)-pF(t,un)]dt+pk∑j=1mIj(un(tj))-k∑j=1m∇Ij(un(tj))un(tj)≥M1∫0kT|un(t)|μdt-C5(p+M3)∫0kTb(t)dt-M1M3μkT=M1∫0kT|un(t)|μdt-C6.
Hence, ∫0kT|un(t)|μdt is bounded. If μ>r, we have
(3.15)∫0kT|un(t)|rdt≤(kT)(μ-r)/μ(∫0kT|un(t)|μdt)r/μ,
which together with (3.10) implies that ∥un∥ is bounded. If μ≤r, then from (2.12), we get
(3.16)∫0kT|un(t)|rdt≤‖un‖∞r-μ(∫0kT|un(t)|μdt)r/μ≤dkr-μ‖un‖r-μ(∫0kT|un(t)|μdt)r/μ.
Since μ>r-p, it follows from (3.10) that ∥un∥ is bounded too. Therefore, ∥un∥ is bounded in WkT1,p. Hence, there exists a subsequence, still denoted by {un}, such that
(3.17)un⇀u0weakly  inWkT1,p,(3.18)un⟶u0strongly inC([0,kT];RN),(3.19)un⟶u0strongly inLp([0,kT];RN).
From (2.11), we have
(3.20)〈ϕk′(un),un-u0〉=∫0kT[(|u̇n(t)|p-2u̇n(t),u̇n(t)-u̇0(t))+(L(t)|un(t)|p-2un(t),un(t)-u0(t))]dt-∫0kT(∇F(t,un(t)),un(t)-u0(t))dt+k∑j=1m(∇Ij(un(tj)),un(tj)-u0(tj)).
From (3.3) and (3.18), we have
(3.21)|〈ϕk′(un),un-u0〉|≤‖ϕk′(un)‖‖un-u0‖⟶0asn⟶∞.
By (3.8), we know that c1≤∥L∥≤c2, which together with the boundedness of {un} and (3.19) implies that
(3.22)∫0kT(L(t)|un(t)|p-2un(t),un(t)-u0(t))dt≤‖L‖‖un‖Lpp-1‖un-u0‖Lp⟶0asn⟶∞.
From the boundedness of {un}, the continuity of ∇Ij, and (3.18), we have
(3.23)∑j=1m(∇Ij(un(tj)),un(tj)-u0(tj))⟶0asn⟶∞.
It follows from (A), (3.18) and the boundedness of {un} that
(3.24)∫0kT(∇F(t,un(t)),un(t)-u0(t))dt⟶0asn⟶∞,
which together with (3.20), (3.21), (3.22), and (3.23) implies that
(3.25)∫0kT(|u̇n(t)|p-2u̇n(t),u̇n(t)-u̇0(t))dt⟶0asn⟶∞.
It is easy to see from the boundedness of {un} and (3.18) that
(3.26)∫0kT(|un(t)|p-2un(t),un(t)-u0(t))dt⟶0asn⟶∞.
Let f(u)=(1/p)(∫0kT|u(t)|pdt+∫0kT|u̇(t)|pdt). Then, we have
(3.27)〈f′(un),un-u0〉=∫0kT(|u̇n(t)|p-2u̇n(t),u̇n(t)-u̇0(t))dt+∫0kT(|un(t)|p-2un(t),un(t)-u0(t))dt,(3.28)〈f′(u0),un-u0〉=∫0kT(|u̇0(t)|p-2u̇0(t),u̇n(t)-u̇0(t))dt+∫0kT(|u0(t)|p-2u0(t),un(t)-u0(t))dt.
It follows from (3.25) and (3.26) that
(3.29)〈f′(un),un-u0〉⟶0asn⟶∞.
From (3.17), we get
(3.30)〈f′(u0),un-u0〉⟶0asn⟶∞.
By (3.27), (3.28), and Hölder's inequality, we have
(3.31)〈f′(un)-f′(u0),un-u0〉=∫0kT(|u̇n(t)|p-2u̇n(t),u̇n(t)-u̇0(t))dt+∫0kT(|un(t)|p-2un(t),un(t)-u0(t))dt-∫0kT(|u̇0(t)|p-2u̇0(t),u̇n(t)-u̇0(t))dt-∫0kT(|u0(t)|p-2u0(t),un(t)-u0(t))dt=‖un‖p+‖u0‖p-∫0kT(|u̇n(t)|p-2u̇n(t),u̇0(t))dt-∫0kT(|un(t)|p-2un(t),u0(t))dt-∫0kT(|u̇0(t)|p-2u̇0(t),u̇n(t))dt-∫0kT(|u0(t)|p-2u0(t),un(t))dt≥‖un‖p+‖u0‖p-(‖un‖Lpp-1‖u0‖Lp+‖u̇n‖Lpp-1‖u̇0‖Lp)-(‖u0‖Lpp-1‖un‖Lp+‖u̇0‖Lpp-1‖u̇n‖Lp)≥‖un‖p+‖u0‖p-(‖un‖Lpp+‖u̇n‖Lpp)(p-1)/p(‖u0‖Lpp+‖u̇0‖Lpp)1/p-(‖u0‖Lpp+‖u̇0‖Lpp)(p-1)/p(‖un‖Lpp+‖u̇n‖Lpp)1/p=‖un‖p+‖u0‖p-(‖un‖p-1‖u0‖+‖u0‖p-1‖un‖)=(‖un‖p-1-‖u0‖p-1)(‖un‖-‖u0‖).
Hence, from (3.29) and (3.30), we obtain
(3.32)0≤(‖un‖p-1-‖u0‖p-1)(‖un‖-‖u0‖)≤〈f′(un)-f′(u0),un-u0〉⟶0asn⟶∞.
That is, ∥un∥→∥u0∥ as n→∞. Since WkT1,p has the Kadec-Klee property, we have un→u0 in WkT1,p. Therefore, the functional ϕk satisfies condition (C).Step 2.
From (H2), for any smallε=ε(k)>0, there exists small enough δ>0 such that
(3.33)F(t,u)≤ε|u|pfor|u|≤δ,a.e.t∈[0,kT].
For u∈W̃kT1,p and ∥u∥p=ρkp=δp/(kT)p/q, it follows from (2.13) that
(3.34)‖u‖∞p≤(kT)p/q‖u̇‖Lpp≤(kT)p/q‖u‖p=δp,
which implies that |u(t)|≤δ. Then from (I1), (3.8), and (3.33), we have
(3.35)φk(u)=1p∫0kT|u̇(t)|pdt+1p∫0kT(L(t)|u(t)|p-2u(t),u(t))dt-∫0kTF(t,u)dt+k∑j=1mIj(u(tj))≥1p∫0kT|u̇(t)|pdt+1p∫0kTc1|u(t)|pdt-∫0kTε|u(t)|pdt≥min⁡⁡{1p,c1p}‖u‖p-kTεδp=C4‖u‖p-kTεδp.
Let ε=ε(k)∈(0,C4/2(kT)p); then from (3.24), we have
(3.36)φk(u)≥C4ρkp-kTεδp≥C42ρkp≡α>0
for all u∈W̃T1,p and ∥u∥=ρk. This implies that condition (a) of Lemma 2.2 holds.Step 3.
Letc=max{cj}, j∈B. Choose C7>(c2/p)+(mc/T); then from (H3), there exists M4>0 such that
(3.37)F(t,x)≥C7|x|p,|x|≥M4,a.e.t∈[0,T].
By assumption (A), for |x|≤M4, there exists C8=max|x|≤M4a(|x|)>0 such that
(3.38)|F(t,x)|≤C8b(t),a.e.t∈[0,T],
which together with (3.37) implies that
(3.39)F(t,x)≥C7|x|p-C8b(t),∀x∈RN,a.e.t∈[0,T].
Thus, from (H1), (I1), (3.8), and (3.39), we have
(3.40)ϕk(u)=1p∫0kT(L(t)|u|p-2u,u)dt-∫0kTF(t,u)dt+k∑j=1mIj(u)=kp∫0T(L(t)|u|p-2u,u)dt-k∫0TF(t,u)dt+k∑j=1mIj(u)≤c2kp∫0T|u|pdt-k∫0TC7|u|pdt+k∫0TC8b(t)dt+mc|u|pforu∈RN.
From (H3), we can choose C7 suitable large such that
(3.41)ϕk(u)≤0,∀u∈RN.
Let W¯kT1,p=span{ek}+ℝN, where ek=(k-1sin(k-1ωt)), ω=2π/T. Since W¯T1,p is finite dimensional, there exists a constant d>0 such that
(3.42)(∫0T|x|pdt)1/p≥d(∫0T|x|2dt)1/2,∀x∈W¯T1,p.
By (I1), we have
(3.43)|ψ(u+rek)|=|k∑j=1mIj(u+rek(tj))|≤∑j=1mcj|u+rek(tj)|p≤2pmc|u|p+2pmcrp|ek(tj)|p≤2pmc|u|p+2pmcrpk-p≤2pmc|u|p+2pmcrp,u∈RN.
From (3.39), (3.42), and (3.43), we obtain
(3.44)ϕk(u+rek)=1p∫0kT|rėk(t)|pdt-∫0kTF(t,u+rek(t))dt+1p∫0kT(L(t)|u+rek(t)|p-2(u+rek(t)),u+rek(t))dt+k∑j=1mIj(u+rek(tj))≤1pk-2prpωp∫0kT|cos⁡⁡(k-1ωt)|pdt+c2p∫0kT|u+rek(t)|pdt+∫0kTC8b(t)dt-∫0kTC7|u+rek(t)|pdt+2pmc|u|p+2pmcrp≤1pk-2p+1rpωp∫0T|cos⁡⁡(ωt)|pdt-k∫0T(C7-c2p)|u+re1(t)|pdt+∫0TC8kb(t)dt+2pmc|u|p+2pmcrp≤(Tpk-2p+1ωp+2pmc)rp-kdp(C7-c2p)(∫0T|u+re1(t)|2dt)p/2+2pmc|u|p+C9k≤(Tpk-2p+1ωp+2pmc)rp-kdp(C7-c2p)(∫0T(|u|2+r2|e1(t)|2d))p/2+2pmc|u|p+C9k≤(Tpk-2p+1ωp+2pmc)rp-kdp(C7-c2p)(T|u|2+Tr22)p/2+2pmc|u|p+C9k,∀r≥0,u∈RN.
From (H3), we can choose C7 suitable such that
(3.45)dp(C7-c2p)(T2)p/2-23pmc>0,dp(C7-c2p)Tp/2-2p+1mc>0.
If k≥2(Tp)1/2pω1/2/[dp(C7-c2/p)(T/2)p/2-23pmc]:=C10, then we get
(3.46)k-1ϕk(u+rek)≤[Tpk-2prpωp+2pmck-dp(C7-c2p)(T2)p/2]rp+C9≤[Tk-2pωp+2pmc-dp(C7-c2p)(T2)p/2]rp+C9≤-12dp(C7-c2p)(T2)p/2rp+C9,k-1ϕk(u+rek)≤-12dp(C7-c2p)Tp/2|u|p+C9.
It follows from (3.46) that
(3.47)φk(u+rek)≤0,eitherr≥r1or|u|≥r2,
where r1=2(2C9)1/p/(C7-c2/p)1/pdT1/2, r2=(2C9)1/p/d(C7-c2/p)1/pT1/2. Notice that for any u∈ℝN, we have
(3.48)‖u‖=‖u‖Lp=(∫0kT|u|pdt)1/p=(kT)1/p|u|≥(C10T)1/pr2≔r3.
Hence, (3.47) holds for all ∥u∥≥r3 whenever u∈ℝN. Set
(3.49)Qk={rek∣0≤r≤r1,ek∈W̃kT1,p}⊕{u∈RN∣‖u‖≤r3};
then ∂Qk=Q1k⋃Q2k⋃Q3k, where
(3.50)Q1k={u∈RN∣‖u‖≤r3},Q2k={u+rek∣‖u‖=r3,r∈[0,r1],ek∈W̃kT1,p},Q3k={u+rek∣‖u‖≤r3,r=r1,ek∈W̃kT1,p}.
By (3.41) and (3.47), we have
(3.51)φ(u)≤0,u∈∂Qk=Q1k⋃Q2k⋃Q3k.
Furthermore, for all u+rek∈Qk, it follows from (H1), (3.8), and (3.43) that
(3.52)ϕk(u+rek)=1p∫0kT|rėk(t)|pdt-∫0kTF(t,u+rek(t))dt+1p∫0kT(L(t)|u+rek(t)|p-2(u+rek(t)),u+rek(t))dt+k∑j=1mIj(u+rek(tj))≤1prp∫0kT|ėk(t)|pdt+c2p∫0kT|u+rek(t)|pdt+2pmc|u|p+2pmcrp≤1pk-2prpωp∫0kT|cos⁡⁡(k-1ωt)|pdt+2p-1c2p∫0kT(|u|p+rpk-p|sin⁡(k-1ωt)|p)dt+2pmc|u|p+2pmcrp≤1pk-2p+1rpωp∫0T|cos⁡⁡(ωt)|pdt+2p-1c2p(‖u‖p+rpk-p+1∫0T|sin⁡(ωt)|pdt)+2pmcT‖u‖p+2pmcrp≤Tpk-2p+1rpωp+2p-1c2p(‖u‖p+rpk-p+1T)+2pmcT‖u‖p+2pmcrp≤Tpr1pωp+2p-1c2p(r3p+r1pT)+2pmc(1Tr3p+r1p).
Then by Lemma 2.2, ϕk has at least a critical point uk whose critical value ck satisfies
(3.53)0<α≤ck=ϕk(uk)≤Tpr1pωp+2p-1c2p(r3p+r1pT)+2pmc(1Tr3p+r1p).
Similar to the proof of [28], let uk1 be a k1T-periodic solution; we can prove that there exists a positive integer k2>k1 such that ukk1≠uk1 for all kk1≥k2. Otherwise, φk(ukk1)=kφk(uk1)→∞ as k→∞, which contradicts to (3.53). Repeating this process, we can obtain a sequence {ukj} of distinct periodic solutions of problem (1.1). From (3.41), we know that ukj is nonconstant. The proof is complete.


## 4. Examples

In this section, we give an example to illustrate our result.Example 4.1.
Letp=3, r=5, μ=4, and consider the following p-Laplacian system with impulsive effects
(4.1)ddt(|u̇(t)|u̇(t))-L(t)|u(t)|u(t)+∇F(t,u(t))=0,a.e.t∈R,u(0)-u(T)=u̇(0)-u̇(T)=0,Δ(|u̇(ti)|u̇(ti))=|u̇(ti+)|u̇(ti+)-|u̇(ti-)|u̇(ti-)=∇Ii(u(ti)),i=1,2,…,m.
Let
(4.2)L(t)=diag⁡(1+exp⁡(1-sin(k-1ωt)),…,1+exp⁡(1-sin(k-1ωt))),Ii(x)=cik|x|p,F(t,x)=1+e3(2+sin(k-1ωt))|x|5,
where ci>0, i∈B. It is easy to check that F satisfies (A), (H1), and (H2). By a direct computation, we have
(4.3)lim⁡|x|→∞⁡F(t,x)|x|3=+∞,lim⁡⁡sup⁡|x|→∞⁡F(t,x)|x|5≤1+e,lim⁡⁡inf⁡|x|→∞⁡(∇F(t,x),x)-3F(t,x)|x|4≥2(1+e)3,
which show that (H3), (H4), and (H5) hold. On the other hand,
(4.4)0≤Ii(x)≤ck|x|p,∇Ii(x)x=pcik|x|p=pIi(x),
where c=max{ci}, i∈B. It is easy to see that Ii satisfies (I1) and (I2). Hence, from Theorem 3.1, problem (4.1) has a sequence of distinct nonconstant periodic solutions with period kjT satisfying kj∈ℕ and kj→∞ as j→∞.



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*Source: 434938-2011-08-21.xml*