A note on the existence and multiplicity of solutions for sublinear fractional problems

R E S E A R C H

Open Access

A note on the existence and multiplicity of

solutions for sublinear fractional problems

Yongqiang Fu

*_{Correspondence:} [email protected]

Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, P.R. China

Abstract

In this paper, we study the existence of weak solutions for fractionalp-Laplacian equations with sublinear growth and oscillatory behavior as the following

Lp

Ku=

λ

u= 0 inRN\

whereLp_Kis a nonlocal operator with singular kernel,

equations to fractionalp-Laplacian equations.

MSC: 35R11; 35A15; 35J60; 47G20

Keywords: fractionalp-Laplacian problem; variational method; multiple solutions

1 Introduction

Recently, a great attention has been devoted to the research of problems involving frac-tional and nonlocal operators. This type of operators finds many applications in a lot of fields, such as continuum mechanics, phase transition phenomena, population dynam-ics and game theory, as they are the typical outcome of stochastic stabilization ofLévy

processes; see, for instance, [–] and the references therein. There are many works on nonlocal fractional operators and their applications which are very interesting; we refer the interested reader to [–] and the references therein. Here we want to generalize the multiplicity existence results for fractional Laplacian equations in [] and at the same time fix some bugs there.

In this paper we deal with the following fractional problem in a bounded smooth domain

⊂RN:

Lp

Ku=λf(x,u) in,

u=  inRN\,

(.)

whereλ> ,  <p< +∞.Lp_Kis a nonlocal fractional operator defined as follows:

Lp

Kϕ= _εlim_→

RN_\Bε(x)

ϕ(x) –ϕ(y)p–ϕ(x) –ϕ(y)K(x–y)dy,

provided that the limit exists andKis a measurable function having the following prop-erty:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

γK∈L_(RN_{) whereγ(x) =min{|x|}p_{, },}

there existλ,> ,

such thatλ<K(x)|x|N+ps_{< for anyx∈R}N_{\ {},}

K(x) =K(–x) for anyx∈RN_{\ {}.}

(.)

Throughout this paper we assumeN>pswiths∈(, ). A typical example forKis the singular kernel|x|–(N+ps)_{, in which case problem (.) becomes}

(–)s_pu=λf(x,u) in,

u=  inRN_\, (.)

where (–)s

pis the fractionalp-Laplacian operator defined as

(–)s_pϕ(x) = lim ε→

RN_\Bε(x)

|ϕ(x) –ϕ(y)|p–_{(ϕ(x) –ϕ(y))} |x–y|N+sp dy

for allx∈RN_{. In the casep= , problem (.) becomes the fractional Laplacian problem}

(–)su=λf(x,u) in,

u=  inRN_\. (.)

Following [, ], we present the main structural assumptions on the nonlinear termf. (f) f ∈C(RN×R),f(x, ) = ,lim inft→_|tf_|(px–,t)_t> –∞.

(f) lim|t|→∞_|f_t(_|xp,–t) = uniformly inx.

(f) There existt–< andt+> such thatmin{f(x,t) : (x,t)∈×{t–}}> andmax{f(x,t) : (x,t)∈× {t+_{}}< .}

(f) There existtˆ–andtˆ+, withˆt–<t–andˆt+>t+, such that

minF(x,t) : (x,t)∈׈t–,ˆt+>maxf(x,t) : (x,t)∈×t–,t+,

whereF(x,t) =_tf(x,ξ)dξ.

A typical example off satisfying (f)-(f) is

f(x,t) = ⎧ ⎨ ⎩

|t|p–_{(|t|– )sgn(t), |t|< ,} (|t|– )p––ε_{sgn(t), |t|> ,}

whereεis small.

The natural solution space of problem (.) is

X() =

whereX(RN_{) is the fractional Sobolev space given by}

XRN=u∈LpRN: [u]s,p<∞

, [u]s,p=

RN

u(x) –u(y)pK(x–y)dx dy

/p

and endowed with the norm

u_X(RN₎=

up

Lp_(RN₎+ [u]

p s,p

/p

.

For the basic properties of fractional Sobolev spaces, we refer to []. We seek the solutions of (.) as the critical points of the functional given by

Iλ(u) =



p[u]

p s,p–λ

F(x,u)dx.

Set

αλ=infIλ|[t–_,t+_],

wheret–_andt+_{are the numbers given in (f} ).

Theorem . Assume that(f)-(f)hold.Then there exists> such that,for everyλ≥

,problem(.)admits two positive,two negative and two sign-changing solutions in Y\

[t–_,t+_{],where Y:=X}

()∩Cα/().

Here, for two real numberst<s, the symbol [t,s] denotes either an order interval inY

or a usual interval inR. In this paper there will be no ambiguous meaning concerning this symbol, for example, in Theorem .Y\[t–_,t+_{] denotes the difference ofY and [t}–_,t+_], where [t–,t+] is an order interval inY.

Theorem . Assume that(f)-(f)hold and f(x,t)is odd with respect to t.Then,for every

k∈N,there exists positivek such that,for everyλ≥k,problem(.)admits a

sign-changing solution uiwith Iλ(ui) <αλand a sign-changing solution viwith Iλ(vi)≥αλ,where

i= , . . . ,n.

Next consider the case thatf(x,t) is not odd with respect tot, but oscillates around  in the following manner.

(f) There existti– and ti+, i= , . . . ,n, with t–n <· · ·<t– <  <t+ <· · ·<t+n, such that min{f(x,t) : (x,t)∈× {t–

i}}>  >max{f(x,t) : (x,t)∈× {t+i}}for alli= , . . . ,n.

(f) min{F(x,t) : (x,t)∈× {ti–,t+i}}>max{F(x,t) : (x,t)∈×[ti––,t+i–]}fori= , . . . ,n. Theorem . Assume that(f)-(f)and(f)-(f)hold.Then there exists> such that,for

everyλ≥,problem(.)admits positive solutions ui,ui,negative solutions vi,vi,and

sign-changing solutions wi,wiin[ti–+,ti++]\[ti–,ti+],i= , . . . ,n– .

(f) There exist a decreasing (an increasing) sequence(ti–)iand an increasing (a decreasing)

sequence(t_i+)isuch thatti–<  <ti+, andf(x,ti+) <  <f(x,t–i)andmax{F(x,t) : (x,t)∈ ×[t–

i–,t+i–]}<min{F(x,t) : (x,t)∈× {t–i,ti+}}for everyi∈N.

Theorem . Assume that(f)-(f)and(f)hold.Then,for every arbitrarily chosen k∈N,

there existsk> such that,for everyλ≥k,problem(.)admits at leastk positive, k negative,andk sign-changing solutions.

Theorem . is immediate in view of Theorem ..

Remark . Bartsch and Liu obtained Theorems .-. forp-Laplacian Dirichlet prob-lems in []. Fu and Pucci obtained Theorems .-. for fractional Laplacian Dirichlet problems in []. Theorems .-. above are generalizations of the corresponding results in [].

This article is organized as follows. In Section , we introduce the fractional Sobolev spaces and some preliminary results. In Section , in a suitably chosen framework, we verify that the conditions in the abstract critical point theorems in [] are satisfied, then we generalize the existence of multiplicity solutions for fractional Laplacian problems to the one for fractionalp-Laplacian problems.

2 Preliminaries

Let us now recall some inequalities.

Lemma .(See []) There exist positive constants C-Csuch that,for allξ,η∈RN, _|ξ|p–ξ–|η|p–η≤C

|ξ|+|η|p–|ξ–η|, (.)

|ξ|p–ξ–|η|p–η(ξ–η)≥C

|ξ|+|η|p–|ξ–η|, (.) _|ξ|p–ξ–|η|p–η≤C|ξ–η|p–, if <p≤, (.)

|ξ|p–ξ–|η|p–η(ξ–η)≥C|ξ–η|p, if p> . (.)

The Hölder regularity up to the boundary, strong maximum principles and the Hopf lemma are important in the proof of our results.

Lemma .(See []) Let u∈Xsatisfy|LpKu| ≤K weakly infor some K> .Then

u(x)≤(CK)/p–δ(x)s a.e. in

for some C=C(N,p,s,),δ(x) =dist(x,C).Furthermore,there existsα∈(,s]such that,

for all weak solutions u∈Xof problem(.),u∈Cα()and uCα₍₎≤Cf/_L∞p–₍₎.

Lemma .(See []) If u∈Xis such that u(x)≥a.e.inand

RN

u(x) –u(y)p–u(x) –u(y)ϕ(x) –ϕ(y)K(x–y)dx dy≥

Lets∈(, ),p∈(,∞). Consider the problem ⎧

⎨ ⎩

(–)s

pu=c(x)|u|p–u in,

u=  inRN\. (.)

Lemma . (see []) Let be bounded and satisfy the interior ball condition on∂, ≥c∈C(),and u∈Xbe a weak solution of(.),then either u= a.e.inRN or

lim inf BRx→x

u(x)

δR(x)s

> ,

whereδR(x) =dist(x,BCR).

In the sequel, for any fixed parameterλ> , the spaceX() is endowed with the norm uλ=

[u]p_s,p+λmup_Lp₍₎

/p

,

and, for brevity, we put Xλ= (X(), · λ). The number m>  will be determined in

Section .Y=Xλ∩Cα/() is endowed with theCα/-norm andZ=Xλ∩Cα() with the

Cα_{-norm, whereα>  is from Lemma ..}

3 Multiplicity of weak solutions

We say thatu∈Xλis a (weak) solution of problem (.) if

u,ϕs,p=λ

f(x,u)ϕ(x)dx

holds for anyϕ∈Xλ, where

u,ϕs,p=

RN

u(x) –u(y)p–u(x) –u(y)ϕ(x) –ϕ(y)K(x–y)dx dy.

Fixλ>  andm> . DefineL(v) =L_Kpv+λm|v|p–_vand

Aλ:Xλ→Xλ, Aλ(u) =v,

wherevis the solution of the linear problem ⎧

⎨ ⎩

Lp

Kv+λm|v|p–v=λ(f(x,u) +m|u|p–u) in,

v=  inRN _\, (.)

which is uniquely determined.

In the following we give a series of lemmas to show that the conditions in the abstract critical point theorems in [] are satisfied. First we study the properties of the operatorAλ.

Proof First, from

L(v),v_X∗ λ,Xλ=

Lp

Kv+λm|v|p–v,v

X_λ∗,Xλ= [v] p s,p+λm

|v|pdx,

we know that

lim

vXλ→∞

L(v),vX∗_λ,Xλ

vXλ

=∞,

i.e., the operatorLis coercive. Second, from

L(v) –L(v),v–v

X_λ∗,Xλ

=

RN

v(x) –v(y)p –

v(x) –v(y)

–v(x) –v(y)p– ×v(x) –v(y)

v(x) –v(y)

–v(x) –v(y)

K(x–y)dx dy

+λm

|v|p–v–|v|p–v

(v–v)dx

> , (.)

wheneverv=v, we get thatLis monotone. It is easy to show thatLis weakly continuous. Thus, by the monotone operator theory, see [, ], we conclude that problem (.) has a unique (weak) solution inXλ. Therefore,Aλis well defined.

Next we prove thatAλ∈C(Xλ,Xλ). By (f) and (f), we have|f(x,u)| ≤C(|u|p–+ ) for some constantCdepending only onf. Suppose thatun→uinXλ. Then (un)nis bounded

inXλ. By (.), Hölder’s and Young’s inequalities, we have

[vn]ps,p+λm

|vn|pdx=

f(x,un)vndx+λm

|un|p–unvndx

≤CunpL–p₍₎+ 

vnLp₍₎

≤CεvnpLp₍₎+C(ε)

unpLp₍₎+ 

.

Takingε=λm/C, we conclude that (vn)nis bounded inXλ. Hence, (vn)nadmits a weakly

convergent subsequence, still denoted by (vn)n. Suppose thatvnvweakly in Xλ. By

Lemma . in [], we havevn→vinLp() sinceis bounded. By [], we getf(x,un)→

f(x,u) inLp_{() by going to a further subsequence if necessary. By (.) and Lemma .,}

using an argument similar to (.), we also have

λf(x,u) +λm|u|p–u–λf(x,un) –λm|un|p–un

Aλ(u) –vn

dx

=Lp_KAλ(u) +λmAλ(u) p–

Aλ(u) –Lp_Kvn–λm|vn|p–vn,Aλ(u) –vn

X_λ∗,Xλ

=

RN

v(x) –v(y)p–v(x) –v(y)–vn(x) –vn(y) p–

×vn(x) –vn(y)

v(x) –v(y)–vn(x) –vn(y)

+λm

|v|p–v–|vn|p–vn

(v–vn)dx

≥C

RN

v(x) –v(y)+vn(x) –vn(y) p–

×v(x) –v(y)–vn(x) –vn(y)K(x–y)dx dy

+Cλm

|v|+|vn|

p–

|v–vn|dx

≥C

[v–vn]ps,p+λm

|v–vn|pdx

.

Lettingn→ ∞, we obtain

vn→v=Aλ(u) inXλ.

This shows thatAλ∈C(Xλ,Xλ), as claimed.

Ifu∈Y, then by (f) and (f) the solutionvof (.) is inL∞() thanks to Lemma .. Therefore,x→g(x) =λ(f(x,u(x)) +m|u(x)|p–_{u(x)) is inL}∞_{() by (f}

), (f) and the fact thatu∈L∞(). Hence,v∈Cα_{() by Lemma ., sincev∈X}

λ, and sov∈Z by the

definition ofZ.

Second, we show that conditions (J) and (J) in [] are satisfied. Lemma .

(i) If <p≤,then the functionalIλsatisfies

I_λ(u),u–v_X∗

λ,Xλ≥Cu–v



Xλ

uXλ+vXλ

p– , Iλ(u)_X∗

λ≤Cu–v p–

Xλ.

(ii) Ifp≥,then the functionalIλsatisfies

Iλ(u),u–v

X_λ∗,Xλ≥Cu–v p Xλ,

Iλ(u)X_λ∗≤Cu–vXλ

uXλ+vXλ

p– .

Proof Letu∈Xλ. Thusv=Aλ(u) implies that

Iλ(u),u–v

X_λ∗,Xλ

=

RN

u(x) –u(y)p–u(x) –u(y)u(x) –u(y)–v(x) –v(y)K(x–y)dx dy

–λ

f(x,u)(u–v)dx

=

RN

u(x) –u(y)p–u(x) –u(y)–v(x) –v(y)p–v(x) –v(y)

×u(x) –u(y)–v(x) –v(y)K(x–y)dx dy

+λm

Ifp≥, it follows by Lemma . that

Iλ(u),u–v

X_λ∗,Xλ≥Cu–v p Xλ.

If  <p≤, then from Lemma . we have

I_λ(u),u–v_X∗ λ,Xλ

≥C

RN

u(x) –u(y)+v(x) –v(y)p–u(x) –u(y)–v(x) –v(y)

×K(x–y)dx dy+Cλm

_u(x)+v(x)p–u(x) –v(x)dx.

By Hölder’s inequality

u–vp_Xλ≤C(C,p)

Iλ(u),u–v

p



X_λ∗,Xλ

uXλ+vXλ

p(–p_) ,

and therefore

I_λ(u),u–v_X∗

λ,Xλ≥Cu–v



Xλ

uXλ+vXλ

p– .

Forw∈Xλ, we have

Iλ(u),w

X_λ∗,Xλ=

RN

_{u(x) –u(y)}p–

u(x) –u(y)w(x) –w(y)K(x–y)dx dy

–λ

f(x,u)w dx

=

RN

_{u(x) –u(y)}p–

u(x) –u(y)–v(x) –v(y)p–v(x) –v(y) ×w(x) –w(y)K(x–y)dx dy+λm

|u|p–u–|v|p–vw dx.

If  <p≤, then by Lemma . Iλ(u)X∗_λ≤Cu–v

p–

Xλ.

Ifp≥, then by Lemma . and Hölder’s inequality, we have Iλ(u)_X∗

λ≤Cu–vXλ

uXλ+vXλ

p– .

Now we conclude the result.

Third, we establish the regularity of the critical points ofIλ.

Lemma . Let K={u∈X() :Iλ(u) = }.Then K⊂Y.

Proof Let u be fixed in K. Thenu∈L∞() by (f), (f) and Lemma .. Hence,

x→λf(x,u(x)) is inL∞() again by (f), (f) and the fact thatuis inL∞(). Therefore, Lemma . givesu∈Cα_{() sinceu∈X}

Forth, we get a comparison principle for the operatorL.

Lemma . If u,v∈Xλ,with v≥u inRN \,are such thatLv≥Lu in,i.e.,Lv–

Lu,wX_λ∗,Xλ≥for all w∈Xλ,with w≥in,then v≥u in.

Proof Letu,v∈Xλ, withv≥uinRN\, be such thatLv≥Luin. Takeϕ=u–v=

[u–v]+_{– [u–v]}–_{,w= [u–v]}+_{= [ϕ]}+_{. Clearly,w∈X}

λandw≥ in. As

|b|p–b–|a|p–a= (p– )(b–a) 



a+t(b–a)p–dt,

we have

Lv–Lu,wX_λ∗,Xλ

=

RN(p– )

u(y) –v(y)–u(x) –v(x)Q(x,y)w(x) –w(y)K(x–y)dx dy

+λm

(p– )(v–u)R(x)w dx

=

RN(p– )Q(x,y)

–ϕ+(x) –ϕ+(y)–ϕ–(x)ϕ+(y) –ϕ–(y)ϕ+(x)K(x–y)dx dy

–λm

(p– )R(x)ϕ+dx

≥,

where

Q(x,y) = 



u(x) –u(y)+tv(x) –v(y)–u(x) –u(y)p–dt,

R(x) = 



u+t(v–u)p–dt.

We see thatQ(x,y)≥ andQ(x,y) =  only ifv(y) =v(x) andu(y) =u(x), andR(x)≥.

Thusϕ+_{=  in, i.e.,v≥uin.}

Foru,v∈Y, defineuvprovidedu(x) <v(x) for anyx∈and

lim inf x=x–dν∈,d→

u(x) –v(x)

dα/ < 

for anyx∈∂, whereνis the unit outward normal vector of∂atx. Ifu≤vin, then [u,v] denotes the order interval{w∈Y:u≤w≤v}.

By (f)-(f), we can choose suitable positivemso that

tf(x,t) +m|t|p>  fort= ,

f(x,t) +m|t|p–t

⎧ ⎨ ⎩

>m|t–_|p–_t– _fort≥t–_, <m|t+|p–t+ fort≤t+.

(.)

Fifth, we give the following three technical lemmas which are necessary to verify that the conditions in the abstract critical point theorems in [] (especially (P)-(P) there) are satisfied.

Lemma . Assume that(f)and(f)also hold. Then there existφ,ψ ∈Y having the

following properties. () t–_{≤φψ≤t}+_.

() For everyu∈Xλ∩L∞()withφ≤u,φAλ(u).

() For everyu∈Xλ∩L∞()withu≤ψ,Aλ(u)ψ.

() For everyu∈Xλ∩L∞()witht–≤u,φAλ(u).

() For everyu∈Xλ∩L∞()withu≤t+,Aλ(u)ψ.

() max{F(x,t) : (x,t)∈×[–φL∞(),ψL∞()]}<min{F(x,t) : (x,t)∈× {s–,s+}}.

Proof Letψ be the solution of Lp

Ku+λm|u|p–u=λmt+ p–

t+ in,

u=  inRN\.

Thus ψ ≤t+_{again by Lemma . and the strong maximum principle Lemma .} sinceLp_K(ψ)≥. By Lemmas . and ., we also havelim inf_{BRx→x δ}ψ(x)

R(x)s > . Letφbe

the solution of

Lp

Ku+λm|u|

p–_u=λmt–p–

t– in,

u=  inRN\.

Thereforet–≤φ by Lemma . and by the strong maximum principle Lemma . sinceLp_K(–φ)≥. By Lemmas . and ., we also havelim infBRx→x

–ϕ(x)

δR(x)s> .

Next we prove the results onAλ. By (.), we can chooseδ∈(, ) small enough such

that

f(x,t) +m|t|p–t

⎧ ⎨ ⎩

>δm|t–_|p–_t– _fort≥t–_, <δm|t+|p–t+ fort≤t+.

Letu∈Xλ∩L∞() such thatt–≤u. Denotev=Aλ(u). Then

Lp

Kv+λm|v|p–v=λf(x,u) +λm|u|p–u≥δλmt– p–

t–=δLp_Kφ+λm|φ|p–φ.

Hence,v≥δp–_{φφby Lemma ..}

Similarly, ifu∈Xλ∩L∞() withu≤t+, thenv≤δ 

p–_{ψψ. This completes the proof}

of ()-().

Conclusion () follows directly from (f) and properties ()-(). Lemma . Suppose that(f)and(f)also hold.Then there exists> such that,for

everyλ≥,there exists h∈C([, ],Y˜)satisfying h()≤≤h()and

max

≤t≤Jλ

h(t)< inf

whereY˜ =X∩C_{(),φandψ are the ones in Lemma..Furthermore,if f(x,t)is odd}

with respect to t,then,for every k∈N,there existsk> such that,for everyλ≥k,there exists an odd map hk∈C(Sk,Y˜)satisfying

max θ∈SkJλ

hk(θ)

< inf

u∈[φ,ψ]Jλ(u). (.)

The proof is a minor modification of the corresponding argument given in order to prove Lemma . of [].

Lemma . For any bounded set B⊂Y and any order interval[φ,ψ],there existφ,ψ∈Y

such that

B∪[φ,ψ]⊂int

Y[φ,ψ], (.)

and

Aλ(u)∈[φ,ψ] for every u∈X satisfyingφ≤u≤ψ. (.)

Proof By (f) there existε,C> , which depend onλ, such that

λf(x,t)≤(λ– ε)|t|p–+C for everyx∈,t∈R,

whereλis the first eigenvalue inofLpKwith zero Dirichlet boundary condition. By the

monotone operator theory, see [, ], there exists a unique solutionψto

Lp

Ku– (λ–ε)|u|p–u=

C

μ in,

u=  inRN\,

where

μ=λ– ε+λm

λ–ε+λm < .

Letφ= –ψ. Then we getφψby Lemma . and the strong maximum principle lemma. Provided thatC>  is large enough, (.) follows.

Next, fixu∈Xλwhich satisfiesφ≤u≤ψ. Denotev=Aλ(u). We have

Lp

Kv+λm|v|p–v=λ

f(x,u) +m|u|p–u

≤(λ– ε)|ψ|p–ψ+C+λm|ψ|p–ψ =μ(λ–ε)|ψ|p–ψ+λm|ψ|p–ψ+C/μ

=μLp_Kψ+λm|ψ|p–ψ

.

Hence,v≤μp– _ψ

ψ by Lemma .. Similarly,vφ. This completes the proof of

At last, we show thatIλsatisfies the Palais-Smale condition which is crucial to

guaran-teeing the existence of critical points.

Lemma . Let(un)n⊂Xλbe a Palais-Smale sequence of Iλ,i.e., (Iλ(un))nis bounded and

Iλ(un)→as n→ ∞.Then(un)nadmits a strongly convergent subsequence in Xλ.

Proof By (f)-(f), forε>  small enough, _f(x,t)≤C(ε) + 

pε|t|

p–_{, F(x,t)≤C(ε) +ε|t|}p_{. (.)}

By Lemma . in [],

uLp₍₎≤C_[u]_s,p, (.)

sinceis bounded.

Let (un)nbe a Palais-Smale sequence ofIλinXλ. Then there existsC>  such that, for

alln,

C≥Iλ(un)≥



p[un]

p

s,p–λεunLp₍₎–C(ε)|| ≥

 p[un]

p s,p–C,

whereε= /pC

λin (.). Thus (un)nis bounded inXλby (.). So, up to a subsequence,

still denoted by (un)n, we haveunuweakly inXλ. ThenI(un),un–u →, and further

we obtain

I(un),un–u

=

RN

un(x) –un(y) p–

un(x) –un(y)

un(x) –u(x)

–un(y) –u(y)

×K(x–y)dx dy–λ

f(x,un)(un–u)dx

→ (.)

asn→ ∞. Moreover, by Lemma . in [], up to a subsequence,un→ustrongly inLp()

and a.e. in. Thus,f(x,un)(un–u)→ a.e. inasn→ ∞. It is easy to show that the

sequence (f(x,un)(un–u))nis uniformly bounded and equi-integrable inL(). Hence, by

the Vitali convergence theorem (see Rudin []), we get

lim n→∞

f(x,un)(un–u)dx= .

Therefore, by (.), we have

RN

un(x) –un(y)p

–

un(x) –un(y)

un(x) –u(x)

–un(y) –u(y)

K(x–y)dx dy

asn→ ∞. Thus, by the weak convergence of (un)ninXλ, we get

RN

_{un(x) –un(y)}p–

un(x) –un(y)

–u(x) –u(y)p–u(x) –u(y) ×un(x) –u(x)

–un(y) –u(y)

K(x–y)dx dy

→

asn→ ∞. By Lemma ., we obtain, forp> ,

RN

un(x) –u(x)

–un(y) –u(y) p

K(x–y)dx dy

≤C

RN

_{un(x) –un(y)}p–

un(x) –un(y)

–u(x) –u(y)p–u(x) –u(y) ×un(x) –u(x)

–un(y) –u(y)

K(x–y)dx dy

→ (.)

asn→ ∞. For  <p< , we have

RN

un(x) –u(x)

–un(y) –u(y)pK(x–y)dx dy

≤C

RN

_{un(x) –un(y)}p–

un(x) –un(y)

–u(x) –u(y)p–u(x) –u(y)

×un(x) –u(x)

–un(y) –u(y)

K(x–y)dx dy

p/

×

RN

(un(x) –un(y)–u(x) –u(y) p

K(x–y)dx dy

–p/

≤C

RN

un(x) –un(y) p–

un(x) –un(y)

–u(x) –u(y)p–u(x) –u(y)

×un(x) –u(x)

–un(y) –u(y)

K(x–y)dx dy

p/

→ (.)

asn→ ∞. Combining (.) and (.), we get thatun→ustrongly inXλasn→ ∞.

Therefore,Iλsatisfies the (PS) condition.

Taking inspiration from [], we apply Lemmas .-. in order to prove Theorems .-.. Let intD(D) refer to theY-topology onD.

Proof of Theorem. Letφ,ψandbe the ones in Lemma . and Lemma .. Fixλ≥. Lethbe the one in Lemma .. ChooseB=h([, ]) so thatB⊂ ˜Y⊂Yby Lemma .. Let the correspondingφandψbe the ones stated in Lemma ..

DefineD+_ = [,ψ] andD+= [,ψ]. Then

intD+ D

+

where intD+(D

+

) refers to theY-topology onD+. By Lemmas . and . and Lemma ., conditions (P)-(P) in Section  are satisfied forD+⊂D+ andA=Aλ. By Lemmas .-.

and Theorem . in [], there are two positive and two negative solutions of problem (.). ChooseD= [φ,ψ],D= [φ,ψ],D= [φ,ψ], andD= [φ,ψ]. By Lemmas .-.,Di,

i= , . . . , , andA=Aλsatisfy all the assumptions of Theorem . in []. So there are two

sign-changing solutions of problem (.).

At last, in view of () of Lemma ., we complete the proof.

Proof of Theorem. LetDi,i= , . . . , , be the ones in the proof of Theorem .. First, in

view of Lemmas .-., all the assumptions of Theorem . in [] are satisfied. Second, by () and () of Lemma .,

inf

u∈[t–_,t+_]Iλ(u)≤_uinf_∈D 

Iλ(u) (.)

and

Aλ

t–_,t+_⊂int

Y(D). (.)

Hence, (.)-(.) and Proposition . of [] yield the result.

Proof of Theorem. Definefi(x,t),i= , . . . ,n– , as in []. Consider

⎧ ⎨ ⎩

Lp

ku=λfi(x,u) in,

u(x) = , inRN_\. (.)

By Theorem ., problem (.) admits two positive, two negative and two sign-changing solutions. Replacingt–,t+byt_i–,t_i+, respectively, the six solutions are outside of the order interval [t_i–,t_i+]. According to the choice ofm, we have

mt–_i+p–mt_i–_+<f(x,t) +mt=fi(x,t) +m|t|p–t<mti++

p–

t+_i+ fort_i–_+≤t≤t+_i+. Thus, by the definition offi, we get

Lp

Ku+λm|u|p–>λmt–i+

p–

t–_i+=Lp_Kt_i–_++λmt_i–_+p–t–_i+ in,

u=  >t_i–_+ inRN\.

Furthermore, by Lemma . the six solutions of problem (.) are inside the order interval [t–_i+,t_i+_+] and are obviously solutions of problem (.). In this way, we manage to get (n– ) positive, (n– ) negative and (n– ) sign-changing solutions of problem (.).

4 Conclusions

The purpose of this paper is to study the existence of multiplicity solutions for fractional

Acknowledgements

The author is grateful to anonymous reviewers for carefully reading this paper and for their comments and suggestions which have improved the paper.

Funding

The work is financially supported by the National Natural Science Foundation of China (Grant No. 11371110). Conflict of interest

The author declares that there is no conflict of interest regarding the publication of this paper. Competing interests

The author declares that he has no competing interests. Authors’ contributions

The author contributed wholly in writing this paper. He read and approved the final manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 6 July 2017 Accepted: 10 November 2017

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