Variational method to a fractional impulsive \((p,q)\) Laplacian coupled systems with partial sub \((p,q)\) linear growth

R E S E A R C H

Open Access

Variational method to a fractional impulsive

(

p

,

q

)

-Laplacian coupled systems with partial

sub-

(

p

,

q

)

linear growth

Cuiling Liu

_{, Xingyong Zhang}

_{and Junping Xie}

*_{Correspondence:}

[email protected]

1_{Faculty of Science, Kunming}

University of Science and Technology, Kunming, P.R. China 3_{School of Mathematics and}

Statistics, Central South University, Changsha, P.R. China

Full list of author information is available at the end of the article

Abstract

In this paper, by using the least action principle, an existence result of nontrivial weak solutions for a class of fractional impulsive coupled systems with (p,q)-Laplacian is obtained if the nonlinear term has sub-(p,q) linear growth, and by using an extension of Clark’s theorem, infinitely many solutions of the system are obtained if the

nonlinear term has partial sub-(p,q) linear growth.

Keywords: Fractional coupled systems; (p,q)-Laplacian; Multiplicity; Clark’s theorem; Impulsive effects; The least action principle

1 Introduction and main results

In recent years, because of the important applications of fractional differential equations to engineering, physics, chemistry and biology, the existence and multiplicity of solutions for fractional differential equations have been investigated extensively by different meth-ods such as fixed point theory, degree theory, monotone iterative technique and upper and lower solutions method (for example, see [1–4] and the references therein). It is well known that the variational method is an effective tool to deal with existence and multi-plicity of solutions for integer-order ordinary differential equations which have variational structures. For the fractional ordinary differential equation, a pioneering work by a vari-ational method was presented by Jiao and Zhou in [5], where they studied the following fractional differential equations with the left and right Riemann–Liouville fractional inte-grals:

⎧ ⎨ ⎩ d dt(

1 2 0D

–β

t (u(t)) +12tD –β

T (u(t))) +∇F(t,u(t)) = 0, a.e.t∈[0,T],

u(0) =u(T) = 0,

(1)

whereT> 0,0D–tβandtD–Tβdenote the left and right Riemann–Liouville fractional

inte-grals of order 0≤β< 1, respectively,F: [0,T]×RN_{→Rand∇F(t,x) is the gradient ofF}

atx. They established the variational structure of system (1), some embedding relations of working spaces, and some existence results of solutions for system (1) under subquadratic and superquadratic conditions, respectively. Subsequently, some authors applied a varia-tional method to different kinds of fracvaria-tional differential equations and some interesting

results were given (for example, see [6–15] and the references therein). Especially, in [9], Zhang and Li considered the following fractional differential equation:

⎧ ⎨ ⎩

tDαT(c0Dαtu(t)) =∇W(t,u(t)), t∈[0,T],

u(0) =u(T) = 0, (2)

wherec

0Dαt is the left Caputo fractional derivative, and they obtained the following

theo-rem.

Theorem A([9], Theorem 1.1) Suppose that the following conditions hold:

(A1) W(t, 0) = 0for allt∈[0,T],W(t,u)≥a(t)|u|θ_{and|∇W(t,u)| ≤b(t)|u|}θ–1_{for all} (t,u)∈[0,T]×RN_{,where1 <θ< 2is a constant,a: [0,T]→R}+_{is a continuous}

function andb: [0,T]→R+_{is a continuous function;}

(A2) there is a constant1 <σ≤θ< 2such that

∇W(t,u),u≤σW(t,u) for allt∈[0,T]andu∈RN;

(A3) W(t,u) =W(t, –u)for allt∈[0,T]andu∈RN.

Then(2)has infinitely many nontrivial solutions.

Impulse phenomena exist extensively in the real world and impulsive differential equa-tions are often used to describe these phenomena. In the last decade, by using the varia-tional methods, the problems on existence and multiplicity of solutions for integer-order impulsive differential equations with different boundary value conditions have been stud-ied deeply. We refer to the papers in [16–21] and the references therein. In comparison to the integer-order impulsive differential equations, there are less results for the frac-tional impulsive differential equations by variafrac-tional methods. In 2014, Rodrìguez-López et al. [22] and Bonanno et al. [23] considered the fractional impulsive differential equation with the right Riemann–Liouville fractional derivative and left Caputo fractional deriva-tive:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

tDαT(c0Dαtu(t)) +a(t)u(t) =λf(t,u(t)), t=ti, a.e.t∈[0,T],

(tDαT–1(c0Dαtu))(ti) =μQi(u(ti)), i= 1, 2, . . . ,l,

u(0) =u(T) = 0,

(3)

whereα∈(1₂, 1], bothλandμare positive parameters,f ∈C([0,T]×R,R),Qi∈C(R,R)

anda∈C([0,T],R). By using variational methods, they found that Eq. (3) has at least one or three solutions. Subsequently, in [24–27], several results are given along this direc-tion by variadirec-tional methods. For example, recently, Heidarkhani etc. [12] considered the following perturbed impulsive fractional differential system:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

tD

αi

T(ai(t)0D αi

t ui(t)) =λFui(t,u) +μGui(t,u) +hi(ui), t=tj,t∈(0,T),

(tDαTi–1(c0D αi

t ui))(tj) =Iij(ui(tj)), j= 1, 2, . . . ,m,

ui(0) =ui(T) = 0,

for 1≤i≤N, whereu= (u1, . . . ,uN), 0 <αi≤1, λ> 0, μ≥0,T > 0, ai∈L∞([0,T]),

F,G: [0,T]×RN _{→R are measurable with respect to t for all u ∈R}N _and

contin-uously differentiable in u for almost every t ∈[0,T], and hi : R→ R is a Lipschitz

continuous function for 1≤i ≤N. By using a three critical point theorem due Bo-nanno and Candito [28], they found that system (4) has at least three distinct weak so-lutions.

As a natural extension of Eq. (3), Zhao and Tang [29] considered thep-Laplacian frac-tional impulsive differential equations:

critical point theory, they obtained two multiplicity results of solutions for Eq. (5) whenf

satisfies the superquadratic conditions.

Motivated by the work in [5,6,11] and [29], Xie-Zhang [30] investigated the existence of infinitely many solutions for the following (p,q)-Laplacian fractional impulsive differential system:

W: [0,T]×RN_×RN_{→Rsatisfies the following assumption:}

(W0) W(t,x,y)is continuously differentiable in(x,y)∈RN_×RN _{for a.e.t∈[0,T],} mea-surable in t for every (x,y)∈RN ×RN, and there are a1,a2 ∈C(R+,R+) and b∈L∞([0,T];R+_{)such that}

W(t,x,y)+∇W(t,x,y)≤ a1

|x|+a2

|y|b(t),

Ii(x)+∇Ii(x)≤a1

|x|, i= 1, 2, . . . ,l,

Hj(y)+∇Hj(y)≤a2

|y|, j= 1, 2, . . . ,m

for all(x,y)∈RN×RN and a.e.t∈[0,T]. They assumedW(t,x,y) = –K(t,x,y) +

F(t,x,y)for a.e.t∈[0,T]and all(x,y)∈RN _×RN_{, whereKhas sub-(p,q)linear} growth andFhas super-(p,q)liear growth. By using the symmetric mountain pass theorem, they obtained system (6) has infinitely many weak solutions.

In this paper, we investigate the existence and multiplicity of solutions for system (6). Different from [30], we use the least action principle and an extension of Clark’s theorem to prove our results. We assumeW has sub-(p,q) linear growth on (x,y)∈RN_×RN _so

that system (6) has at least one weak solution, and we assumeW has partially sub-(p,q) linear growth on (x,y)∈RN×RN and is even for all (x,y)∈RN×RN so that system (6) has infinitely many weak solutions. To be precise, we obtain the results below.

Theorem 1.1 Suppose that(W0)and the following conditions hold:

(A) ρ–_:=essinf

[0,T]ρ(t) > 0,γ–:=essinf[0,T]γ(t) > 0;

(W1) there exist constantsd1∈[0, ρ –

pCpα),d2∈[0, γ–

qCq_β),γ1∈(0,p],γ2∈(0,q]andg1,g2,g3∈

L1_([0,T],R+_{)such that}

W(t,x,y)≤d1|x|p+d2|y|q+g1(t)|x|p–γ1+g2(t)|y|q–γ2+g3(t)

for a.e.t∈[0,T]and all(x,y)∈RN_×RN_,whereCα= Tα

Γ(α+1)andCβ=

Tβ

Γ(β+1);

(I1) there exist positive constantsk1,k2andν1∈[0,p)such that

Ii(x)≥–k1|x|ν1–k2

for a.e.t∈[0,T]and allx∈RN_{,i= 1, . . . ,l;}

(H1) there exist positive constantsk3,k4andν2∈[0,q)such that

Hj(y)≥–k3|y|ν2–k4

for a.e.t∈[0,T]and ally∈RN_{,j= 1, . . . ,m.}

Then system(6)has at least one weak solution.

Remark1.1 There exist examples of functions satisfying the assumptions in Theorem1.1.

For example, letp> 1,q> 1,ρ(t) =γ(t) =et_{+ 1,I}

i(x) = –ln(1 +|x|p),i= 1, . . . ,l,Hj(y) =

–ln(1 +|y|q_{),j= 1, . . . ,m, and}

W(t,x,y) =et+ 1 ln1 +|x|p+ln1 +|y|q

Theorem 1.2 Assume that(W0), (A), (I1), (H1)and the following conditions hold:

(W1) there exist constantsδ1> 0,δ2> 0,μ1∈[0,p),μ2∈[0,q)andf1,f2∈L1([0,T];R+)

such that

W(t,x,y)≤f1(t)|x|μ1+f2(t)|y|μ2

for a.e.t∈[0,T]and all(x,y)∈RN×RN with|x| ≤δ1and|y| ≤δ2;

(W2) W(t, 0, 0) = 0andW(t,x,y) =W(t, –x, –y)for a.e.t∈[0,T]and all(x,y)∈RN_×

RN_; (W3)

lim

|x|+|y|→0

W(t,x,y)

|x|p_+|y|q = +∞ uniformly for a.e.t∈[0,T];

(I1) there exist positive constantsl1,l2,δ3andν3∈[0,p)such that

Ii(x)≥–l1|x|ν3–l2

for a.e.t∈[0,T]and allx∈RN _{with|x| ≤δ}

3,i= 1, . . . ,l;

(H1) there exist positive constantsl3,l4,δ4andν4∈[0,q)such that

Hj(y)≥–l3|y|ν4–l4

for a.e.t∈[0,T]and ally∈RNwith|y| ≤δ4,j= 1, . . . ,m;

(I2) there exists a constantb1> 0such that

lim

|x|→0 Ii(x)

|x|p <b1, i= 1, . . . ,l;

(H2) there exists a constantb2> 0such that

lim

|y|→0 Hj(y)

|y|q <b2, j= 1, . . . ,m;

(I3) Ii(0) = 0andIi(x)is even inx∈RN,i= 1, . . . ,l; (H3) Hj(0) = 0andHj(y)is even iny∈RN,j= 1, . . . ,m.

Then system(6)has infinitely many weak solutions.

Remark1.2 There exist examples of functions satisfying the assumptions in Theorem1.2.

For example, letp= 4,q= 5,ρ(t) =γ(t) =et_+1,I

i(x) = –e|x|

4

+1,i= 1, . . . ,l,Hj(y) = –e|y|

5 +1,

j= 1, . . . ,m, andW(t,x,y) = (t2_{+ 1)(|x|}3_+|y|3_{) for all (x,y)∈R}N_×RN _{and a.e.t∈[0,T].}

It is easy to obtain similar theorems to Theorem1.1and Theorem1.2for thep-Laplacian fractional impulsive differential system:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

tDαT(ρ(t)Φp(c0Dαtu(t))) =∇uW(t,u(t)), a.e.t∈[0,T],

(tDαT–1(ρΦp(c0Dαtu)))(ti) =∇Ii(u(ti)), i= 1, 2, . . . ,l,

u(0) =u(T) = 0.

Theorem 1.3 Suppose that the following conditions hold:

(W0) W(t,x)is continuously differential inx∈RN _{for a.e.t∈[0,T],measurable int} for eachx∈RN,and there are functionsa∈C(R+,R+)andb∈L1([0,T];R+)such that

W(t,x)+∇W(t,x)≤a|x|b(t),

Ii(x)+∇Ii(x)≤a

|x|, i= 1, 2, . . . ,l,

for allx∈RN_{and a.e.t∈[0,T];} (A) ρ–:=essinf[0,T]ρ(t) > 0;

(W1) there exist constantsd∈[0, ρ–

pCαp),γ∈(0,p], andg1,g2∈L

1_([0,T],R+_{)such that}

W(t,x,y)≤d|x|p+g1(t)|x|p–γ+g2(t)

for a.e.t∈[0,T]and all(x,y)∈RN_×RN_;

(I1) there exist positive constantsk1,k2andν1∈[0,p)such that

Ii(x)≥–k1|x|ν1–k2

for a.e.t∈[0,T]and allx∈RN_{,i= 1, . . . ,l.}

Then system(7)has at least one weak solution.

Theorem 1.4 Assume that(W0), (A), (I1)and the following conditions hold:

(W1) there exist constantsδ1> 0,μ∈[0,p)andf ∈L1([0,T];R+)such that

W(t,x)≤f(t)|x|μ

for a.e.t∈[0,T]and allx∈RN_{with|x| ≤δ}

1;

(W2) W(t, 0) = 0andW(t,x) =W(t, –x)for a.e.t∈[0,T]and allx∈RN_; (W3)

lim

|x|→0 W(t,x)

|x|p = +∞;

(I1) there exist positive constantsl1,l2,δ2andν2∈[0,p)such that

Ii(x)≥–l1|x|ν2–l2

for a.e.t∈[0,T]and allx∈RN_{with|x| ≤δ}

2,i= 1, . . . ,l;

(I2) there exists a constantb> 0such that

lim

|x|→0 Ii(x)

|x|p <b, i= 1, . . . ,l;

(I3) Ii(0) = 0,Ii(x)is even inx∈RN,i= 1, . . . ,l.

Remark1.3 Ifp= 2,ρ(t)≡1 andIi(x)≡0 for a.e.t∈[0,T] and allx∈RN,i= 1, . . . ,l, then

Theorem1.3reduces to Theorem 5.46 in [31]. Hence, Theorem1.3generalizes Theorem 5.46 in [31]. In TheoremA(Theorem 1.1 in [9]),Wis required to satisfy the subquadratic condition for allx∈RN _{while (W1)is a partial sub-plinear growth condition, which is}

only for allx∈RN_{with|x| ≤δ. Hence, Theorem1.4is still different from TheoremAeven}

ifp= 2 andIi(x)≡0 for allx∈RN,i= 1, . . . ,l. Moreover, in Theorem1.4, a condition like

(A2) in TheoremA((W2) in Theorem 1.1 in [9]) is not required. There exist examples of functions satisfying the assumptions in Theorem1.4withp= 2 but not satisfying those in TheoremA. For example, letN= 1,p= 2 and

W(t,x) =

⎧ ⎨ ⎩

(t2_{+ 1)|x|}32_{, if}|_x| ≤_1,

(t2_{+ 1)|x|}3_{, if|x|> 1.}

2 Preliminaries

LetEbe a real Banach space andΦ∈C1_{(E,R). If any sequence{u}

k}possesses a convergent

subsequence, where{uk}satisfiesΦ(uk) is bounded andΦ(uk)→0 ask→ ∞, then one

say thatΦsatisfies the Palais–Smale (PS) condition (see [32]). We use the following two lemmas to prove our results.

Lemma 2.1([32]) Let E be a real Banach space andΦ∈C1_{(E,R)satisfy the(PS)condition.} IfΦis bounded from below,then c=infEΦis a critical value ofΦ.

Lemma 2.2([33]) Assume that(E, · )is a Banach space andΦ∈C1_{(X,R).Suppose that}

Φ satisfies the(PS)condition,is even and bounded from below,andΦ(0) = 0.If for any k∈N,there exists a k-dimensional subspace Xk_{of E andρ}

k> 0such thatsupXk_∩S

ρ_kΦ< 0, where Sρ={u∈E|u=ρ},then at least one of the following conclusions holds:

(i) there exists a sequence of critical points{uk}satisfyingΦ(uk) < 0for allkand

uk →0ask→ ∞;

(ii) there existsr> 0such that for any0 <a<rthere exists a critical pointusuch that

u=aandΦ(u) = 0.

Next we recall the definitions of Riemann–Liouville fractional derivatives and Caputo fractional derivatives and some related lemmas. Assumea,b∈R. Suppose that AC([a,b]) denote the space which consists of all absolutely continuous functions u: [a,b]→RN_.

Set

C₀∞[0,T],RN_:=u|u∈C∞_[0,T],RN_{,u(0) =u(T) = 0}

and the normu∞=max[0,T]|u(t)|and fors> 1,

Ls[0,T],RN:=

u|u: [0,T]→RN,

T

0

u(t)sdt<∞

and the normuLs= (T

Definition 2.1([31,34]) Assume thatf ∈AC[a,b] andθ∈(0, 1). Define

aDθtf(t) =

1

Γ(1 –θ)

d dt

t

a

(t–s)–θf(s)ds, t>a,

tDθbf(t) = –

1

Γ(1 –θ)

d dt

t

a

(s–t)–θf(s)ds, t<b.

ThenaDθt andtDθbare called the left and right Riemann–Liouville fractional derivatives of

orderθ of the functionf, respectively.

Definition 2.2([31,34]) Assume thatf ∈AC[a,b] andθ∈(0, 1). Define

c aD

θ

tf(t) =aDθt–1f(t) =

1

Γ(1 –θ)

t

a

(t–s)–θf(s)ds,

c tD

θ

bf(t) =tDθb–1f(t) =

1

Γ(1 –θ)

b

t

(s–t)–θf(s)ds.

Thenc

aDθt andctDθbare called the left and right Caputo fractional derivatives of orderθ of

the functionf, respectively.

LetE₀θ,s(0,T) be the closure ofC₀∞([0,T],RN) with the norm:

us=

T

0

c

0D θ

tu(t) s

dt+

T

0

_u(t)s

dt

1/s

, ∀u∈E₀θ,s(0,T),

whereθ∈(0, 1] ands> 1. ThenE0θ,sis reflexive and separable Banach space andu,c0Dθtu∈

Ls([0,T],R) ifu∈Eθ0,s(0,T) (see [5]).

Proposition 2.1([5]) Assume thatθ∈(0, 1]and s> 1.For any u∈Eθ₀,s(0,T),

uLs≤Cθc 0D

θ

tuLs,

where Cθ=_Γ(T_θθ₊₁₎.Ifθ>1_s,then

u∞≤Cθ,s,∞c0D θ

tuLs,

where Cθ,s,∞:= T

θ– 1_s

Γ(θ)(θs–s+1) 1 s

and s=_s–1s .

Proposition 2.2([5]) Assume that 1

p <θ≤1and1 <p<∞,and the sequence{uk}

con-verges weakly to u in Eθ₀,p.Then uk→u in C([0,T],RN).

DefineE=E₀α,p(0,T)×Eβ₀,q(0,T) with the norm(u,v)E=up+vqfor all (u,v)∈E,

≥ρ–

Proof of Theorem1.1 By combining Lemma3.1, Lemma3.2with Lemma2.1, the proof is

Proof of Theorem1.2 We follow the basic idea of the proof in [33]. We first investigate the

Then the solutions of system (11) correspond to the critical points of the functional ˆ

and then it is bounded from below. Similar to the argument of Lemma3.2,Φˆ satisfies the Palais–Smale condition.

LetXk_{be ak-dimensional subspace ofXfor anyk∈N. Then all norms are equivalent}

inXk_{. Hence, there exist positive constantsC}

2andC3such that

up_Lp≥C2upp, v q

Lq≥C3vqq. (13)

It follows from (W3), (I2) and (H2) that, for any given

for all (x,y)∈RN_×RN_{with|x|+|y| ≤δ}

0, and

Ii(x)≤b1|x|p, Hj(y)≤b2|y|q, (16)

for allx∈RN _{with|x| ≤δ}

0and ally∈RN with|y| ≤δ0, wherei= 1, . . . ,landj= 1, . . . ,m. For any given 0 <ρk≤maxmin{{Cδ0,α,pδ1,∞δ,2,Cδβ3,,q,δ∞4}}, define

Sρ_k=(u,v)∈E|(u,v)_E=ρk

.

Then, by Proposition2.1, we have

u∞+v∞≤Cα,p,∞up+Cβ,q,∞vq≤max{Cα,p,∞,Cβ,q,∞}(u,v)_E

≤min{δ0,δ1,δ2,δ3,δ4} (17)

for all (u,v)∈Sρk. Then (13), (15), (16), (17) and the definitions ofWˆ ,IˆiandHˆj(i= 1, . . . ,l, j= 1, . . . ,m) imply that

ˆ

Φ(u,v)≤ρ +

p

T

0

c

0D α

tu(t) p

dt+γ +

q

T

0

c

0D β

tv(t) q

dt+b1

l

i=1

u(ti)p

+b2

m

j=1

_v(sj) q

–M

T

0

_u(t)p+v(t)qdt

≤ρ+

p u

p p+

γ+

q v

q

q+b1lup∞+b2mvq∞–MupLp–Mvq_Lq

≤ρ+

p u

p p+

γ+

q v

q

q+b1lCpα,p,∞upp+b2mCβq,q,∞vqq–MC2upp

–MC3vqq (18)

for any (u,v)∈Sρ_k. So (14) and (18) imply thatΦˆ(u,v)|_Sρ

k∩Xk< 0. The conditions (i)–(iii), (W2), (I3) and (H3) imply thatΦˆis even andΦˆ(0, 0) = 0. Hence, Lemma2.2and Lemma3.2

show that system (11) has infinitely many solutions{(uk,vk)}such that(uk,vk)E→0 as

k→ ∞. Then (17) implies thatuk∞→0 andvk∞→0 ask→ ∞. So there exists a

sufficiently large integerk0> 0 such that|uk(t)|+|vk(t)| ≤min{δ0,δ1,δ2,δ3,δ4}for allk≥k0. Then the definitions ofWˆ ,Iˆi(i= 1, . . . ,l) andHˆj(j= 1, . . . ,m) imply that (uk,vk) are also

solutions of system (6) for allk≥k0.

Acknowledgements

The authors would like to express their sincerely thanks to the reviewers for the reviewers’ valuable comments.

Funding

This project is supported by the National Natural Science Foundation of China (No: 11301235) and Candidate Talents Training Fund of Yunnan Province (Project No: 2016PY027).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Author details

1_{Faculty of Science, Kunming University of Science and Technology, Kunming, P.R. China.}2_{Faculty of Transportation}

Engineering, Kunming University of Science and Technology, Kunming, P.R. China. 3_{School of Mathematics and Statistics,} Central South University, Changsha, P.R. China.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 12 June 2018 Accepted: 21 February 2019

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