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Solitary Wave and Periodic Wave Solutions for a Class of Singular p-Laplacian Systems with Impulsive Effects.

1 Introduction

In this paper, we consider a class of second-order singular p-Laplacian systems with impulsive effects described by

[mathematical expression not reproducible], (1.1)

where p [greater than or equal to] 2, u = [([u.sub.1], [u.sub.2], ..., [u.sub.N]).sup.T] [member of] [R.sup.N], f [member of] C([R.sup.N], [R.sup.N]) and f may be singular at u = 0, [g.sub.j](u) = [grad.sub.u][G.sub.j](u) for some [mathematical expression not reproducible]. It is assumed that there exist an m [member of] N and a T > 0 such that 0 = [t.sub.0] < [t.sub.1] < [t.sub.2] < ... < [t.sub.m-1] < [t.sub.m] = T, [t.sub.j+m] = [t.sub.j] + T, [g.sub.j+m] = [g.sub.j], j [member of] Z.

During the past years, different types of impulsive differential equations have been studied by many authors. Some classical tools have been widely used to get the solutions of impulsive differential equations, such as fixed point theorems in cones, topological degree theory (including continuation method and coincidence degree theory), the method of lower and upper solutions, and the critical point theory. For the theory and classical results, we refer the readers to the references, [4], [17], [19], [26], [32] and books [2], [22], [31].

Recently, the study on the existence of homoclinic solutions for the impulsive differential equations has attracted many researchers' attention. See, to name a few, [8], [29], [32]. For example, in [32], by applying the variational methods, Zhang and Li established the existence result of homoclinic solutions of the following second order impulsive differential equations

[mathematical expression not reproducible],

where k [member of] Z, q [member of] [R.sup.N], [DELTA]q'([s.sub.k]) = q'([s.sup.+.sub.k]) - q'[s.sup.-.sub.k] with [mathematical expression not reproducible] for each k [member of] N, and there exist an m [member of] N and a T [member of] [R.sup.+] such that 0 = [s.sub.0] < [S.sub.1] < ... < [S.sub.m] = T, [S.sub.k+m] = [S.sub.k] + T and [g.sub.k+m] = [g.sub.k] for all k [member of] Z (that is, [g.sub.k] is m-periodic in k).

Singular equations appear in a lot of physical models, see [9], [15], [18], [21], [30] and the references therein. The existence periodic solutions and homoclinic solutions of different kinds of singular equations has been proposed by many authors, see [3], [5], [6], [7], [11], [13], [20], [27], [28], [33] and the references therein. Singular problems with impulsive effects have been scarcely studied, see [1], [12], [23], [24]. For example, in [12], the author and Luo considered the following first-order singular problems:

x'(t) + [x.sup.-[alpha](t)] = e(t)

under impulsive conditions

[DELTA](x([t.sub.k])) = x([t.sup.+.sub.k]) - x([t.sup.-.sub.k]) = [I.sub.k](x([t.sub.k])), k = 1, 2, ..., q - 1,

where [alpha] > 0, e : R [right arrow] R is continuous and T-periodic, [I.sub.k] : R [right arrow] R(k = 1, 2, ..., q - 1) are continuous and [I.sub.k+q] [equivalent to] [I.sub.k]. [t.sub.k], k = 1, 2, ... q - 1, are the instants where the impulses occur and 0 = [t.sub.0] < [t.sub.1] < [t.sub.2] < [t.sub.3] < ... < [t.sub.q-1] < [t.sub.q] = T, [t.sub.k+q] = [t.sub.k] + T. By applying the continuation theorem due to Mawhin and Gaines, the authors proved that the positive periodic solution was generated by impulses.

However, to the best of our knowledge, few researchers have studied the existence of periodic wave solutions and the nonexistence of solitary wave solutions for the singular p-Laplacian problems with impulsive effects. Inspired by the works mentioned above, in this paper, by means of the mountain pass theorem and an approximation technique, we establish the existence results of periodic wave solutions and nonexistence results of solitary wave solutions for system (1.1).

Definition 1. Suppose that u(s) is a solution of the system (1.1) for s [member of] R, u(s) is called a solitary wave solution if [mathematical expression not reproducible]. Usually, a solitary wave solution of system (1.1) corresponds to a homoclinic solution of system (1.1). Similarly, aperiodic wave solution of system (1.1) corresponds to a periodic solution of system (1.1).

Thus, in order to investigate the existence of periodic wave solutions and nonexistence of solitary wave solutions of system (1.1), we only need to prove the existence of periodic solutions and nonexistence of homoclinic solutions of system (1.1).

A function u [member of] C(R, [R.sup.N]) is a solution of system (1.1) if function u satisfies (1.1). A solution u of system (1.1) is homoclinic to 0 if u(t) [right arrow] 0 and u' ([t.sup.[+ or -]]) [right arrow] 0 as t [right arrow] [+ or -] [infinity], and the corresponding orbit is called a homoclinic orbit.

In general, it is very difficult (if not impossible) to construct a suitable functional such that the existence of its critical point implies to that of a homoclinic orbit of system (1.1). First of all, we assume that there exists homoclinic orbits of system (1.1). Then, by applying an approximation technique, we show that the existence of homoclinic orbit is obtained as a limit of 2kT-periodic solutions of the following sequence of impulsive differential equations

[mathematical expression not reproducible], (1.2)

where [e.sub.k] : R [right arrow] [R.sup.N] is a 2kT periodic extension of the restriction of e to the interval [-kT, kT].

Remark 1. Note that the domain under consideration is unbounded, and thus, there is a lack of compactness for the Sobolev embedding. To overcome this difficulty, we show the existence of a homoclinic orbit of system (1.1) by proving that (1.2) has 2kT periodic solutions whose limit gives a homoclinic orbit of system (1.1). Another difficulty is that we have to deal with the impulsive perturbations in system (1.2).

For the sake of convenience, we list the following assumptions:

* [H1] There exist constants a > 0 and [gamma] [member of] (1,p] such that for all u [member of] [R.sup.N],

- F(u) [greater than or equal to] a[absolute value of u][gamma],

where F [member of] [C.sup.1]([R.sup.N], R) and f(u) = [grad.sub.u]F(u).

* [H2] There exists a constant [theta] > p such that

-F(u) [less than or equal to] -f (u)u [less than or equal to] -[theta]F(u), for all u [member of] [R.sup.n].

* [H3] [G.sub.j](0) = 0, [g.sub.j](u) = o([[absolute value of u].sup.p-1]) as [absolute value of u] [right arrow] 0, j = 1, 2, ..., m.

* [H4] There are two constants [mu], [beta] with [mu] > [theta] and 0 < [beta] < [mu] - [theta] such that

0 <[mu][G.sub.j](u) [less than or equal to] [g.sub.j](u)u + [beta]a[absolute value of u][gamma], u [member of] [R.sup.N]\{0}, j = 1, 2, ..., m.

* [H5] e [member of] C(R, [R.sup.N]) [intersection] [L.sup.p](R, [R.sup.N]) [intersection] [L.sup.q](R, [R.sup.N]) with

[mathematical expression not reproducible],

where M = sup {[G.sub.j](u) : j = 1, 2, ..., m, [absolute value of u] = 1}, 1/p +1/q = 1, [delta] [member of] (0,1] such that

[mathematical expression not reproducible]

and C > 0 is a constant.

* [H6] [M.sub.1] > [beta]a/([mu] - [gamma]), where [M.sub.1] = inf {[G.sub.j](u) : j = 1, 2, ...,m, [absolute value of u] = 1}.

The remainder part of this paper is organized as follows. Section 2 is devoted to state some necessary definitions, lemmas and the variational structure. Section 3 is devoted to state the main results and an example is given to support the established results. Section 4 is devoted to prove the main results.

2 Preliminary

Throughout this paper, we adopt the convention that [mathematical expression not reproducible].

Define the space

[mathematical expression not reproducible].

Then [H.sub.2kT] is a separable and reflexive Banach space with the norm defined by

[mathematical expression not reproducible].

Denote [L.sup.[infinity].sub.2kT] (R, R)by the space of 2kT periodic essentially bounded measurable functions from R into R with norm given by

[mathematical expression not reproducible].

Let [[OMEGA].sub.k] = {- km +1, -km + 2, ..., 0, 1, 2, ..., km - 1, km} and define a functional [[phi].sub.k]

[mathematical expression not reproducible], (2.1)

where

[mathematical expression not reproducible]. (2.2)

Then [[phi].sub.k] is Frechet differentiable at any u [member of] [H.sub.2kT]. For any v [member of] [H.sub.2kT], by a simple calculation, we have

[mathematical expression not reproducible].

Thus, by [H2], we get

[mathematical expression not reproducible]. (2.3)

It is evident that critical points of the functional [f.sub.k] are classical 2kT periodic solutions of system (1.2).

Lemma 1. [10] There is a positive constant C such that for each k [member of] N and u [member of] [H.sub.2kT] the following inequality holds:

[mathematical expression not reproducible].

Lemma 2. [14] There exists [r.sub.p] > 0, for any x, y [member of] [R.sup.N] such that

[mathematical expression not reproducible].

Lemma 3. Suppose u : R [right arrow] [R.sup.N] is a continuous 'mapping such that

u' [member of] [L.sup.p.sub.loc] (R,Rn) := {u : R [right arrow] R |for any finite interval [a, b], u|sub.[a,b]] [member of] [L.sup.p]([a,b], [R.sup.N])}.

Then for a, b [greater than or equal to] 0 with a + b > 0, the following inequality holds:

[mathematical expression not reproducible].

In particular,

[mathematical expression not reproducible].

Proof. Fix t [member of] R, for any given [delta] [member of] R, we can have

[absolute value of u(t)] [less than or equal to] [absolute value of u([delta])] + [[integral]sup.t.sub.[delta]] u'(s)ds]. (2.4)

Integrating (2.4) on the interval [t - a, t + b] with respect to [delta], then it follows from the Jensen and Holder inequalities that

[mathematical expression not reproducible],

then, we obtain

[mathematical expression not reproducible].

In particular, if a = b and a = b = 1, respectively, we have

[mathematical expression not reproducible].

Therefore, the proof is completed.

3 Main results

Theorem 1. Assume that [H1]-[H6] hold, then the system (1.1) possesses at least one 2kT-periodic wave solution.

Theorem 2. Assume that [H1]-[H6] hold, then the system (1.1) possesses no solitary wave solution.

We conclude this section considering the following example.

Example 1. Consider system (1.1) with [mathematical expression not reproducible]. Then, it is easy to verify that F, f, [G.sub.j], [g.sub.j], e satisfy the assumptions of Theorem 1-2 with a = 1/4, [gamma] = 4, [mu] = 6, [theta] = 5 and [beta] = 1/2. Therefore, system (1.1) possesses at least one periodic wave solution and no solitary wave solution, which are induced by impulses.

4 Proofs of main results

Now, we give the proof of Theorem 1 by using the mountain-pass theorem [16].

4.1 Proof of Theorem 1

Proof. For any given sequence {[u.sub.n]} [member of] [H.sub.2kT] such that {[[phi].sub.k]([u.sub.n])} is bounded and [mathematical expression not reproducible], there exists a constant [C.sub.1] > 0 such that

[mathematical expression not reproducible],

where [H.sup.*.sub.2kT] is the dual space of [H.sub.2kT.] The rest of the proof is divided into three steps.

Step 1. We show that {un} is bounded. In fact, by (2.1) and [H4]

[mathematical expression not reproducible]. (4.1)

By (2.3) and [mu] > [theta] > p, we have

[mathematical expression not reproducible]. (4.2)

From (4.1) and (4.2), we can obtain

[mathematical expression not reproducible], (4.3)

where q > 1 satisfying 1/p + 1/q =1.

On the other hand, from [H1], (2.2) and Lemma 1, we have

[mathematical expression not reproducible]. (4.4)

It follows from (4.3) and (4.4) that

[mathematical expression not reproducible].

Since p [greater than or equal to] [gamma] > 1 and 0 < [beta] < [mu] - [theta], then we can see that [mathematical expression not reproducible] is bounded.

Because [H.sub.2kT] is a reflexive Banach space, we can pick {[u.sub.n]} be a weakly convergent sequence to u in [H.sub.2kT], and {[u.sub.n]} converges uniformly to u in C[-kT,kT]. So, we have

[mathematical expression not reproducible]. (4.5)

Then

[mathematical expression not reproducible]. (4.6)

Thus, it follows from (4.5), (4.6) and Lemma 2 that [mathematical expression not reproducible]. Therefore, the functional [[phi].sub.k] satisfies the Palais-Smale condition.

Step 2. Define

[phi](s) = [s.sup.[mu]][G.sub.j] (u/s) - [beta][as.sup.[mu]-[gamma]/[mu] - [gamma][absolute value of u][gamma], j = 1, 2, ..., m, s > 0.

From [H4], we can have

[mathematical expression not reproducible],

which implies that [phi](s) is non-increasing for s > 0. Thus,

[mathematical expression not reproducible], (4.7)

for 0 < [u([t.sub.j])] [greater than or equal to] 1, j [member of], and

[mathematical expression not reproducible], (4.8)

for [u([t.sub.j])] [greater than or equal to] 1, j [member of] [[OMEGA].sub.k]. Note that [H1] implies that p [greater than or equal to] [gamma] > 0, thus, we have

[mathematical expression not reproducible]

If [mathematical expression not reproducible], then from (2.1), [H1], [H2], (4.6) and (4.7), we have

[mathematical expression not reproducible].

Further, we have

[mathematical expression not reproducible].

Set [rho] = [delta]/C, and

[mathematical expression not reproducible],

where C is defined in Lemma 1. Let [mathematical expression not reproducible]. Therefore, [[phi].sub.k](u) [greater than or equal to] [alpha] > 0.

Step 3. We choose [zeta] [member of] R, w = [pi]/T, Q(t) = (sin(wt), 0,..., 0) [member of] [H.sup.2T]\{0}. Then, we can see that Q([+ or -]T) = 0. Let

[m.sub.1]=min{F(u) : [absolute value of u(t)] [less than or equal to] 1, t [member of] [0,T]}, [m.sub.2]=min{F(u) : [absolute value of u(t)] = 1,t [member of] [0,T]},

then 0 > [m.sub.2] > [m.sub.1] > - [infinity].

Let h(s) = [s.sup.-[theta]]F(su), s > 0. It follows from [H2] that

h'(s) = (f (su)su - [theta]F(su))/[s.sup.[theta]+1] [greater than or equal to] 0.

Then, we can get

F(u) [greater than or equal to] [[absolute value of u].sup.[theta]]F (u/[absolute value of u]), [absolute value of u] [greater than or equal to] 1, (4.9)

which implies that

F(u) [greater than or equal to] [m.sup.2][[absolute value of u].sup.[theta]] + [m.sup.1], u [member of] R.

Set [absolute value of [zeta]u([t.sub.j])] [greater than or equal to] 1, j [member of] [[OMEGA].sub.k]. From (4.8), we can have

[mathematical expression not reproducible].

Define

[mathematical expression not reproducible]. (4.10)

It follows from (2.1), (2.2) and (4.9)-(4.10) that

[mathematical expression not reproducible].

Clearly,

[[phi].sub.k]([zeta][??]) [right arrow] [infinity] as [absolute value of [zeta]] [right arrow] [infinity].

Consequently, [[phi].sub.k] possesses a critical value [c.sub.k] > [alpha] > 0. Let [u.sub.k] denote the corresponding critical point of [[phi].sub.k] on [H.sub.2kT], that is,

[[phi].sub.k] ([u.sub.k]) = [c.sub.k], [[phi]'.sub.k] ([u.sub.k]) = 0. (4.11)

Hence, system (1.2) possesses a 2kT-periodic solution [u.sub.k]. Therefore, the system (1.1) possesses at least one 2kT-periodic wave solution.

Theorem 3. Let {[u.sub.k]} be the sequence defined in (4.11). Then there exist a subsequence {[u.sub.k,k]} of {[u.sub.k]} and a function [u.sub.0] [member of] [W.sup.1-p.sub.loc][intersection] [L.sup.[infinity].sub.loc](R, [R.sup.N]) such that {[u.sub.k,k]} converges to [u.sub.0] weakly in [W.sup.1-p.sub.loc] and strongly in [L.sup.[infinity].sub.loc] (R, [R.sup.N]).

Proof. We claim that there is a constant [M.sub.3] > 0 independent of k such that [mathematical expression not reproducible]. Let [e.sup.1] [member of] [H.sub.2T] \ {0} such that [e.sub.1]([+ or -]T) = 0, [e.sup.1]([t.sub.k)] [not equal to] 0 for some [t.sub.k] [member of] (-T, T) and [[phi].sub.1]([e.sub.1]) < 0. Define

[mathematical expression not reproducible].

We then extend [e.sub.k ](k = 1, 2,...) to be 2kT periodic, which, for convenience, we denote also by [e.sub.k]. It is clear that [e.sub.k] [member of] [H.sub.2kT] and [[phi].sub.k]([e.sub.k]) = f [[phi].sub.1]([e.sub.1]) [less than or equal to] 0.

Define [g.sub.k] : [0,1] [right arrow] [H.sub.2kT] by [g.sub.k](s) = [se.sub.k] for s [member of] [0,1]. Then, we can have

[mathematical expression not reproducible]

independently of k, where [c.sub.k] is a constant of (4.11).

As in Step 1 in the proof of Theorem 1, we can prove that {[u.sub.k]} is a bounded sequence in [W.sub.1,p]((- T, T), [R.sup.N]). Hence, we can choose a subsequence {[u.sub.1,k]} such that {[u.sub.1,k]} converges weakly in [W.sup.-1,p]((-T, T), [R.sup.N]) and strongly in [L.sup.[infinity]]((-T, T), [R.sup.N]). Note that {[u.sub.1,k]} is a bounded sequence in [W.sup.-1,p]((-2T, 2T), [R.sup.N]), we can choose a subsequence {[u.sub.2,k]} such that {[u.sub.2,k]} converges weakly in [W.sub.1,p]((-2T, 2T), [R.sup.N]) and strongly in [L.sup.[infinity]]((-2T; 2T), [R.sup.N]). Repeating this process, we obtain, for any positive integer n, a sequence {[u.sub.n,k]} that converges weakly in [W.sup.1,p]((-nT, nT), [R.sup.N]) and strongly in [L.sup.[infinity]]((-nT, nT), [R.sup.N]) satisfying

{[u.sub.k]} [contains] {[u.sub.1,k]} [contains] {[u.sub.2,k]} [contains] ... {[u.sub.n,k]} [contains] ....

Therefore, for any positive integer n, the sequence {[u.sub.j,j]} converges weakly in [W.sup.1,p]((-nT, nT), [R.sup.N]) and the sequence {[u.sub.j,j]} converges strongly in [L.sup.[infinity]]((-nT, nT), [R.sup.N]). Therefore, there exists a function [u.sub.0] [member of] [W.sup.1,p.sub.loc] (R, [R.sup.N]) [intersection] [L.sup.[infinity].sub.loc] (R, [R.sup.N]) such that the sequence {[u.sub.j,j]} converges weakly [u.sub.0] in [W.sup.1,p.sub.loc], [R.sup.N]) and strongly in [L.sup.[infinity].sub.loc](R, [R.sup.N]).

4.2 Proof of Theorem 2

Proof. We divide the proof into three steps.

Step 1. We show that [u.sub.0] is a solution to system (1.1). Here, for simplicity, we denote {[u.sub.k,k]} by {[u.sub.k}]. For any given interval (a, b) [contains] (-kT, kT) and any v [member of] [W.sup.1,p.sub.0]((a,b), [R.sup.N]), define

[mathematical expression not reproducible].

For any v [member of] [W.sup.1,p.sub.0]((a,b), [R.sup.N]), we get

[mathematical expression not reproducible]

then, it follows that

[mathematical expression not reproducible].

By using a similar argument as the proof of Lemma 2.5 in [25], we can show that [u.sub.0] is a solution to system (1.1).

Step 2. We prove that [u.sub.0](t) [right arrow] 0 as t [right arrow] [+ or -] [infinity]. Since {[u.sub.k]} is weakly convergent in [W.sup.1,p.sub.loc], it follows from Step 1 of the proof of Theorem 1 that there exists a constant [M.sub.3] > 0 such that

[mathematical expression not reproducible].

So, we can have

[mathematical expression not reproducible], (4.12)

which together with Lemma 3 yields that [u.sub.0](t) [right arrow] 0 as t [right arrow] [+ or -] [infinity].

Step 3. We prove that [mathematical expression not reproducible] as t [right arrow] [+ or -][infinity].

Note that [mathematical expression not reproducible]. By means of the Holder inequality, we have

[mathematical expression not reproducible]. (4.13)

From (4.12), we can see that

[mathematical expression not reproducible]. (4.14)

It follows from e [member of] [L.sup.p] (R, [R.sup.N]) that

[mathematical expression not reproducible]. (4.15)

In Step 1, we have prove that [u.sub.0] is a solution to system (1.1). Then,

[mathematical expression not reproducible]. (4.16)

However, f (0) [not equal to] 0. So that, from [mathematical expression not reproducible], (4.15) and (4.16), we can see that

[mathematical expression not reproducible]. (4.17)

Substituting (4.17) and (4.14) into (4.13), we can obtain

[mathematical expression not reproducible].

Hence, by the definition of homoclinic solutions, we can see that there are no existence of homoclinic solutions for system (1.1). Therefore, the system (1.1) possesses no solitary wave solution.

https://doi.org/10.3846/mma.2018.002

Acknowledgements

The authors would like to express their great thanks to the reviewers who carefully reviewed the manuscript. The research was supported by the National Natural Science Foundation of China (Grant No.11471109), the Construct Program of the Key Discipline in Hunan Province and Hunan Provincial Innovation Foundation for Postgraduate (CX2017B172).

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Fanchao Kong (a), Zhiguo Luo (b) and Hongjun Qiu (c)

(a) School of Mathematics and Big Data, Anhui University of Science and Technology Huainan, 232001 Anhui, China

(b) Department of Mathematics, Hunan Normal University Changsha, 410081 Hunan, China

(c) College of Science, Jiujiang University Jiujiang, 332005 Jiangxi, China

E-mail(corresp.): fanchaokong88@sohu.com

Received March 8, 2017; revised December 5, 2017; accepted December 6, 2017
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Author:Kong, Fanchao; Luo, Zhiguo; Qiu, Hongjun
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