# Three Homoclinic Solutions for Second-Orderp-Laplacian Differential System

Authors: Jia Guo; Bin-Xiang Dai
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider second-orderp-Laplacian differential system. By using three critical points theorem, we establish the new criterion to guarantee that this
p-Laplacian differential system has at least three homoclinic solutions. An example is presented to illustrate the main result.


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## Body

## 1. Introduction

Let us consider the following second-orderp-Laplacian differential system:
(P)(ρ(t)Φp(u′(t)))′-s(t)Φp(u(t))+λf(t,u(t))=0,
whereΦp(x):=|x|p-2x, p>1, ρ,s∈L∞ withessinfρ>0andessinfs>0,f:R×Rn→Rnis continuous,t∈R, and λ∈[0,+∞). As usual, we say that a solutionu(t)of (P) is nontrivial homoclinic (to 0) ifu(t)≠0, u(t)→0 andu˙(t)→0ast→±∞.In the past two decades, many authors have studied homoclinic orbits for the second-order Hamiltonian systems(1)q¨(t)+∇V(t,q(t))=f(t),
and the existence and multiplicity of homoclinic solutions for (1) have been extensively investigated via critical point theory (see [1–15]). For instance, Yang et al. [5] have shown the existence of infinitely many homoclinic solutions for (1) by using fountain theorem.Theorem A (see [5]).
Assume thatfandVsatisfy the following conditions: (H1)
f(t)=0and∇V(t,q(t))=-L(t)q(t)+∇W(t,q(t));(H2)
L∈C(R,Rn×n)is a symmetric and positive definite matrix for allt∈Rand there is a continuous functionα:R→Rsuch thatα(t)>0for allt∈Rand
(2)(L(t)u,u)≥α(t)|u|2,α(t)⟶+∞
as|t|→∞;(H3)
consider the following(3)W(t,u)=m(t)|u|γ+d|u|p,
wherem:R→R+is a positive continuous function such thatm∈L2/(2-γ)(R,R+)and1<γ<2, d≥0, and p>2are constants.
Then (1) possesses infinitely many homoclinic solutions.
Moreover, Tang and Xiao [10] prove the existence of homoclinic solution of (1) as a limit of the2kT-periodic solutions of the following extension of system (1):
(4)q¨(t)=-∇V(t,q(t))+fk(t),
and they established the following theorem.Theorem B (see [10]).
Assume thatfandVsatisfy the following conditions: (H4)
V,f(t)≠0and V(t,x)=-K(t,x)+W(t,x), where V∈C1(R×Rn,R) is T-periodic with respect to t, and T>0;(H5)
∇W(t,x)=o(|x|), as|x|→0 uniformly with respect to t;(H6)
there is a constantμ>2such that
(5)0<μW(t,x)≤(x,∇W(t,x))∀(t,x)∈R×(Rn∖{0});(H7)
f:R×Rn is a continuous and bounded function;(H8)
there exist constantsb>0 and γ∈(1,2]such that
(6)K(t,0)=0,K(t,x)≥b|x|γfor(t,x)∈[0,T]×Rn;(H9)
there is a constantϱ∈[2,μ)such that
(7)(x,∇K(t,x))≤ϱK(t,x),for(t,x)∈[0,T]×Rn;(H10)
consider the following(8)∫R‍|f(t)|2dt<2(min{δ2,bδγ-1-Mδμ-1})2.
Then system (1) possesses a nontrivial homoclinic solution.Forp-Laplacian problem, Tian and Ge [16] obtained sufficient conditions that guarantee the existence of at least two positive solutions of p-Laplacian boundary value problem with impulsive effects. Two key conditions of the main results of [16] are listed as follows: (H11)
there existμ>p, h∈C([a,b]×[0,+∞),[0,+∞)), η>0, r∈C([a,b]×[0,+∞)), g∈C([0,+∞),[0,+∞)), and
(9)∫ab‍r(s)ds+η>0,
such that
(10)f(t,x)=r(t)Φμ(x)+h(t,x),Ii(x)=ηΦμ(x)+g(x);(H12)
there existc∈L1([a,b],[0,+∞)), d∈C([a,b],[0,+∞)), ξ≥0, such that
(11)h(t,x)≤c(t)+d(t)Φp(x).In [17], Ricceri established a three critical points theorem. After that, several authors used it to obtain some interesting results (see [18–22]).Existence and multiplicity of solutions forp-Laplacian boundary value problem have been studied extensively in the literature (see [23–26]). However, to our best knowledge, the existence of at least three homoclinic solutions for p-Laplacian differential system has attracted less attention.Motivated by the aforementioned facts, in this paper we are devoted to study the multiplicity homoclinic solutions of(P) via three critical points theorem obtained by Ricceri [17].In order to receive the homoclinic solution of(P), similar to [10] we consider a sequence of system of differential equations as follows:
(Pk)(ρ(t)Φp(u′(t)))′-s(t)Φp(u(t))+λf(k)(t,u(t))=0,
wheref(k):R×Rn→Rnis a 2kT-periodic extension of restriction offto the interval[-kT,kT), k∈N. We will prove the existence of three homoclinic solutions of (P) as the limit of the 2kT-periodic solutions of (Pk) as in [10]. However, many technical details in our paper are different from [10, 12].


## 2. Preliminaries

For eachk∈N, letE(k)=W2kT1,p(R,Rn)denote the Sobolev space of 2kT-periodic functions onRwith values inRnunder the norm
(12)∥u∥≔∥u∥E(k)=[∫-kTkT‍(ρ(t)|u′(t)|p+s(t)|u(t)|p)dt]1/p,
which is equivalent to the usual one. We define the norm inC([-kT,kT])as ∥u∥C[-kT,kT]=max{|u(t)|:t∈[-kT,kT]}.ConsiderJ(k):E(k)×[0,+∞)→Rndefined by
(13)J(k)(u,λ)=ϕ1(u)+λϕ2(u),
where
(14)ϕ1(u)=∥u∥pp,ϕ2(u)=-∫-kTkT‍F(k)(t,u(t))dt,F(k)(t,x)=∫0x‍f(k)(t,y)dy∀(t,x)∈[-kT,kT]×Rn.
Using the continuity of f(k), one has that J(k)(u,λ)is (strongly) continuous inE(k)×[0,+∞), J(k)(·,λ)∈C1(E(k),Rn)and for anyu,v∈E(k),
(15)〈J′(k)u(u,λ),v〉=∫-kTkT‍ρ(t)Φp(u′(t))v′(t)dt+∫-kTkT‍s(t)Φp(u(t))v(t)dt-λ∫-kTkT‍f(k)(t,u(t))v(t)dt.In order to prove our main result, we list some basic facts in this section.Definition 1.
A function(16)u∈{u∈E(k):ρΦp(u′)(·)∈W2kT1,∞(R,Rn)}
is said to be a 2kT-periodic solution of (Pk) ifusatisfies the equation in (Pk).Lemma 2.
Ifu∈E(k)is a critical point ofJ(k)(·,λ); thenuis a 2kT-periodic solution of  (Pk).Proof.
Assume thatu∈E(k)is a critical point ofJ(k)(·,λ); then for allv∈E(k),  one has
(17)0=∫-kTkT‍ρ(t)Φp(u′(t))v′(t)dt+∫-kTkT‍s(t)Φp(u(t))v(t)dt-λ∫-kTkT‍f(k)(t,u(t))v(t)dt.
It follows that
(18)∫-kTkT‍ρ(t)Φp(u′(t))v′(t)dtm=-∫-kTkT‍s(t)Φp(u(t))v(t)dtmm+λ∫-kTkT‍f(k)(t,u(t))v(t)dt.
By the definition of weak derivative, (18) implies that
(19)(ρ(t)Φp(u′(t)))′=s(t)Φp(u(t))-λf(t,u(t)).
ThusρΦp(u′)(·)∈W2kT1,∞(R,Rn)andusatisfies the (Pk). Therefore,uis a solution of (Pk).Lemma2 motivates us to apply three critical points theorem to discuss the multiplicity of the 2kT-periodic solution of (Pk). Here, at the end of this section, let us recall some important facts.Definition 3.
LetXbe a Banach space andf:X→(-∞,+∞].fis said to be sequentially weakly lower semi-continuous ifliminfk→+∞f(xk)≥f(x) as xk⇀x inX.Definition 4.
SupposeEis a real Banach space. Forϕ∈C1(E,Rn), we say that ϕsatisfies PS condition if any sequence{uk}⊂Efor whichϕ(uk)is bounded andϕ′(uk)→0ask→∞possesses a convergent subsequence.Lemma 5 (see [16]).
Foru∈E(k), one then has∥u∥C[-kT,kT]≤M∥u∥E(k), where
(20)M=21/qmax{1(2kT)1/p(essinf[-kT,kT]s)1/p,mmmmmmmmm(2kT)1/q(essinf[-kT,kT]ρ)1/p},1p+1q=1.Lemma 6 (see [27]).
LetXbe a nonempty set, andΦ,Ψare two real functions onX. Assume that there arer>0, x0,x1∈Xsuch that
(21)Φ(x0)=Ψ(x0)=0,Φ(x1)>r,supx∈Φ-1(-∞,r]Ψ(x)<rΨ(x1)Φ(x1).
Then, for eachρsatisfying
(22)supx∈Φ-1(-∞,r]Ψ(x)<ρ<rΨ(x1)Φ(x1),
one has
(23)supλ≥0infx∈X(Φ(x)+λ(ρ-Ψ(x)))m<infx∈Xsupλ≥0(Φ(x)+λ(ρ-Ψ(x))).Lemma 7 (see [17]).
LetXbe a separable and reflexive real Banach space,I⊆Ran interval, andf:X×I→Ra function satisfying the following conditions: (i)
for eachx∈X, the functionf(t,·)is continuous and concave;(ii)
for eachλ∈I, the functionf(t,·)is sequentially weakly lower semicontinuous and Dâteaux differentiable, andlim∥x∥→∞f(x,λ)=+∞;(iii)
there exists a continuous concave functionh:I→Rsuch that
(24)supλ∈Iinfx∈X(f(x,λ)+h(λ))<infx∈Xsupλ∈I(f(x,λ)+h(λ)).Then, there exist an open intervalJ⊆Iand a positive real numberρ, such that, for eachλ∈J, the equation
(25)fx′(x,λ)=0
has at least two solutions inXwhose norms are less thanρ. If, in addition, the functionfis (strongly) continuous inX×I, and, for eachλ∈I, the functionf(t,·)isC1and satisfies the PS condition, then the above conclusion holds with “three” instead of “two.”Lemma 8.
Letu∈W1,p(R,Rn). Then for everyt∈R, the following inequality holds:
(26)|u(t)|≤(∫t-1/2t+1/2‍|u(s)|pds)1/p+12(∫t-1/2t+1/2|u′(s)|p‍ds)1/p.Proof.
Fixt∈R. For everyτ∈R,
(27)|u(t)|≤|u(τ)|+|∫τt‍u′(s)ds|.
Integrating (27) over[t-1/2,t+1/2]and using the Hölder inequality, we get
(28)|u(t)|≤∫t-1/2t+1/2‍[|u(τ)|+|∫τt‍u′(s)ds|]dτ≤∫t-1/2t+1/2‍|u(τ)|dτ+∫t-1/2t‍∫t-1/2t‍|u′(s)|dsdτm+∫tt+1/2‍∫tt+1/2‍|u′(s)|dsdτ≤(∫t-1/2t+1/2‍|u(s)|pds)1/pm+12(∫t-1/2t+1/2‍|u′(s)|pds)1/p.


## 3. Main Result

In this section, our main result of this paper is presented. First, we introduce the following three conditions:(V1)
there exist constantsc1,δ1,δ2,η1>0andη2>0,  withδ12+δ22≠0, η1+η2<η1η2  and
(29)0<c1M<(K2)1/p
such that 2kT  max(t,x)∈[-kT,kT]×[-c1,c1]F(k)(t,x)<EΩ, where
(30)E=(c1/M)pK2+K3p∫-kTkT‍s(t)dt,Ω=∫-kT-kT+2kT/η1‍F(k)(t,g1(t))dtm+∫-kT+2kT/η1kT-2kT/η2‍F(k)(t,g2(t))dtm+∫kT-2kT/η2kT‍F(k)(t,g3(t))dt,K1=δ1η2+δ2η1η1+η2-η1η2,K2=|δ1|p∫-kT-kT+2kT/η1‍ρ(t)dtm+|K1|p∫-kT+2kT/η1kT-2kT/η2‍ρ(t)dtm+|δ2|p∫kT-2kT/η2kT‍ρ(t)dt,k∈N,K3=max{2kTη1|δ1|,2kTη2|δ2|},g1(t)=δ1(t+kT),g3(t)=δ2(t-kT),g2(t)=K1(t+kT-2kTη1),k∈N;(V2)
there exist constantμ∈[0,p)and functionsτ1(t),τ2(t)∈L([-kT,kT])with essinf[-kT,kT]τ1>0such that
(31)F(k)(t,x)≤τ1(t)|x|μ+τ2(t)∀t∈[-kT,kT]andx∈Rn;(V3)
ρ,s∈L∞andf:R×Rn→Rn are continuous functions.Remark 9.
If there exist constantμ∈[0,p)and functionsτ3(t)∈C([-kT,kT])withmin[-kT,kT]τ3>0such that
(32)limsup|x|→∞F(k)(t,x)|x|μ<τ3(t)uniformly∀t∈[-kT,kT],
then (V2) holds.
In fact, (32) implies that there existsc2>0such that
(33)F(k)(t,x)≤τ3(t)|x|μ∀t∈[-kT,kT],|x|≥c2,
which combining the continuity of F(k)(t,x)-τ3(t)|x|μon[-kT,kT]×[-c2,c2]yields that there exists constantc3>0such that
(34)F(k)(t,x)≤τ3(t)|x|μ+c3∀t∈[-kT,kT],x∈Rn.Lemma 10.
Assume that (V1) holds; then, for eachk∈N, there exists a continuous concave functionh(k):[0,+∞)→Rnsuch that
(35)supλ≥0infu∈E(k)(J(k)(u,λ)+h(λ))<infu∈E(k)supλ≥0(J(k)(u,λ)+h(λ)).Proof.
We define(36)r=1p(c1M)p,u1(t)={g1(t),t∈[-kT,-kT+2kTη1),g2(t),t∈[-kT+2kTη1,kT-2kTη2],g3(t),t∈(kT-2kTη2,kT].
It is clear thatu1∈E(k). It follows from
(37)∫-kTkT‍ρ(t)|u1′(t)|pdt=K2,0≤∫-kTkT‍s(t)|u1(t)|pdt≤K3p∫-kTkT‍s(t)dt
that
(38)K2≤∥u1∥p≤K2+K3p∫-kTkT‍s(t)dt.
Letg(x)=(1/p)xp, x≥0. It is clear thatg(x)has the following properties: (1)g(x) strictly increases forx≥0 and (2)g(x)=whas unique solutionQ(w)for eachw>0.
In view of (29), (38), and (1), one has
(39)1p∥u1∥p≥1pK2>1p(c1M)p=r>0,
which yields that
(40)ϕ1(u1)=1p∥u1∥p>r>0.
It follows from Lemma 5, (1), and (2) that
(41)ϕ1-1(-∞,r]⊆{u∈E(k):1p∥u1∥p≤r}⊆{u∈E(k):∥u∥≤Q(r)}⊆{u∈E(k):maxt∈[-kT,kT]|u(t)|≤MQ(r)},k∈N.
Let G=MQ(r); then G/M is a solution ofg(x)=r.From the definition ofg(x)andr, we haveg(c1/M)=r. Thus, (2) impliesG=c1, which combining (41) yields that
(42)ϕ1-1(-∞,r]⊆{u∈E(k):maxt∈[-kT,kT]|u(t)|≤c1},k∈N.
Therefore,(43)supu∈ϕ1-1(-∞,r](-ϕ2(u))=supu∈ϕ1-1(-∞,r]∫-kTkT‍F(k)(t,u(t))dt≤sup|u(t)|≤c1∫-kTkT‍F(k)(t,u(t))dt≤2kTmax(t,x)∈[-kT,kT]×[-c1,c1]F(k)(t,x),k∈N.
SinceF(k)(t,0)=0, we obtain
(44)2kTmax(t,x)∈[-kT,kT]×[-c1,c1]F(k)(t,x)≥0,k∈N.
It follows fromδ12+δ22≠0 thatK2+K3p∫-kTkT‍s(t)dt>0, which combiningc1,M>0yields thatE>0. Therefore, in view of (V1) and (44), we getΩ>0. Thus, it follows from (38) and (40) that
(45)r∫-kTkT‍F(k)(t,u1(t))dtϕ1(u1)≥rΩ(1/p)∥u1∥p≥Ω(c1/M)pK2+K3p∫-kTkT‍s(t)dt,k∈N.
From (43), (45), and (V1), we have
(46)supu∈ϕ1-1(-∞,r](-ϕ2(u))<r-ϕ2(u1)ϕ1(u1).
It is obvious thatϕ1(0)=-ϕ2(0)=0. Owing to Lemma 6, choosingh(λ)=ρλ, we obtain
(47)supλ≥0infu∈E(k)(ϕ1(u)+λϕ2(u)+h(λ))m<infu∈E(k)supλ≥0(ϕ1(u)+λϕ2(u)+h(λ)),
which combining J(k)(u,λ)=ϕ1(u)+λϕ2(u) implies the conclusion.Lemma 11.
If (V2) holds, then for eachk∈N,lim∥u∥→∞J(k)(u,λ)=+∞andJ(k)(·,λ)satisfies thePScondition.Proof.
Let{un(k)}be a sequence inE(k)such thatlimn→+∞J(k)u′(un(k),λ)=0andJ(k)′(un(k),λ)is bounded, for eachk∈N.
Lemma5 implies that
(48)|u(t)|≤∥u∥C[-kT,kT]∞≤M∥u∥E(k)∀t∈[-kT,kT].
It follows from (V2) and (48) that
(49)∫-kTkT‍F(k)(t,u(t))dt≤∫-kTkT‍τ1(t)|u(t)|μdt+∫-kTkT‍τ2(t)dt≤Mμ∥u(t)∥μ∫-kTkT‍τ1(t)dtm+∫-kTkT‍τ2(t)dt,
which yields that
(50)J(k)(u,λ)m≥1p∥u∥p-λMμ∥u(t)∥μ∫-kTkT‍τ1(t)dt-λ∫-kTkT‍τ2(t)dt,
for eachk∈N. Noting thatμ∈[0,p), the above inequality implies thatlim∥u∥→∞J(k)(u,λ)=+∞and{un(k)}is bounded inE(k). Next, we will prove that{un(k)}converges strongly to someu(k)inE(k). The proof is similar to [22]. Since{un(k)}is bounded inE(k), there exists a subsequence of{un(k)}(for simplicity denoted again by{un(k)}) such that{un(k)}converges weakly to someu(k)inE(k). Then{un(k)}converges uniformly tou(k)on[-kT,kT](see [28]). Therefore,
(51)∫-kTkT‍(f(k)(t,un(k)(t))-f(k)(t,u(k)(t)))(un(k)(t)-u(k)(t))dt⟶0
asn→+∞,  for eachk∈N. In view thatlimn→+∞J(k)u′(un(k),λ)=0 and{un(k)}converges weakly to someu(k), we get
(52)〈J(k)u′(un(k),λ)-J(k)u′(u(k),λ),un(k)-u(k)〉⟶0
as n→+∞, for eachk∈N. Then, from (15), one has
(53)〈J(k)u′(un(k),λ)-J(k)u′(u(k),λ),un(k)-u(k)〉m=∫-kTkT‍ρ(t)(Φp(un′(k)(t))-Φp(u′(k)(t)))mmmmm×(un′(k)(t)-u′(k)(t))dtmm+∫-kTkT‍s(t)(Φp(un(k)(t))-Φp(u(k)(t)))mmmmmm×(un(k)(t)-u(k)(t))dtmm-λ∫-kTkT‍(f(k)(t,un(k)(t))-f(k)(t,u(k)(t)))mmmmmm×(un(k)(t)-u(k)(t))dt
for eachk∈N. By [29], for eachk∈N, there existcp,dp>0such that
(54)∫-kTkT‍ρ(t)(Φp(un′(k)(t))-Φp(u′(k)(t)))mmm×(un′(k)(t)-u′(k)(t))dtm+∫-kTkT‍s(t)(Φp(un(k)(t))-Φp(u(k)(t)))mmmm×(un(k)(t)-u(k)(t))dt≥{cp∫-kTkT(ρ(t)|un′(k)(t)-u′(k)(t)|p+s(t)|un(k)(t)-u(k)(t)|p)dt,ifp≥2,dp∫-kTkT(ρ(t)|un′(k)(t)-u′(k)(t)|2(|un′(k)(t)|+|u′(k)(t)|)2-p+s(t)|un(k)(t)-u(k)(t)|2(|un(k)(t)|+|u(k)(t)|)2-p)dt,if1<p<2.
Ifp≥2, it follows from (51)–(54) that ∥un(k)-u(k)∥→0 as n→+∞.
If1<p<2, by Holder’s inequality, we obtain
(55)∫-kTkT‍ρ(t)|un′(k)(t)-u′(k)(t)|pdtm≤(∫-kTkT‍ρ(t)|un′(k)(t)-u′(k)(t)|2(|un′(k)(t)|+|u′(k)(t)|)2-pdt)p/2mm×(∫-kTkT‍ρ(t)|un′(k)(t)|+|u′(k)(t)|pdt)(2-p)/2m≤2(p-1)(2-p)/2mm×(∫-kTkT‍ρ(t)|un′(k)(t)-u′(k)(t)|2(|un′(k)(t)|+|u′(k)(t)|)2-pdt)p/2mm×(∫-kTkT‍ρ(t)|un′(k)(t)|+|u′(k)(t)|pdt)(2-p)/2m≤2(p-1)(2-p)/2mmm×(∫-kTkT‍ρ(t)|un′(k)(t)-u′(k)(t)|2(|un′(k)(t)|+|u′(k)(t)|)2-pdt)p/2mmm×(∥un(k)∥+∥u(k)∥)(2-p)p/2
for eachk∈N. Similarly,
(56)∫-kTkT‍s(t)|un(k)(t)-u(k)(t)|pdtm≤2(p-1)(2-p)/2(∫-kTkT‍s(t)|un(k)(t)-u(k)(t)|2(|un(k)(t)|+|u(k)(t)|)2-pdt)p/2mm×(∥un(k)∥+∥u(k)∥)(2-p)p/2.
It follows from1<p<2 and (54)–(56) that
(57)∫-kTkT‍ρ(t)(Φp(u′n(k)(t))-Φp(u′(k)(t)))mmm×(u′n(k)(t)-u′(k)(t))dtmm+∫-kTkT‍s(t)(Φp(un(k)(t))-Φp(u(k)(t)))mmmmmm×(un(k)(t)-u(k)(t))dtm≥2(p-1)(2-p)/2dp(∥un(k)∥+∥u(k)∥)2-pmm×[(∫-kTkT‍ρ(t)|un′(k)(t)-u′(k)(t)|pdt)2/pmmm+(∫-kTkT‍s(t)|un(k)(t)-u(k)(t)|pdt)2/p]m≥dp2p-2∥un(k)-u(k)∥2(∥un(k)∥+∥u(k)∥)2-p.
In view of (51)–(53) and (57), we have∥un(k)-u(k)∥→0asn→+∞,  for eachk∈N.
Therefore,{un(k)}converges strongly tou(k)inE(k), for eachk∈N. Thus, for eachk∈N, J(k)(·,λ)satisfies thePScondition.Lemma 12.
Assume that (V1) and (V2) hold; then there exist an open intervalΛ⊆[0,+∞)and a positive real numberσ, such that, for eachλ∈Λ  andk∈N,  (Pk) has at least three 2kT-periodic solutions inE(k)whose norms are less thanσ.Proof.
Let{un(k)}be a weakly convergent sequence to u(k)inE(k); then{un(k)}converges uniformly sequence tou(k)on[-kT,kT]. The continuity and convexity of(1/p)∥u(k)∥p imply that(1/p)∥u(k)∥pis sequentially weakly lower continuous [28, Lemma 1.2], for eachk∈N, which combining the continuity off(k)yields that
(58)liminfn→+∞[1p∥un(k)∥p-λ∫-kTkT‍F(k)(t,un(k))dt]m≥1p∥u(k)∥p-λ∫-kTkT‍F(k)(t,u(k))dt.
Hence,J(k)(·,λ)is sequentially weakly lower semi-continuous, for eachk∈N.
It is obvious thatJ(k)(u,·)is continuous and concave for eachu∈E(k). In view of Lemmas 10 and 11, it follows from Lemma 7 that there exist an open intervalΛ⊆[0,+∞)and a positive real numberσ, such that, for eachλ∈Λ andk∈N, J(k)(·,λ)has at least three critical points inE(k)whose norms are less thanσ. Therefore, we can reach our conclusion by using Lemma 2.Lemma 13.
Assume that (V3) holds. Letu~(k)∈E(k)be one of the three 2kT-periodic solutions of system  (Pk) obtained by Lemma 12 for eachk∈N. Then there exists a subsequence{u~(kj)}of{u~(k)}k∈Nconvergent to a certainu~0∈C1(R,Rn)inCloc1(R,Rn).Proof.
From Lemma12, we have
(59)∥u~(k)∥E(k)<σ,
which combining Lemma 5 yields that there exists a positive constantM1independent ofksuch that
(60)∥u~(k)∥L2kT∞≤M1.
Thus, we obtain that{u~(k)}k∈N is a uniformly bounded sequence. Next, we will show that{u~(k)′}k∈Nand{ρΦp(u~(k)′)′}k∈Nare also uniformly bounded sequences. Since{u~(t)(k)}is a 2kT-periodic solutions of system (Pk) for everyt∈[-kT,kT),  we have
(61)(ρ(t)Φp(u~(k)′(t)))′m=s(t)Φp(u~(k)(t))-λf(k)(t,u~(k)(t)).
By (60), (61), and (V3), we get
(62)|(ρ(t)Φp(u~(k)′(t)))′|≤|s(t)Φp(u~(k)(t))|m+λ|f(k)(t,u~(k)(t))|≤sup0≤t<kT,|x|≤M1|s(t)Φp(x)|m+λsup0≤t<kT,|x|≤M1|f(t,x)|≡M2,t∈[-kT,kT),
which yields that
(63)∥(ρ(t)Φp(u~(k)′(t)))′∥L2kT∞≤M0,k∈N.
Then, from (63), (V3), and the definition ofΦp(x), we obtain
(64)∥u~(k)′′(t)∥L2kT∞≤M3,k∈N.
Fori=-k,-k+1,…,k-1, by the continuity ofu~(k)′(t), we can chooseti(k)∈[iT,(i+1)T]such that
(65)u~(k)′(ti(k))=1T∫iT(i+1)T‍u~(k)′(s)ds=T-1[u(k)((i+1)T)-u(k)(iT)];
it follows that fort∈[iT,(i+1)T], i=-k,-k+1,…,k-1(66)|u~(k)′(t)|=|∫ti(k)t‍u~(k)′′(s)ds+u~(k)′(ti(k))|≤∫iT(i+1)T‍|u~(k)′′(s)|ds+|u~(k)′(ti(k))|≤M3T+T-1|u(k)((i+1)T)-u(k)(iT)|≤M3T+2M1T-1≡M4.
Consequently,
(67)∥u~(k)′(t)∥L2kT∞≤M4,k∈N.
Now we prove that the sequences{u~(k)}k∈Nand{u~(k)′}k∈N are uniformly bounded and equicontinuous. In fact, for everyk∈Nandt1,t2∈R, we have by (67)
(68)|u~(k)(t1)-u~(k)(t2)|=|∫t1t2‍u~(k)′(s)ds|≤∫t1t2‍|u~(k)′(s)|ds≤M4|t1-t2|.
Similarly, from (64), we have
(69)|u~(k)′(t1)-u~(k)′(t2)|≤M3|t1-t2|.
Then, by application of the Arzelà-Ascoli Theorem, we obtain the existence of a subsequence{u~(kj)}of{u~(k)}k∈N and a functionu~0such that
(70)u~(kj)⟶u~0,asj⟶∞inCloc1(R,Rn).
Thus, Lemma13 is proved.Lemma 14.
Letu~0∈C1(R,Rn)be determined by Lemma 13. Thenu~0is a nontrivial homoclinic solution of system (P).Proof.
The first step is to show thatu~0is a solution of system (P). By Lemma 13, one has
(71)(ρ(t)Φp(u~(kj)′(t)))′=s(t)Φp(u~(kj)(t))-λf(kj)(t,u~(kj)(t)),
fort∈[-kjT,kjT), j∈N. Takea,b∈Rwitha<b. There existsj0∈Nsuch that for allj>j0one has
(72)(ρ(t)Φp(u~(kj)′(t)))′=s(t)Φp(u~(kj)(t))-λf(t,u~(kj)(t)),fort∈[a,b].
Integrating (72) fromatot∈[a,b], we obtain
(73)ρ(t)Φp(u~(kj)′(t))-ρ(a)Φp(u~(kj)′(a))m=∫at‍[s(v)Φp(u~(kj)(v))-λf(v,u~(kj)(v))]dv,
fort∈[a,b]. Since (70) shows thatu~(kj)→u~0uniformly on[a,b]andu~(kj)′→u~0′uniformly on[a,b]asj→∞. Letj→∞in (73), we get
(74)ρ(t)Φp(u~0′(t))-ρ(a)Φp(u~0′(a))m=∫at‍[s(v)Φp(u~0(v))-λf(v,u~0(v))]dv,
fort∈[a,b]. Sinceaandbare arbitrary, (74) yields thatu~0is a solution of system (P). It is easy to see thatu=0is not a solution of system (P) forf(t,0)≠0and sou~0≠0.
Secondly, we will prove thatu~0(t)→0ast→±∞. By (59), we have
(75)∫-kTkT‍(ρ(t)|u~(k)′(t)|p+s(t)|u~(k)(t)|p)dt≤σp,k∈N.
For everyl∈N, there existsj1∈Nsuch that forj>j1(76)∫-lTlT‍(ρ(t)|u~(kj)′(t)|p+s(t)|u~(kj)(t)|p)dt≤σp.
Letj→∞in the above and use (70), and it follows that for eachl∈N,
(77)∫-lTlT‍(ρ(t)|u~0′(t)|p+s(t)|u~0(t)|p)dt≤σp.
Letl→∞in the above, and we get
(78)∫-∞∞‍(ρ(t)|u~0′(t)|p+s(t)|u~0(t)|p)dt≤σp.
Thus
(79)∫|t|≥r‍(ρ(t)|u~0′(t)|p+s(t)|u~0(t)|p)dt⟶0,asr⟶∞.
Combining the above with (V3) we have
(80)∫|t|≥r‍(|u~0′(t)|p+|u~0(t)|p)dt⟶0,asr⟶∞.
By (26), we obtain
(81)|u~0(t)|≤p1/p(∫t-1/2t+1/2‍(|u~0(s)|p+|u~0′(s)|p)ds)1/p.
Combining (80) with (81), we getu~0(t)→0ast→±∞.
Finally, we show that(82)u~0′(t)⟶0ast⟶±∞.
From (60) and (70), one has
(83)|u~0(t)|≤M1,fort∈R.
From this and (64), we have
(84)∥u~0′′(t)∥≤M3,fort∈R.
If (82) does not hold, then there existε0∈(0,1/2)and a sequence{tk}such that
(85)|t1|<|t2|<|t3|<⋯<|tk|<|tk+1|,k=1,2,…,|u~0′(tk)|≥2ε0,k=1,2,…,
which yield that fort∈[tk,tk+ε0/(1+M3)](86)|u~0′(t)|=|u~0′(tk)+∫tkt‍u~0′′(s)ds|≥|u~0′(tk)|-∫tkt‍|u~0′′(s)|ds≥ε0.
It follows that
(87)∫-∞∞‍|u~0′(t)|pdt≥∑k=1∞‍∫tktk+ε0/(1+M3)‍|u~0′(t)|pdt=∞,
which contradicts to (78) and so (82) holds. The proof is completed.Lemmas13 and 14 imply that the limit of the 2kT-periodic solutions of system (Pk) is a nontrivial homoclinic solution of system (P). Combining this with Lemma 10–Lemma 12, we can get the following.Theorem 15.
Assume that (V1), (V2), and (V3) hold. Then system(P) possesses three nontrivial homoclinic solutions.


## 4. Example

Example 1.
Consider the followingp-Laplacian problem:
(88)((t+3)Φ3(u′(t)))′-(2t+2)Φ3(u(t))+λf(t,u(t))=0,
whereλ∈[0,+∞),kT=2, and
(89)f(k)(t,x)=tx+1,∀(t,x)∈[-2,2]×(-∞,+∞).
It is obvious that (V3) holds and for everyt∈[-2,2],(90)F(k)(t,x)=t2x2+x-2,∀(t,x)∈[-2,2]×(-∞,+∞).
Then,
(91)limx→+∞F(k)(t,x)x2=t2
for eacht∈[-2,2].Thus, there existsc4>0such that
(92)F(k)(t,x)≤2|x|2∀t∈[-2,2],|x|≥c4,
which combining the continuity of F(k)(t,x)-2|x|2on[-2,2]×[-c4,c4]yields that there exists constantc5>0such that
(93)F(k)(t,x)≤2|x|2+c5foreach(t,x)∈[-2,2]×[-∞,+∞).
Therefore, (V2) is satisfied. Furthermore, in view of Lemma 5,M=4. Letη1=η2=4, δ1=1, δ2=1, andc1=(2-1)/2;  thenK1=0, K2=6, K3=1, E=1.112×10-3, Ω=1/2, and [2-(-2)]max(t,x)∈[-2,2]×[-(2-1)/2,(2-1)/2]F(k)(t,x)≤0. Thus (V1) is satisfied. Moreover, f(k)(t,0)=1≠0. In view of Theorem 15, we have that Example 1  possesses three nontrivial homoclinic solutions.



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*Source: 183585-2013-09-18.xml*