# Existence of Homoclinic Orbits for a Singular Differential Equation Involvingp-Laplacian

Authors: Honghui Yin; Bo Du; Qing Yang; Feng Duan
Journal: Journal of Function Spaces
 (2020)

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## Abstract

The efficient conditions guaranteeing the existence of homoclinic solutions to second-order singular differential equation withp-Laplacian ϕpx′t′+fx′t+gxt+ht/1−xt=et are established in the paper. Here, ϕps=sp−2s,p>1,f,g,h,e∈Cℝ,ℝ with ht+T=ht. The approach is based on the continuation theorem for coincidence degree theory.


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## Body

## 1. Introduction

In the last years, homoclinic solutions for Hamiltonian systems and differential and difference systems have been studied by several authors. Based on variational methods and critical points theory, Rabinowitz [1] has given fundamental contributions to homoclinic solutions for Hamiltonian systems. Carriãro and Miyagaki [2] obtained the existence of homoclinic orbits for second-order time-dependent Hamiltonian systems. Izydorek and Janczewska [3] obtained that a homoclinic orbit is obtained as a limit of 2kT-periodic solutions of a certain sequence of the second-order differential equations. By means of an extension of Mawhin’s continuation theorem, Lu et al. [4] obtained the existence of homoclinic solutions for a class of second-order neutral functional differential systems. Ding and Guo [5] showed that there exists at least one homoclinic solution for the anomalous diffusion system. For more results about homoclinic solutions, see, e.g., [6–10] and relevant references.In recent years, homoclinic solution problems of second-order singular differential equation have raised concerns. Bonheure and Torres [11] studied the existence of homoclinic solutions for the model scalar second-order boundary value problem
(1)−u′′+cxu′+axu=bxupx,where a,b,c∈Cℝ,ℝ,p>0. When c=0, Equation (1) has a good variational structure and can be studied by variational method for Equation (1), see [1, 12, 13]. When c≠0, variational method cannot be used to study Equation (1) because of the no good variational structure. Hence, based on the method of the upper and lower solutions and fixed point theorem on cones, the authors obtained the existence of homoclinic solutions for Equation (1) which is different from the variational methods used in [14–16].Motivated by the above work, this paper is devoted to the study of the existence of homoclinic solutions to second-order singular differential equation withp-Laplacian:
(2)ϕpx′t′+fx′t+gxt+ht1−xt=et,where ϕps=sp−2s,p>1,f,g,h,e∈Cℝ,ℝ with ht+T=ht>0. As in the literature, a solution ut of Equation (2) is called a homoclinic solution if ut⟶0 as ∣t∣⟶+∞. When such a solution satisfies in addition to u′t⟶0 as ∣t∣⟶+∞, it is usually called a homoclinic solution or a pulse, although here, 0 is not a stationary solution of Equation (2). Since Equation. (2) is a strongly nonlinear equation, the traditional methods (including fixed point theorem and lower and upper solutions) are no longer applicable to study homoclinic solutions to Equation (2), so a new continuation theorem due to Mana’sevich and Mawhin will be developed for studying Equation (2).The distinctive contributions of this paper are outlined as follows:(1)
The problem (2) is a more general form compared with existing problems (see [1, 11–13]). Hence, the results of this paper can be extended to other more specific problems(2)
Due to singularity, it is very difficult for estimating priori bound. In order to overcome this difficulty, we develop a new technique introduced in [17] for continuation theorem(3)
A unified framework is established to handle second-order equations with singularity term andp-Laplacian operatorThe following sections are organized as follows: In Section2, some useful lemmas and notations are given. In Section 3, sufficient conditions are established for the existence of homoclinic solutions of (2). In Section 4, two examples are given to show the feasibility of our results. Finally, Section 5 concludes the paper.


## 2. Preliminary and Some Lemmas

In this section, we give some notations and lemmas which will be used in this paper. The set of all positive integers is denoted by N. Let(3)CT=x∣x∈Cℝ,ℝ,xt+T≡xt,∀t∈ℝwith the norm φ0=maxt∈0,T∣φt∣,∀φ∈CT. When p≠2,p-Laplacian ϕps′′ in (2) is a nonlinear operator, the famous Mawlin’s continuation theorem [18] cannot be directly applied to (2). In order to generalize Mawlin’s continuation theorem, Mana’sevich and Mawhin [17] obtained the following continuation theorem for nonlinear systems with p-Laplacian-like operators:
(1)
For eachλ∈0,1, the problem
(4)ϕpu′=λft,u,u′,u0=uT,u′0=u′T,has no solution on ∂Ω(2)
The equation(5)Fa=1T∫0Tft,a,0dt=0,has no solution on ∂Ω∩ℝN(3)
The Brouwer degree(6)dBF,Ω∩ℝN,0≠0Theorem 1.
Assume thatΩ is an open bounded set in CT such that the above conditions (4) – (6) hold.Then, problem(7)ϕpu′=ft,u,u′,u0=uT,u′0=u′T,has a solution in Ω¯.Lemma 2 (see [19]).
Ifu∈Cℝ,ℝ, a>0,μ1,μ2>1are constants, then for everyt∈ℝ, the following inequality holds:(8)ut≤2a−1/μ1∫t−at+ausμ1ds1/μ1+a2a−1/μ2∫t−at+au′sμ2ds1/μ2.Lemma 3 (see [4]).
Letuk∈C2kT1 be a sequence of 2kT-periodic functions, such that for each k∈N, uk satisfies
(9)uk0≤A0,uk′0≤A1,where A0,A1 are constants independent of k∈N. Then, there exist u0∈Cℝ,ℝ and a subsequence ukj of uk such that for each j∈N(10)maxt∈−jT,jT∣ukit−u0t∣⟶0asi⟶+∞.For investigating the existence of homoclinic solutions to (2), for each k∈N, we firstly consider the existence of 2kT-periodic solutions ukt for the following equation:
(11)ϕpx′t′+fx′t+gxt+ht1−xt=ekt,where ek:ℝ⟶ℝ is a 2kT-periodic extension such that
(12)ekt=et,t∈−kT,kT−ε0,ekT−ε0+h−kT−hkT−ε0ε0t−kT+ε0,t∈kT−ε0,kT,here ε0∈0,T/2 is a constant.In the present paper, we list the following assumptions:(H1). f:ℝ⟶ℝ is a continuous bounded nonnegative function(H2). g:ℝ⟶ℝ is strictly monotone increasing and there are positive constants σ and n such that
(13)xgx≥σxn+1forx∈ℝ(H3).
(14)supt∈ℝ∣et∣=ρ1<+∞,∫ℝetn+1ndt=ρ2<+∞,n>0


## 3. Main Results

Letyt=1−xt, then (11) is changed into the following form:
(15)ϕp−y′t′+f−y′t+g1−yt+htyt=ekt.Obviously, the existence of2kT-periodic solutions to (2) is a transfer to the existence of 2kT-periodic solutions to (15). For (15), consider the corresponding parameter equation:
(16)ϕp−y′t′+λf−y′t+λg1−yt+λhtyt=λekt,λ∈0,1.Here, we give the main results of the present paper in the following theorem.Theorem 4.
Assume that the assumptions (H1)–(H3) hold. Then, Equation (2) has at least one positiveT-periodic solution, ifρ1>f0,hl/ρ1−f0<1,hl=mint∈ℝht.Proof.
Let(17)Ω=x:ϕp−y′t′+λf−y′t+λg1−yt+λhtyt=λekt,λ∈0,1,xt+2kT=xt,xt>0,t∈−kT,kT,k∈Nwith the norm
(18)φ∞=maxt∈−kT,kTφ0,φ′0.
Letu∈Ω, then u satisfies
(19)ϕp−u′t′+λf−u′t+λg1−ut+λhtut=λekt.
There existt1,t2∈−kT,kT such that
(20)ut1=maxt∈−kT,kTut,ut2=mint∈−kT,kTut.
This implies that(21)u′t1=u′t2=0.
By (19), we have
(22)f0+g1−ut1+ht1ut1=ekt1,(23)g1−ut1≥−ek0−f0=−ρ1−f0.
In view of monotonicity ofg, it follows by (22) that
(24)ut1<1−g−1−ρ1−f0≔A1.
On the other hand,(25)f0+g1−ut2+ht2ut2=ekt2.
We claim that(26)ut2>hlρ1−f0≔A2.
In fact, if (26) is not true, then
(27)ut2≤hlρ1−f0<1.
By (25), we have
(28)g1−ut2≤ρ1−f0−hlut2,≤ρ1−f0−ρ1−f0,=0.Thus, we haveut2≥1 which is a contradiction to (27). From (24) and (26), we have
(29)A2<ut<A1forallt∈−kT,kT.Now, we estimate the bound ofu′t. For i∈−k,−k+1,⋯,k−1, there exists ti∈iT,i+1T such that
(30)u′ti=1T∫iTi+1Tu′sds.It follows from (29) and (30) that
(31)∣u′ti∣=1T∫iTi+1Tu′sds,=1TuiT−ui+1T<2A1T.Integrating (19) over ti,t, we have
(32)ϕp−u′t=ϕp−u′ti−λ∫titf−u′sds,−λ∫titg1−usds−λ∫tithsusds+λ∫titeksds,u′tp−1≤2A1Tp−1+∫iTi+1T∣f−u′s∣ds,+∫iTi+1T∣g1−us∣ds+∫iTi+1T∣hsus∣ds+∫iTi+1T∣eks∣ds,≤2A1Tp−1+Tfm+TgA2,A1+ThmA2+Tρ1≔A3,where gA2,A1=maxA2≤u≤A1∣g1−u∣,fm=maxs∈ℝ∣fs∣,hm=maxs∈ℝ∣hs∣. Thus,
(33)u′0=maxt∈−kT,kT∣u′t∣≤A31/p−1.Letu∞=maxA1,A2,A31/p−1+1 for u∈Ω. Then, condition (1) of Theorem 1 holds. Next, let
(34)Fa=f0+g1−a+h¯a−ek¯=0,a∈ℝ.Clearly, equation (34) has no solution on ∂Ω∩ℝ. Hence, condition (2) of Theorem 1 holds. Furthermore, by h¯>0 and (29), we have the following inequalities:
(35)f0+g1−u+h¯u−ek¯>0foru∈0,A2,f0+g1−u+h¯u−ek¯<0foru∈A1,+∞.Thus(36)f0+g1−A2+h¯A2−ek¯f0+g1−A1+h¯A1−ek¯<0,and dBF,γ0,M1∩ℝ,0≠0, i.e., condition (3) of Theorem 1 holds. By using Theorem 1, we see that Equation (15) exists at least one positive 2kT-periodic solution ukt such that
(37)A2≤ukt≤A1,∣u′kt∣≤A31/p−1,k∈N.Sinceyt=1−xt, there exist positive constants B1, B2, and B3 such that
(38)B1≤ωkt≤B2,∣ω′kt∣≤B3,k∈N,where ωkt is 2kT-periodic solution to (11). Thus,
(39)ϕpωk′t′+fωk′t+gωkt+ht1−ωkt=ekt,In view of Lemma3, there exist ω0∈C1ℝ,ℝ and a subsequence ωki of ωk such that
(40)maxt∈−jT,jT∣ωkit−ω0t∣⟶0,maxt∈−jT,jT∣ωki′t−ω0′t∣⟶0ast⟶+∞,j∈N.From (39), (40), and the standard argument, ω0t is a solution of (11), i.e.,
(41)ϕpω0′t′+fω0′t+gω0t+ht1−ω0t=et,t∈ℝ.Now, we will show(42)ω0t⟶0,ω0′t⟶0as∣t∣⟶+∞.Multiplying (39) by ωkt and integrating it over −kT,KT, we have
(43)∫−kTkTωk′tp+∫−kTkTfωk′sωksds+∫−kTkTgωksωksds+∫−kTkThsωks1−ωksds=∫−kTkTeksωksds.From (43), assumptions (H1) and (H2), we have
(44)∫−kTkTωk′t∣p+σ∫−kTkTωksn+1ds≤∫−kTkT∣eksωks∣ds≤∫−kTkTeksn+1/ndsn/n+1∫−kTkTωksn+1ds1/n+1.In view of (H3) and (12), we have
(45)∫−kTkTeksn+1/nds=∫−kTkT−ε0eksn+1/nds+∫kT−ε0kTeksn+1/nds≤∫−kTkT−ε0esn+1/nds+ε0ρ1n+1/n≤∫ℝesn+1/nds+ε0ρ1n+1/n≤ρ2+ε0ρ1n+1/n≔C1.In view of (44) and (45), then
(46)σ∫−kTkTωksn+1ds≤C1n/n+1∫−kTkTωksn+1ds1/n+1,(47)∫−kTkTωk′tp≤C1n/n+1∫−kTkTωksn+1ds1/n+1.It follows by (46) that
(48)∫−kTkTωksn+1ds≤C1σ−n+1/n,k∈N.In view of (47) and (48), then
(49)∫−kTkTωk′tp≤C1σ−1/n,k∈N.From (48), (49) and standard limit analysis, we have
(50)∫−∞+∞ω0s∣n+1ds+∫−∞+∞ω0′spds≤C1σ−n+1/n+C1σ−1/n,(51)∫t>rω0sn+1+ω0′spds⟶0asr⟶+∞,which together with Lemma 2 yields that
(52)∣ω0t∣≤2a−1/n+1∫t−at+aω0sn+1ds1/n+1+a2a−1/p∫t−at+aω0′spds1/p≤max2a−1/n+1,a2a−1/p∫t−at+aω0sn+1ds1/n+1+∫t−at+aω0′spds1/p⟶0as∣t∣⟶+∞.Thus,(53)∣ω0t∣⟶0as∣t∣⟶+∞.Next, we prove(54)∣ω0′t∣⟶0as∣t∣⟶+∞.Furthermore, by (38) we have
(55)B1≤ω0t≤B2,∣ω′0t∣≤B3,t∈ℝwhich together with (41) yields that
(56)∣ϕpω0′t′∣≤fB3+gB1,B2+hmB1−1+e0≔C2,where fB3=max∣s∣≤B3∣fs∣,gB1,B2=maxB1≤s≤B2∣gs∣. If (54) does not hold. Then, there are constant δ∈0,1/2 and a sequence tk with ∣t1∣<∣t2∣<⋯ such that
(57)∣ϕpω0′tk∣≥2δ,k=1,2,⋯,∣ϕpω0′t∣=∣ϕpω0′tk+∫tktϕpω0′s′ds∣,≥∣ϕpω0′tk∣−∫tkt∣ϕpω0′s′∣ds,≥δfort∈tk,tk+δ1+C2,∫−∞+∞∣ϕpω0′t∣dt≥∑k=1+∞∫tktk+δ/1+C2∣ϕpω0′t∣dt=+∞,which contradicts to (50). It is easy to see that (54) holds. Thus, ω0t is just a homoclinic solution to Eq. (2).


## 4. Examples

This section presents two examples that demonstrate the validity of our theoretical results.Example 5.
Consider the following equation:(58)u′t2u′t′+1+2x′2t+2x3t+3−1/2cost1−xt=5+12sin2t,where
(59)p=4,fx′t=1+2x′2t,gxt=2x3t,ht=3−12cost,et=5+12sin2t.Obviously,ρ1=5.5,f0=1,hl=2.5,hl/ρ1−f0=5/7<1. We also easily check that assumptions (H1)-(H3) hold. Based on Theorem 4, Equation (58) has at least one nontrivial homoclinic solution.Example 6.
Consider the following equation:(60)u′′t+3+2x′2t+4x3t+5−1/2sint1−xt=8+12cos2t,where
(61)p=2,fx′t=3+2x′2t,gxt=4x3t,ht=5−12sint,et=8+12cos2t.
Obviously,ρ1=8.5,f0=3,hl=4.5,hl/ρ1−f0=9/11<1. We also easily check that assumptions (H1)-(H3) hold. Based on Theorem 4, Equation (60) has at least one nontrivial homoclinic solution.


## 5. Conclusions

In this paper, we study a class of second-order singular equation withp-Laplacian. By employing some analytic techniques and continuation theorem due to Mana’sevich and Mawhin, we have presented some new sufficient criteria for the existence of homoclinic solutions for the above singular equation. These criteria possess adjustable parameters which are important in some applied fields. Finally, two examples are given to demonstrate the effectiveness of the obtained theoretical results. However, there exist many problems for further study such as heteroclinic orbits of second-order singular equations.



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