# Periodic Solutions for Autonomous(q,p)-Laplacian System with Impulsive Effects

Authors: Xiaoxia Yang; Haibo Chen
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

By using the variational method, some existence theorems
are obtained for periodic solutions of autonomous(q,p)-Laplacian system with impulsive effects.


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## Body

## 1. Introduction

LetB={1,2,…,l}, C={1,2,…,k}, l,k∈ℕ.In this paper, we consider the following system:(1.1)ddtΦq(u̇1(t))=∇u1F(u1(t),u2(t)),a.e.t∈[0,T],ddtΦp(u̇2(t))=∇u2F(u1(t),u2(t)),a.e.t∈[0,T],u1(0)-u1(T)=u̇1(0)-u̇1(T)=0,u2(0)-u2(T)=u̇2(0)-u̇2(T)=0,ΔΦq(u̇1(tj))=Φq(u̇1(tj+))-Φq(u̇1(tj-))=∇Ij(u1(tj)),j∈B,ΔΦp(u̇2(sm))=Φp(u̇2(sm+))-Φp(u̇2(sm-))=∇Km(u2(sm)),m∈C,
where p>1,q>1, T>0, u(t)=(u1(t),u2(t))=(u11(t),u12(t),…,u1N(t),u21(t),u22(t),…,u2N(t))τ,  tj(j=1,2,…,l), and sm(m=1,2,…,k) are the instants where the impulses occur and 0=t0<t1<t2<⋯<tl<tl+1=T,0=s0<s1<s2<⋯<sk<sk+1=T, Ij:ℝN→ℝ(j∈B), and Km:ℝN→ℝ(m∈C) are continuously differentiable(1.2)Φμ(z)=|z|μ-2z=(∑i=1Nzi2)(μ-2)/2(z1⋮zN),μ∈R,μ>1,∇Ij(x)=(∂Ij∂x1⋮∂Ij∂xN),∇Km(x)=(∂Km∂x1⋮∂Km∂xN),
and F:ℝN×ℝN→ℝ satisfies the following assumption.(A)
F(x)is continuously differentiable in(x1,x2), and there exista1,a2∈C(ℝ+,ℝ+)such that(1.3)|F(x1,x2)|≤a1(|x1|)+a2(|x2|),|∇F(x1,x2)|≤a1(|x1|)+a2(|x2|),|Ij(x1)|≤a1(|x1|),|∇Ij(x1)|≤a1(|x1|),j∈B,|Km(x2)|≤a2(|x2|),|∇Km(x2)|≤a2(|x2|),m∈C,for allx=(x1,x2)∈ℝN×ℝN.Whenp=q=2, Ij≡0(j∈B), Km≡0(m∈C), and F(u1,u2)=F1(u1), system (1.1) reduces to the following autonomous second-order Hamiltonian system:(1.4)ü1(t)=∇u1F1(u1(t)),a.e.t∈[0,T],u1(0)-u1(T)=u̇1(0)-u̇1(T)=0.
There have been lots of results about the existence of periodic solutions for system (1.4) and nonautonomous second order Hamiltonian system(1.5)ü1(t)=∇u1F1(t,u1(t)),a.e.t∈[0,T],u1(0)-u1(T)=u̇1(0)-u̇1(T)=0,
(e.g., see [1–21]). Many solvability conditions have been given, for instance, coercive condition, subquadratic condition, superquadratic condition, convex condition, and so on.Whenp=q=2, ∇Ij≢0(j∈B), Km≡0(m∈C), and F(u1,u2)=F1(u1), system (1.1) reduces to the following autonomous second-order Hamiltonian system with impulsive effects:(1.6)ü1(t)=∇u1F1(u1(t)),a.e.t∈[0,T],u1(0)-u1(T)=u̇1(0)-u̇1(T)=0,u̇1(tj+)-u̇1(tj-)=∇Ij(u1(tj)).
Recently, many authors studied the existence of periodic solutions for impulsive differential equations by using variational methods, and lots of interesting results have been obtained. For example, see [22–28]. Especially, nonautonomous second-order Hamiltonian system with impulsive effects is considered in [25, 26] by using the least action principle and the saddle point theorem.WhenIj≡0(j∈B) and Km≡0(m∈C), system (1.1) reduces to the following system:
(1.7)ddtΦq(u̇1(t))=∇u1F(u1(t),u2(t)),a.e.t∈[0,T],ddtΦp(u̇2(t))=∇u2F(u1(t),u2(t)),a.e.t∈[0,T],u1(0)-u1(T)=u̇1(0)-u̇1(T)=0,u2(0)-u2(T)=u̇2(0)-u̇2(T)=0.
In [29, 30], Paşca and Tang obtained some existence results for system (1.7) by using the least action principle and saddle point theorem. Motivated by [17, 22–30], in this paper, we are concerned with system (1.1) and also use the least action principle and saddle point theorem to study the existence of periodic solution. Our results still improve those in [17] even if system (1.1) reduces to system (1.4).A functionG:ℝN→ℝ is called to be (λ,μ)-quasiconcave if
(1.8)G(λ(x+y))≥μ(G(x)+G(y)),
for some λ,μ>0 and x,y∈ℝN.Next, we state our main results.Theorem 1.1.
Letq′ and p′ be such that  1/q+1/q′=1 and  1/p+1/p′=1. Suppose  F satisfies assumption (A) and the following conditions: (F1)
there exist(1.9)0<r1<(q′+1)q/q′TqΘ(q,q′),0<r2<(p′+1)p/p′TpΘ(p,p′),
such that
(1.10)(∇x1F(x1,x2)-∇y1F(y1,y2),x1-y1)≥-r1|x1-y1|q,∀(x1,x2),(y1,y2)∈RN×RN,(∇x2F(x1,x2)-∇y2F(y1,y2),x2-y2)≥-r2|x2-y2|p,∀(x1,x2),(y1,y2)∈RN×RN,
where
(1.11)Θ(q,q′)=∫01[sq′+1+(1-s)q′+1]q/q′ds,Θ(p,p′)=∫01[sp′+1+(1-s)p′+1]p/p′ds,(F2)
F(x)→+∞,as|x|→∞,wherex=(x1,x2),(I1)
there existsβ∈ℝ such that
(1.12)Ij(x)≥β,∀x∈RN,j∈B,Km(x)≥β,∀x∈RN,m∈C.Then, system (1.1) has at least one solution in WT1,q×WT1,p, where WT1,s={u:[0,T]→ℝN∣u is absolutely continuous,u(0)=u(T) andu̇∈Ls(0,T;ℝN)},s∈ℝ.
Furthermore, ifIj≡0(j∈B), Km≡0(m∈C) and the following condition holds:
(F3)
there existδ>0, a∈[0,(q′+1)q/q′/qTqΘ(q,q′)) and b∈[0,(p′+1)p/p′/(pTpΘ(p,p′))) such that
(1.13)-a|x1|q-b|x2|p≤F(x1,x2)≤0,∀|x1|≤δ,|x2|≤δ,
then system (1.7) has at least two nonzero solutions in WT1,q×WT1,p.Whenp=q=2, F(x1,x2)=F1(x1), by Theorem 1.1, it is easy to get the following corollary.Corollary 1.2.
SupposeF1 satisfies the following conditions: (A)′
F1(z) is continuously differentiable in  z and there exists  a1∈C(ℝ+,ℝ+) such that
(1.14)|F1(z)|≤a1(|z|),|∇F1(z)|≤a1(|z|),|Ij(z)|≤a1(|z|),|∇Ij(z)|≤a1(|z|),j∈B,
for all z∈ℝN.(F1)′
there exists0<r<6/T2 such that
(1.15)(∇zF1(z)-∇wF1(w),z-w)≥-r|z-w|2,∀z,w∈RN,(F2)′
F1(z)→+∞,as|z|→∞,z∈ℝN;(I1)′
there exists β∈ℝ such that
(1.16)Ij(z)≥β,∀z∈RN,j∈B.Then, system (1.6) has at least one solution in WT1,2. Furthermore, if Ij≡0(j∈B) and the following condition holds:(F3)′
there existδ>0 and a∈[0,(3/T2)) such that
(1.17)-a|z|2≤F1(z)≤0,∀z∈RN,|z|≤δ,
then system (1.4) has at least two nonzero solutions in WT1,2.For the Sobolev spaceW̃T1,2, one has the following sharp estimates (see in [3, Proposition  1.2]): (1.18)∫0T|u(t)|2dt≤T24π2∫0T|u̇(t)|2dt(Wirtinger′sinequality),(1.19)‖u‖∞2≤T12∫0T|u̇(t)|2dt(Sobolev′sinequality).
By the above two inequalities, we can obtain the following better results than  by Corollary 1.2.Theorem 1.3.
SupposeF1 satisfies assumption  (A)′, (F2)′, (I1)′ and (F1)′′
there exists0<r<4π2/T2 such that (1.15) holds.Then, system (1.6) has at least one solution in WT1,2. Furthermore, if Ij≡0(j∈B) and the following condition holds:(F3)′′
there existδ>0 and a∈[0,(2π2)/T2) such that
(1.20)-a|z|2≤F1(z)≤0,∀z∈RN,|z|≤δ,
then system (1.4) has at least two nonzero solutions in WT1,2.Moreover, for system (1.6), we have the following additional result.Theorem 1.4.
SupposeF1 satisfies assumption (A)′, (F1)′ ′ and the following conditions: (F4)
F1(z) is (λ,μ)-quasiconcave on ℝN,(F5)
F1(z)→-∞    as |z|→+∞, z∈ℝN,(I2)
there existdj>0(j∈B)  such that
(1.21)|∇Ij(z)|≤dj,∀z∈RN,j∈B,(I3)
there existbj>0,cj>0, γj∈ℝ, αj∈[0,2)(j∈B) such that
(1.22)-bj|z|αj-cj≤Ij(z)≤γj,∀z∈RN,j∈B.Then, system (1.6) has at least one solution in WT1,2.Remark 1.5.
In [17], Yang considered the second-order Hamiltonian system with no impulsive effects, that is, system (1.4). When Ij≡0(j∈B), our Theorems 1.3 and 1.4 still improve those results in [17]. To be precise, the restriction of r is relaxed, and some unnecessary conditions in [17] are deleted. In [17], the restriction of r is 0<r<T/12, which is not right. In fact, from his proof, it is easy to see that it should be 0<r<12/T2. Obviously, our restriction 0<r<4π2/T2 is better. Moreover, in our Theorem 1.4, we delete such conditions (of in [17, Theorem  1]): ∇F1(0)=0, and there exist positive constants M,N such that
(1.23)F1(z)≥-M|z|2-N,z∈RN.
Finally, it is remarkable that Theorems 1.3 and 1.4 are also different from those results in [1–16]. We can find an example satisfying our Theorem 1.3 but not satisfying the results in [1–21]. For example, let
(1.24)F1(z)=π22T2(|z1|4+|z2|4+⋯+|zN|4)-π24T2|z|2,
where z=(z1,…,zN)τ. We can also find an example satisfying our Theorem 1.4 but not satisfying the results in [1–21]. For example, let
(1.25)F1(z)=-r2|z|2,
where 12/T2<r<4π2/T2.


## 2. Variational Structure and Some Preliminaries

The norm inWT1,p is defined by(2.1)‖u‖WT1,p=[∫0T|u(t)|pdt+∫0T|u̇(t)|pdt]1/p.
Set(2.2)‖u‖p=(∫0T|u(t)|pdt)1/p,‖u‖∞=max⁡t∈[0,T]|u(t)|.
Let(2.3)W̃T1,p={u∈WT1,p∣∫0Tu(t)dt=0}.
Obviously, WT1,p is a reflexive Banach space. It is easy to know that W̃T1,p is a subset of WT1,p and WT1,p=ℝN⊕W̃T1,p. In this paper, we will use the space W defined by
(2.4)W=WT1,q×WT1,p,u(t)=(u1(t),u2(t)),
with the norm ∥(u1,u2)∥W=∥u1∥WT1,q+∥u2∥WT1,p. It is clear that W is a reflexive Banach space. Let W̃=W̃T1,q×W̃T1,p. Then, W=(W̃T1,q×W̃T1,p)⊕(ℝN×ℝN).Lemma 2.1 (see [31] or [32]).
Eachu∈WT1,p and each  v∈WT1,q can be written as  u(t)=u¯+ũ(t) and  v(t)=v¯+ṽ(t) with
(2.5)u¯=1T∫0Tu(t)dt,∫0Tũ(t)dt=0,v¯=1T∫0Tv(t)dt,∫0Tṽ(t)dt=0.
Then,
(2.6)‖ũ‖∞≤(Tp′+1)1/p′(∫0T|u̇(s)|pds)1/p,‖ṽ‖∞≤(Tq′+1)1/q′(∫0T|v̇(s)|qds)1/q,(2.7)∫0T|ũ(s)|pds≤TpΘ(p,p′)(p′+1)p/p′∫0T|u̇(s)|pds,∫0T|ṽ(s)|qds≤TqΘ(q,q′)(q′+1)q/q′∫0T|v̇(s)|qds,
where
(2.8)Θ(p,p′)=∫01[sp′+1+(1-s)p′+1]p/p′ds,Θ(q,q′)=∫01[sq′+1+(1-s)q′+1]q/q′ds.Note that ifu∈WT1,p, then u is absolutely continuous. However, we cannot guarantee that u̇ is also continuous. Hence, it is possible that ΔΦp(u̇(t))=Φp(u̇(t+))-Φp(u̇(t-))≠0, which results in impulsive effects.Following the idea in [22], one takes v1∈WT1,q and multiplies the two sides of(2.9)ddt(|u̇1(t)|q-2u̇1(t))-∇x1F(u1(t),u2(t))=0,byv1 and integrate from 0 to T, one obtains(2.10)∫0T[ddt(|u̇1(t)|q-2u̇1(t))-∇x1F(u1(t),u2(t))]v1(t)dt=0.
Note that v1(t) is continuous. So, v1(tj-)=v1(tj+)=v1(tj). Combining u̇1(0)-u̇1(T)=0, one has
(2.11)∫0T(dΦq(u̇1(t))dt,v1(t))dt=∑j=0l∫tjtj+1(d(Φq(u̇1(t)))dt,v1(t))dt=∑j=0l[(Φq(u̇1(tj+1-)),v1(tj+1-))-(Φq(u̇1(tj+)),v1(tj+))]dt-∑j=0l∫tjtj+1(Φq(u̇1(t)),v̇1(t))dt=(Φq(u̇1(T)),v1(T))-(Φq(u̇1(0)),v1(0))-∑j=1l(ΔΦq(u̇1(tj)),v1(tj))-∫0T(Φq(u̇1(t)),v̇1(t))dt=-∑j=1l(∇Ij(u1(tj)),v1(tj))-∫0T(Φq(u̇1(t)),v̇1(t))dt.
Combining with (2.10), one has(2.12)∫0T(Φq(u̇1(t)),v̇1(t))dt+∑j=1l(∇Ij(u1(tj)),v1(tj))+∫0T(∇x1F(u1(t),u2(t)),v1(t))dt=0.
Similarly, one can get(2.13)∫0T(Φp(u̇2(t)),v̇2(t))dt+∑m=1k(∇Km(u2(sm)),v2(sm))+∫0T(∇x2F(u1(t),u2(t)),v2(t))dt=0,
for all v2∈WT1,p. Considering the above equalities, one introduces the following concept of the weak solution for system (1.1).Definition 2.2.
We say that a functionu=(u1,u2)∈WT1,q×WT1,pis a weak solution of system (1.1) if(2.14)∫0T(Φq(u̇1(t)),v̇1(t))dt+∑j=1l(∇Ij(u1(tj)),v1(tj))=-∫0T(∇x1F(u1(t),u2(t)),v1(t))dt,∫0T(Φp(u̇2(t)),v̇2(t))dt+∑m=1k(∇Km(u2(sm)),v2(sm))=-∫0T(∇x2F(u1(t),u2(t)),v2(t))dt
holds for any v=(v1,v2)∈WT1,q×WT1,p.
Define the functionalφ:WT1,q×WT1,p→ℝ by
(2.15)φ(u1,u2)=1q∫0T|u̇1(t)|qdt+1p∫0T|u̇2(t)|pdt+∫0TF(u1(t),u2(t))dt+∑j=1lIj(u1(tj))+∑m=1kKm(u2(sm))=ϕ(u1,u2)+ψ(u1,u2),
where (u1,u2)∈WT1,q×WT1,p,
(2.16)ϕ(u1,u2)=1q∫0T|u̇1(t)|qdt+1p∫0T|u̇2(t)|pdt+∫0TF(u1(t),u2(t))dt,ψ(u1,u2)=∑j=1lIj(u1(tj))+∑m=1kKm(u2(sm)).
By assumption (A) and [33], we know that ϕ∈C1(WT1,q×WT1,p,ℝ). The continuity of Ij(j∈B) and Km(m∈C) implies that ψ∈C1(WT1,p×WT1,p,ℝ). So, φ∈C1(WT1,p,ℝ), and for all (v1,v2)∈WT1,q×WT1,p, we have
(2.17)〈φ′(u1,u2),(v1,v2)〉=∫0T(Φq(u̇1(t)),v̇1(t))dt+∫0T(Φp(u̇2(t)),v̇2(t))dt+∫0T(∇x1F(u1(t),u2(t)),v1(t))dt+∫0T(∇x2F(u1(t),u2(t)),v2(t))dt+∑j=1l(∇Ij(u1(tj)),v1(tj))+∑m=1k(∇Km(u2(sm)),v2(sm)).
Definition 2.2 shows that the critical points of φ correspond to the weak solutions of system (1.1).
We will use the following lemma to seek the critical point ofφ.Lemma 2.3 (see [3, Theorem  1.1]).
Ifφ is weakly lower semicontinuous on a reflexive Banach space  X and has a bounded minimizing sequence, then  φ has a minimum on  X.Lemma 2.4 (see [34]).
Letφ be a  C1 function on  X=X1⊕X2 with  φ(0)=0, satisfying (PS) condition, and assume that  for some  ρ>0,
(2.18)φ(u)≥0,foru∈X1,‖u‖≤ρ,φ(u)≤0,foru∈X2,‖u‖≤ρ.
Assume also that φ is bounded below and infXφ<0, then φ has at least two nonzero critical points.Lemma 2.5 (see [35, Theorem  4.6]).
LetX=X1⊕X2, where  X is a real Banach space and  X1≠{0} and is finite dimensional. Suppose that  φ∈C1(X,ℝ) satisfies (PS)-condition and  (φ1)
there is a constantα and a bounded neighborhood D of 0 in X1 such that φ∣∂D≤α,(φ2)
there is a constantβ>α such that φ∣X2≥β.
Then,φ possesses a critical value c≥β. Moreover, c can be characterized as
(2.19)c=inf⁡h∈Γmax⁡u∈D¯φ(h(u)),
where,
(2.20)Γ={h∈C(D¯,X)∣h=idon⁡∂D}.


## 3. Proof of Theorems

Lemma 3.1.
Under assumption (A),φ is weakly lower semicontinuous on  WT1,q×WT1,p.Proof.
Let(3.1)ϕ1(u1,u2)=1q∫0T|u̇1(t)|qdt+1p∫0T|u̇2(t)|pdt,ϕ2(u1,u2)=∫0TF(u1(t),u2(t))dt.
Since
(3.2)ϕ1(u1+v12,u2+v22)=1q∫0T|u̇1(t)+v̇1(t)2|qdt+1p∫0T|u̇2(t)+v̇2(t)2|pdt≤2q-1q∫0T12q|u̇1(t)|qdt+2q-1q∫0T12q|v̇1(t)|qdt+2p-1p∫0T12p|u̇2(t)|pdt+2p-1p∫0T12p|v̇2(t)|pdt≤12q∫0T|u̇1(t)|qdt+12q∫0T|v̇1(t)|qdt+12p∫0T|u̇2(t)|pdt+12p∫0T|v̇2(t)|pdt=ϕ1(u1,u2)+ϕ1(v1,v2)2,
then ϕ1 is convex. Moreover, by [33], we know that ϕ1 is continuous, and so, it is lower semicontinuous. Thus, it follows from [3, Theorem  1.2] that ϕ1 is weakly lower continuous. By assumption (A), it is easy to verify that ϕ2(u1,u2) is weakly continuous. We omit the details. Let
(3.3)ψ1(u1)=∑j=1lIj(u1(tj)),ψ2(u2)=∑m=1kKm(u2(sm)).
Next, we show that ψ1 and ψ2 are weakly continuous on WT1,q and WT1,p, respectively. In fact, if
(3.4)u1n⇀u1weaklyinWT1,p,asn⟶∞,
then by in [3, Proposition  1.2], we know that
(3.5)u1n⟶u1stronglyinC(0,T;RN),asn⟶∞.
So, there exists M1>0 such that ∥u1∥∞≤M1 and ∥u1n∥∞≤M1, for all n∈ℕ. Thus, we have
(3.6)|ψ1(u1n)-ψ1(u1)|=|∑j=1lIj(u1n(tj))-∑j=1lIj(u1(tj))|≤∑j=1l|Ij(u1n(tj))-Ij(u1(tj))|=∑j=1l|∫01(∇Ij(u1(tj)+s(u1n(tj)-u1(tj))),u1n(tj)-u1(tj))ds|≤‖u1n-u1‖∞∑j=1lmax⁡t∈[0,3M1]a1(t)⟶0.
Hence, ψ1 is weakly continuous on WT1,q. Similarly, we can prove that ψ2 is also weakly continuous on WT1,p. Thus, we complete the proof.Proof of Theorem1.1.
It follows from(F1) and (2.7) that
(3.7)∫0T[F(u1(t),u2(t))-F(u1(t),u¯2)]=∫0T∫011s(∇Fx2(u1(t),u¯2+sũ2(t)),sũ2(t))dsdt=∫0T∫011s(∇Fx2(u1(t),u¯2+sũ2(t))-∇Fx2(u¯1,u¯2),sũ2(t))dsdt≥-r2p∫0T|ũ2(t)|pdt≥-r2TpΘ(p,p′)p(p′+1)p/p′∫0T|u̇2(t)|pdt,∀(u1,u2)∈W,(3.8)∫0T[F(u1(t),u¯2)-F(u¯1,u¯2)]dt=∫0T∫011s(∇x1F(u¯1+sũ1(t),u¯2),sũ1(t))dsdt=∫0T∫011s(∇x1F(u¯1+sũ1(t),u¯2)-∇x1F(u¯1,u¯2),sũ1(t))dsdt≥-r1q∫0T|ũ1(t)|qdt≥-r1TqΘ(q,q′)q(q′+1)q/q′∫0T|u̇1(t)|qdt,∀(u1,u2)∈W.
Hence, by (I1), (3.7), and (3.8), we have
(3.9)φ(u1,u2)=1q∫0T|u̇1(t)|qdt+1p∫0T|u̇2(t)|pdt+∫0T[F(u1(t),u2(t))-F(u1(t),u¯2)]dt+∫0T[F(u1(t),u¯2)-F(u¯1,u¯2)]dt+TF(u¯1,u¯2)+∑j=1lIj(u1(tj))+∑m=1kKm(u2(sm))≥(1p-r2TpΘ(p,p′)p(p′+1)p/p′)∫0T|u̇2(t)|pdt+(1q-r1TqΘ(q,q′)q(q′+1)q/q′)∫0T|u̇1(t)|qdt+TF(u¯1,u¯2)-(l+k)|β|.
Note that for  u∈WT1,p,
(3.10)‖u‖WT1,p⟶∞⟺(|u¯|p+∫0T|u̇(t)|pdt)1/p⟶∞,
and for v∈WT1,q,
(3.11)‖v‖WT1,q⟶∞⟺(|v¯|q+∫0T|v̇(t)|qdt)1/q⟶∞.
So, (F2) and (3.9) imply that
(3.12)φ(u1,u2)⟶+∞,as‖(u1,u2)‖W⟶∞.
Thus, by Lemma 2.3, we know that φ has at least one critical point which minimizes φ on W.
Furthermore, ifIj(u1(tj))≡0(j∈B) and Km(u2(sm))≡0(m∈C), then system (1.1) reduces to (1.7). When (F3) also holds, we will use Lemma 2.4 to obtain more critical points of φ. Let X=W,    X2=ℝN×ℝN and X1=W̃=W̃T1,q×W̃T1,p.
By (3.9), we know that φ(u1,u2)→+∞ as ∥(u1,u2)∥W→∞. So, φ satisfies (PS) condition and is bounded below. Take ρ=δ/c1, where c1 is a positive constant such that ∥u1∥∞≤c1∥u1∥WT1,q≤c1∥u∥W and ∥u2∥∞≤c1∥u2∥WT1,p≤c1∥u∥W for all (u1,u2)∈W. It follows from (F3) and Lemma 2.1 that
(3.13)φ(u1,u2)=1q∫0T|u̇1(t)|qdt+1p∫0T|u̇2(t)|pdt+∫0TF(u1(t),u2(t))dt≥1q∫0T|u̇1(t)|qdt+1p∫0T|u̇2(t)|pdt-a∫0T|u1(t)|qdt-b∫0T|u2(t)|pdt≥1q∫0T|u̇1(t)|qdt+1p∫0T|u̇2(t)|pdt-aTqΘ(q,q′)(q′+1)q/q′∫0T|u̇1(t)|qdt-bTpΘ(p,p′)(p′+1)p/p′∫0T|u̇2(t)|pdt,∀(u1,u2)∈X1.
Since a≤(q′+1)q/q′/(qTqΘ(q,q′)) and b≤(p′+1)p/p′/(pTpΘ(p,p′)), (3.13) implies that φ(u1,u2)≥0 for all (u1,u2)∈X1 with ∥u∥W≤ρ. By (F3), it is easy to obtain that φ(u1,u2)≤0, for all (u1,u2)∈X2 with ∥u∥W≤ρ.
Ifinf{φ(u1,u2):(u1,u2)∈W}=0, then from above, we have φ(u1,u2)=0 for all (u1,u2)∈X2 with ∥(u1,u2)∥W≤ρ. Hence, all (u1,u2)∈X2 with ∥(u1,u2)∥W≤ρ are minimal points of φ, which implies that φ has infinitely many critical points. If inf{φ(u1,u2):(u1,u2)∈W}<0, then by Lemma 2.4, φ has at least two nonzero critical points. Hence, system (1.7) has at least two nontrivial solutions in W. We complete our proof.Proof of Theorem1.3.
We only need to use (1.18) and (1.19) to replace (2.6) and (2.7) in the proof Theorem 1.1 with p=q=2, F(t,u1,u2)=F1(u1) and Km(u2)≡0(m∈C). It is easy. So, we omit it.Lemma 3.2.
Under the assumptions of Theorem1.4, the functional  φ1 defined by
(3.14)φ1(u1)=12∫0T|u̇1(t)|2dt+∫0TF1(u1(t))dt+∑j=1lIj(u1(tj))dt
satisfies (PS) condition.Proof.
Suppose that{u1n} is a (PS) sequence for φ1; that is, there exists D1>0 such that
(3.15)|φ(u1n)|≤D1,∀n∈N,φ′(u1n)⟶0,asn⟶∞.
Hence, for n large enough, we have ∥φ′(u1n)∥≤1. It follows from (F1)′′, (I2), and (1.18) that
(3.16)‖ũ1n‖WT1,2≥〈φ1′(u1n),ũ1n〉=∫0T|u̇1n(t)|2dt+∫0T(∇x1F1(u1n(t)),ũ1n(t))dt+∑j=1l(∇Ij(u1n(tj)),ũ1n(tj))=∫0T|u̇1n(t)|2dt+∫0T(∇x1F1(u1n(t))-∇x1F1(u¯1n(t)),ũ1n(t))dt+∑j=1l(∇Ij(u1n(tj)),ũ1n(tj))≥∫0T|u̇1n(t)|2dt-rT24π2∫0T|u̇1n(t)|2dt-‖ũ1n‖∞∑j=1ldj≥[1-rT24π2]∫0T|u̇1n(t)|2dt-(T12)1/2(∫0T|u̇1n(t)|2dt)1/2∑j=1ldj,
for n large enough. By (1.18), we have
(3.17)‖ũ1n‖WT1,2≤[T24π2+1]1/2(∫0T|u̇1n(t)|2dt)1/2,
and (3.16), (3.17), and r<4π2/T2 imply that there exists D2,D3>0 such that
(3.18)∫0T|u̇1n(t)|2dt≤D2,‖ũ1n‖WT1,2≤D3.
It follows from (F4), (3.15), (I3), (1.18), and (3.18) that
(3.19)-D1≤φ1(u1n)=12∫0T|u̇1n(t)|2dt+∫0TF1(u1n(t))dt+∑j=1lIj(u1(tj))≤12∫0T|u̇1n(t)|2dt+1μ∫0TF1(λu¯1n)dt-∫0TF1(-ũ1n(t))dt+∑j=1lγj=12∫0T|u̇1n(t)|2dt+TμF1(λu¯1n)-TF1(0)-∫0T[F1(-ũ1n(t))-F1(0)]dt+∑j=1lγj=12∫0T|u̇1n(t)|2dt+TμF1(λu¯1n)-TF1(0)+∑j=1lγj-∫0T∫011s(∇F1(-sũ1n(t))-∇F1(0),-sũ1n(t))dsdt≤12∫0T|u̇1n(t)|2dt+TμF1(λu¯1n)+r∫0T∫01s|ũ1n(t)|2dsdt-TF1(0)+∑j=1lγj≤12∫0T|u̇1n(t)|2dt+TμF1(λu¯1n)+r2∫0T|ũ1n(t)|2dt-TF1(0)+∑j=1lγj≤max⁡{1,r}2‖ũ1n‖WT1,22+TμF1(λu¯1n)-TF1(0)+∑j=1lγj≤max⁡{1,r}2D3q+TμF1(λu¯1n)-TF1(0)+∑j=1lγj,
for all n and (3.19) and (F5) imply that {u¯1n} is bounded. Combining (3.18), we know that {u1n} is a bounded sequence. Similar to the argument in [25], it is easy to obtain that φ satisfies (PS) condition.
Proof of Theorem1.4.
From(I3) and (F5), it is easy to see that for x1∈ℝN,
(3.20)φ1(x1)⟶-∞,as|x1|⟶∞.
For all u1∈W̃T1,2, by (1.18), (F1)′′ and (I3), we have
(3.21)φ1(u1)=12∫0T|u̇1(t)|2dt+∫0TF1(u1(t))dt+∑j=1lIj(u1(tj))=12∫0T|u̇1(t)|2dt+∫0T[F1(u1(t))-F1(0)]dt+TF1(0)+∑j=1lIj(u1(tj))=12∫0T|u̇1(t)|2dt+∫0T∫01(∇F1x1(su1(t)),u1(t))dsdt+∑j=1lIj(u1(tj))+TF1(0)=12∫0T|u̇1(t)|2dt+∫0T∫011s(∇F1x1(su1(t))-∇F1x1(0),su1(t))dsdt+∑j=1lIj(u1(tj))+TF1(0)≥12∫0T|u̇1(t)|2dt-r12∫0T|u1(t)|2dt+TF1(0)-∑j=1lbj|u1(tj)|αj-∑j=1lcj≥12∫0T|u̇1(t)|2dt-r1T24π2∫0T|u̇1(t)|2dt+TF1(0)-∑j=1lbj‖u1‖∞αj-∑j=1lcj≥(12-r1T24π2)∫0T|u̇1(t)|2dt+TF1(0)-(T12)αj/2∑j=1lbj(∫0T|u̇1(t)|2dt)αj/2-∑j=1lcj.
Note that for all u1∈W̃T1,2, ∥u1∥WT1,2 is equivalent to ∥u̇1∥L2. Then, r1<4π2/T2, αj<2(j∈B) and (3.21) imply that
(3.22)φ1(u1)⟶+∞,as‖u1‖WT1,2⟶∞,u1∈W̃T1,2.
It follows from (3.20) and (3.22) that φ1 satisfies (φ1) and (φ2) in Lemma 2.5. Combining with Lemma 3.2, Lemma 2.5 shows that φ1 has at least one critical point. Thus, we complete the proof.


## 4. Examples

Example 4.1.
Letq=4, p=2, T=π, t1=1, and s1=2. Consider the following system:
(4.1)ddtΦ4(u̇1(t))=∇u1F(u1(t),u2(t)),a.e.t∈[0,π],ddtΦ2(u̇2(t))=∇u2F(u1(t),u2(t)),a.e.t∈[0,π],u1(0)-u1(π)=u̇1(0)-u̇1(π)=0,u2(0)-u2(π)=u̇2(0)-u̇2(π)=0,ΔΦ4(u̇1(1))=Φq(u̇1(1+))-Φq(u̇1(1-))=∇I1(u1(1)),ΔΦ2(u̇2(2))=Φp(u̇2(2+))-Φp(u̇2(2-))=∇K1(u2(2)),
where F(x1,x2)=x114+x124+⋯+x1N4+(1/π2)(x214+x222+⋯+x2N2)-(1/2π2)|x2|2, x1=(x11,x12,…,x1N), x2=(x21,x22,…,x2N), I1(x)=e|x|2, K1(x)=e|x|2, x∈ℝN. It is easy to verify that all conditions of Theorem 1.1 hold so that system (4.1) has at least one weak solution. Moreover, if F(x1,x2)=(1/π2)(x214+x224+⋯+x2N4)-1/2π2|x2|2, x2=(x21,x22,…,x2N), I1(x)=0 and K1(x)=0, x∈ℝN, then system (4.1) has at least two nonzero solutions.Example 4.2.
LetT=2, t1=1. Consider the following autonomous second-order Hamiltonian system with impulsive effects:
(4.2)ü(t)=∇uF(u(t)),a.e.t∈[0,2],u(0)-u(2)=u̇(0)-u̇(2)=0,u̇(1+)-u̇(1-)=∇I1(u(1)),
where F(z)=z14+z22+⋯+zN2-1/2|z|2,I1(z)=e|z|2, z=(z1,…,zN)τ∈ℝN. It is easy to verify that all conditions of Theorem 1.3 hold so that system (4.2) has at least one weak solution. Moreover, if F(z)=z14+z24+⋯+zN4-1/2|z|2 and I1(z)=0, z∈ℝN, then system (4.2) has at least two nonzero solutions.Example 4.3.
LetT=π, t1=2. Consider the following autonomous second-order Hamiltonian system with impulsive effects:
(4.3)ü(t)=∇uF(u(t)),a.e.t∈[0,π],u(0)-u(π)=u̇(0)-u̇(π)=0,u̇(2+)-u̇(2-)=∇I1(u(2)),
where F(z)=-|z|2,I1(z)=2sinz1, z=(z1,…,zN)τ∈ℝN. It is easy to verify that all conditions of Theorem 1.4 hold so that system (4.3) has at least one weak solution.



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*Source: 378389-2011-12-07.xml*