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Research article

Existence and multiplicity of solutions for a class of damped-like fractional differential system

  • Received: 10 January 2020 Accepted: 20 April 2020 Published: 07 May 2020
  • MSC : 34B15, 34B10

  • In this paper, we obtain some results about the existence and multiplicity of weak solutions for a class of damped-like fractional differential system with a parameter λ. When the nonlinear term is subquadratic only near the origin, we obtain that system has a ground state weak solution uλ if λ is in some given interval, and when the nonlinear term is also even near the origin, then for each λ>0, system has infinitely many weak solutions {uλn} with uλn0 as n. We mainly use Ekeland's variational principle and a variant of Clark's theorem together with a cut-off technique to prove our results.

    Citation: Jie Xie, Xingyong Zhang, Cuiling Liu, Danyang Kang. Existence and multiplicity of solutions for a class of damped-like fractional differential system[J]. AIMS Mathematics, 2020, 5(5): 4268-4284. doi: 10.3934/math.2020272

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  • In this paper, we obtain some results about the existence and multiplicity of weak solutions for a class of damped-like fractional differential system with a parameter λ. When the nonlinear term is subquadratic only near the origin, we obtain that system has a ground state weak solution uλ if λ is in some given interval, and when the nonlinear term is also even near the origin, then for each λ>0, system has infinitely many weak solutions {uλn} with uλn0 as n. We mainly use Ekeland's variational principle and a variant of Clark's theorem together with a cut-off technique to prove our results.


    In this paper, we are concerned with the existence and multiplicity of weak solutions for the damped-like fractional differential system

    {ddt(p(t)(12 0Dξt(u(t))+12 tDξT(u(t))))+r(t)(12 0Dξt(u(t))+12 tDξT(u(t)))+q(t)u(t)=λF(t,u(t)),  a.e. t[0,T],u(0)=u(T)=0, (1.1)

    where 0Dξt and tDξT are the left and right Riemann-Liouville fractional integrals of order 0ξ<1, respectively, p,r,qC([0,T],R), L(t):=t0(r(s)/p(s))ds, 0<meL(t)p(t)M and q(t)p(t)0 for a.e. t[0,T], u(t)=(u1(t),u2(t),un(t))T, ()T denotes the transpose of a vector, n1 is a given positive integer, λ>0 is a parameter, F(t,x) is the gradient of F with respect to x=(x1,,xn)Rn, that is, F(t,x)=(Fx1,,Fxn)T, and there exists a constant δ(0,1) such that F:[0,T]ׯBδ0R (where ¯Bδ0 is a closed ball in RN with center at 0 and radius δ) satisfies the following condition

    (F0) F(t,x) is continuously differentiable in ¯Bδ0 for a.e. t[0,T], measurable in t for every x¯Bδ0, and there are aC(¯Bδ0,R+) and bL1([0,T];R+) such that

    |F(t,x)|a(|x|)b(t)

    and

    |F(t,x)|a(|x|)b(t)

    for all x¯Bδ0 and a.e. t[0,T].

    In recent years, critical point theory has been extensively applied to investigate the existence and multiplicity of fractional differential equations. An successful application to ordinary fractional differential equations with Riemann-Liouville fractional integrals was first given by [1], in which they considered the system

    {ddt((12 0Dξt(u(t))+12 tDξT(u(t))))=F(t,u(t)),  a.e. t[0,T],u(0)=u(T)=0. (1.2)

    They established the variational structure and then obtained some existence results for system (1.2). Subsequently, this topic attracted lots of attention and a series of existence and multiplicity results are established (for example, see [2,3,4,5,6,7,8,9,10,11,12] and reference therein). It is obvious that system (1.1) is more complicated than system (1.2) because of the appearance of damped-like term

    r(t)(12 0Dξt(u(t))+12 tDξT(u(t))).

    In [13], the variational functional for system (1.1) with λ=1 and N=1 has been established, and in [14], they investigated system (1.1) with λ=1, N=1 and an additional perturbation term. By mountain pass theorem and symmetric mountain pass theorem in [15] and a local minimum theorem in [16], they obtained some existence and multiplicity results when F satisfies superquadratic growth at infinity and some other reasonable conditions at origin.

    In this paper, motivated by the idea in [17,18], being different from those in [13,14], we consider the case that F has subquadratic growth only near the origin and no any growth condition at infinity. Our main tools are Ekeland's variational principle in [19], a variant of Clark's theorem in [17] and a cut-off technique in [18]. We obtain that system (1.1) has a ground state weak solution uλ if λ is in some given interval and then some estimates of uλ are given, and when F(t,x) is also even about x near the origin for a.e. t[0,T], for each given λ>0, system (1.1) has infinitely many weak solutions {uλn} with uλn0 as n. Next, we make some assumptions and state our main results.

    (f0) There exist constants M1>0 and 0<p1<2 such that

    F(t,x)M1|x|p1 (1.3)

    for all x¯Bδ0 and a.e. t[0,T].

    (f1) There exist constants M2>0 and 0<p2<p1<2 such that

    F(t,x)M2|x|p2 (1.4)

    for all x¯Bδ0 and a.e. t[0,T].

    (f0) There exist constants M1>0 and 0<p1<1 such that (1.3) holds.

    (f1) There exist constants M2>0 and 0<p2<p1<1 such that (1.4) holds.

    (f2) There exists a constant η(0,2) such that

    (F(t,x),x)ηF(t,x)

    for all x¯Bδ0 and a.e. t[0,T].

    (f3) F(t,x)=F(t,x) for all x¯Bδ0 and a.e. t[0,T].

    Theorem 1.1. Suppose that (F0), (f0), (f1) and (f2) hold. If

    0<λmin{|cos(πα)|2C,(1B)2p2(δ2)2p2|cos(πα)|2C},

    then system (1.1) has a ground state weak solution uλ satisfying

    uλ2p2min{1,(1B)2p2(δ2)2p2},uλ2p2{B2p2,(δ2)2p2}.

    where

    B=T2α12mΓ(α)(2α1)12,C=max{p,η}max{M1,M2}Tmaxt[0,T]eL(t)max{Bp1,Bp2}.

    If (f0) and (f1) are replaced by the stronger conditions (f0) and (f1), then (f2) is not necessary in Theorem 1.1. So we have the following result.

    Theorem 1.2. Suppose that (F0), (f0) and (f1) hold. If

    0<λmin{|cos(πα)|3C,(1B)2p1(δ2)2p1|cos(πα)|3C},

    then system (1.1) has a ground state weak solution uλ satisfying

    uλ2p1min{1,(1B)2p1(δ2)2p1},uλ2p1{B2p1,(δ2)2p1},

    where C=maxt[0,T]eL(t)max{(1+ρ0)a0BT0b(t)dt,M1p1TBp1,ρ0M1TBp1+1}, a0=maxs[0,δ]a(s) and ρ0=maxs[δ2,δ]|ρ(s)| and ρ(s)C1(R,[0,1]) is any given even cut-off function satisfying

    ρ(s)={1,if |s|δ/2,0,if |s|>δ. (1.5)

    Theorem 1.3. Suppose that (F0), (f0), (f1) and (f3) hold. Then for each λ>0, system (1.1) has a sequence of weak solutions {uλn} satisfying {uλn}0, as n.

    Remark 1.1. Theorem 1.1-Theorem 1.3 still hold even if r(t)0 for all t[0,T], that is, the damped-like term disappears, which are different from those in [2,3,4,5,6,7,8,9,10,11,12] because all those assumptions with respect to x in our theorems are made only near origin without any assumption near infinity.

    The paper is organized as follows. In section 2, we give some preliminary facts. In section 3, we prove Theorem 1.1–Theorem 1.3.

    In this section, we introduce some definitions and lemmas in fractional calculus theory. We refer the readers to [1,9,20,21,22]. We also recall Ekeland's variational principle in [19] and the variant of Clark's theorem in [17].

    Definition 2.1. (Left and Right Riemann-Liouville Fractional Integrals [22]) Let f be a function defined on [a,b]. The left and right Riemann-Liouville fractional integrals of order γ for function f denoted by aDγtf(t) and tDγbf(t), respectively, are defined by

    aDγtf(t)=1Γ(γ)ta(ts)γ1f(s)ds,t[a,b],γ>0,tDγbf(t)=1Γ(γ)bt(st)γ1f(s)ds,t[a,b],γ>0.

    provided the right-hand sides are pointwise defined on [a,b], where Γ>0 is the Gamma function.

    Definition 2.2. ([22]) For nN, if γ=n, Definition 2.1 coincides with nth integrals of the form

    aDntf(t)=1(n1)!ta(ts)n1f(s)ds,t[a,b],nN,tDnbf(t)=1(n1)!bt(ts)n1f(s)ds,t[a,b],nN.

    Definition 2.3. (Left and Right Riemann-Liouville Fractional Derivatives [22]) Let f be a function defined on [a,b]. The left and right Riemann-Liouville fractional derivatives of order γ for function f denoted by aDγtf(t) and tDγbf(t), respectively, are defined by

    aDγtf(t)=dndtnaDγntf(t)=1Γ(nγ)dndtn(ta(ts)nγ1f(s)ds),tDγbf(t)=(1)ndndtntDγnbf(t)=(1)nΓ(nγ)dndtn(bt(st)nγ1f(s)ds).

    where t[a,b],n1γ<n and nN. In particular, if 0γ<1, then

    aDγtf(t)=ddtaDγ1tf(t)=1Γ(1γ)ddt(ta(ts)γf(s)ds),t[a,b],tDγbf(t)=ddttDγ1bf(t)=1Γ(1γ)ddt(bt(st)γf(s)ds),t[a,b].

    Remark 2.1. ([9,13]) The left and right Caputo fractional derivatives are defined by the above-mentioned Riemann-Liuville fractional derivative. In particular, they are defined for function belonging to the space of absolutely continuous functions, which we denote by AC([a,b],RN). ACk([a,b],RN)(k=0,1,...) are the space of the function f such that fCk([a,b],RN). In particular, AC([a,b],RN)=AC1([a,b],RN).

    Definition 2.4. (Left and Right Caputo Fractional Derivatives [22]) Let γ0 and nN.

    (ⅰ) If γ(n1,n) and fACn([a,b],RN), then the left and right Caputo fractional derivatives of order γ for function f denoted by caDγtf(t) and ctDγbf(t), respectively, exist almost everywhere on [a,b]. caDγtf(t) and ctDγbf(t) are represented by

    caDγtf(t)=aDγntfn(t)=1Γ(nγ)(ta(ts)nγ1f(n)(s)ds),ctDγbf(t)=(1)n tDγnbfn(t)=(1)nΓ(nγ)(bt(st)nγ1f(n)(s)ds),

    respectively, where t[a,b]. In particular, if 0<γ<1, then

    caDγtf(t)=aDγ1tf(t)=1Γ(1γ)(ta(ts)γf(s)ds),t[a,b],ctDγbf(t)=tDγ1bf(t)=1Γ(1γ)(bt(st)γf(s)ds),t[a,b].

    (ⅱ) If γ=n1 and fACn([a,b],RN), then caDγtf(t) and ctDγbf(t) are represented by

    caDn1tf(t)=f(n1)(t),t[a,b],ctDn1bf(t)=(1)n1f(n1)(t),t[a,b].

    In particular, caD0tf(t)= ctD0bf(t)=f(t), t[a,b].

    Lemma 2.1. ([22]) The left and right Riemann-Liouville fractional integral operators have the property of a semigroup, i.e.

    aDγ1t(aDγ2tf(t))=aDγ1γ2tf(t),tDγ1b(tDγ2bf(t))=tDγ1γ2bf(t),γ1,γ2>0,

    in any point t[a,b] for continuous function f and for almost every point in [a,b] if the function fL1([a,b],RN).

    For 1r<, define

    uLr=(T0|u(t)|rdt)1r (2.1)

    and

    u=maxt[0,T]|u(t)|. (2.2)

    Definition 2.5. ([1]) Let 0<α1 and 1<p<. The fractional derivative space Eα,p0 is defined by closure of C0([0,T],RN) with respect to the norm

    uα,p=(T0|u(t)|pdt+T0|c0Dαtu(t)|pdt)1p. (2.3)

    Remark 2.2. ([9]) Eα,p0 is the space of functions uLp([0,T],RN) having an α-order Caputo fractional derivative c0Dαtu(t)Lp([0,T],RN) and u(0)=u(T)=0.

    Lemma 2.2. ([1]) Let 0<α1 and 1<p<. Eα,p0 is a reflexive and separable Banach space.

    Lemma 2.3. ([1]) Assume that 1<p< and α>1p. Then Eα,p0 compactly embedding in C([0,T],RN).

    Lemma 2.4. ([1]) Let 0<α1 and 1<p<. For all uEα,p0, we have

    uLpTαΓ(α+1)c0DαtuLp. (2.4)

    Moreover, if α>1p and 1p+1q=1, then

    uTα1pΓ(α)((α1)q+1)1qc0DαtuLp. (2.5)

    Definition 2.6. ([13]) Assume that X is a Banach space. An operator A:XX is of type (S)+ if, for any sequence {un} in X, unu and lim supn+A(un),unu0 imply unu.

    Let φ:XR. A sequence {un}X is called (PS) sequence if the sequence {un} satisfies

    φ(un) is bounded, φ(un)0.

    Furthermore, if every (PS) sequence {un} has a convergent subsequence in X, then one call that φ satisfies (PS) condition.

    Lemma 2.5. ([19]) Assume that X is a Banach space and φ:XR is Gˆateaux differentiable, lower semi-continuous and bounded from below. Then there exists a sequence {xn} such that

    φ(xn)infXφ,φ(xn)0.

    Lemma 2.6. ([17]) Let X be a Banach space, φC1(X,R). Assume φ satisfies the (PS) condition, is even and bounded below, and φ(0)=0. If for any kN, there exist a k-dimensional subspace Xk of X and ρk>0 such that supXkSpkφ<0, where Sρ={uX|u=ρ}, then at least one of the following conclusions holds.

    (ⅰ) There exist a sequence of critical points {uk} satisfying φ(uk)<0 for all k and uk0 as k.

    (ⅱ) There exists a constant r>0 such that for any 0<a<r there exists a critical point u such that u=a and φ(u)=0.

    Remark 2.3. ([17]) Lemma 2.6 implies that there exist a sequence of critical points uk0 such that φ(uk)0, φ(uk)0 and uk0 as k.

    Now, we establish the variational functional defined on the space Eα,20 with 12<α1. We follow the same argument as in [13] where the one-dimensional case N=1 and λ=1 for system (1.1) was investigated. For reader's convenience, we also present the details here. Note that L(t):=t0(r(s)/p(s))ds,0<meL(t)p(t)M and q(t)p(t)0 for a.e. t[0,T]. Then system (1.1) is equivalent to the system

    {ddt(eL(t)p(t)(12 0Dξt(u(t))+12 tDξT(u(t))))+eL(t)q(t)u(t)=λeL(t)F(t,u),  a.e. t[0,T],u(0)=u(T)=0. (2.6)

    By Lemma 2.1, for every uAC([0,T],RN), it is easy to see that system (2.6) is equivalent to the system

    {ddt[eL(t)p(t)(12 0Dξ2t(0Dξ2tu(t))+12 tDξ2T(tDξ2Tu(t)))]+eL(t)q(t)u(t)=λeL(t)F(t,u),  a.e. t[0,T],u(0)=u(T)=0, (2.7)

    where ξ[0,1).

    By Definition 2.4, we obtain that uAC([0,T],RN) is a solution of problem (2.7) if and only if u is a solution of the following system

    {ddt(eL(t)p(t)(12 0Dα1t(c0Dαtu(t))12 tDα1T(ctDαTu(t))))+eL(t)q(t)u(t)=λeL(t)F(t,u),u(0)=u(T)=0, (2.8)

    for a.e. t[0,T], where α=1ξ2(12,1]. Hence, the solutions of system (2.8) correspond to the solutions of system (1.1) if uAC([0,T],RN).

    In this paper, we investigate system (2.8) in the Hilbert space Eα,20 with the corresponding norm

    u=(T0eL(t)p(t)(|c0Dαtu(t)|2+|u(t)|2)dt)12.

    It is easy to see that u is equivalent to uα,2 and

    mT0|c0Dαtu(t)|2dtT0eL(t)p(t)|c0Dαtu(t)|2dtMT0|c0Dαtu(t)|2dt.

    So

    uL2TαmΓ(α+1)(T0eL(t)p(t)|c0Dαtu(t)|2dt)12,

    and

    uBu, (2.9)

    where

    B=T2α12mΓ(α)(2α1)12>0.

    (see [13]).

    Lemma 2.7. ([13]) If 12<α1, then for every uEα,20, we have

    |cos(πα)|u2T0eL(t)p(t)(c0Dαtu(t),ctDαTu(t))dt+T0eL(t)p(t)|u(t)|2dtmax{Mm|cos(πα)|,1}u2. (2.10)

    We follow the idea in [17] and [18]. We first modify and extend F to an appropriate ˜F defined by

    ˜F(t,x)=ρ(|x|)F(t,x)+(1ρ(|x|))M1|x|p1,  for all xRN,

    where ρ is defined by (1.5).

    Lemma 3.1. Let (F0), (f0), (f1) (or (f0), (f1)), (f2) and (f3) be satisfied. Then

    (˜F0) ˜F(t,x) is continuously differentiable in xRN for a.e. t[0,T], measurable in t for every xRN, and there exists bL1([0,T];R+) such that

    |˜F(t,x)|a0b(t)+M1|x|p1,|˜F(t,x)|(1+ρ0)a0b(t)+M1p1|x|p11+ρ0M1|x|p1

    for all xRN and a.e. t[0,T];

    (˜f0) ˜F(t,x)M1|x|p1 for all xRN and a.e. t[0,T];

    (˜f1) ˜F(t,x)max{M1,M2}(|x|p1+|x|p2) for all xRN and a.e. t[0,T];

    (˜f2) (˜F(t,x),x)θ˜F(t,x) for all xRN and a.e. t[0,T], where θ=max{p1,η};

    (˜f3) ˜F(t,x)=˜F(t,x) for all xRN and a.e. t[0,T].

    Proof. We only prove (˜f0), (˜f1) and (˜f2). (˜F0) can be proved by a similar argument by (F0). By the definition of ˜F(t,x), (f0) and (f1) (or (f0) and (f1)), we have

    M1|x|p1˜F(t,x)=F(t,x)M2|x|p2, if |x|δ/2,
    ˜F(t,x)=M1|x|p1, if |x|>δ,
    ˜F(t,x)F(t,x)+M1|x|p1M1|x|p1+M2|x|p2, if δ/2<|x|δ

    and

    ˜F(t,x)ρ(|x|)M1|x|p1+(1ρ(|x|))M1|x|p1=M1|x|p1, if δ/2<|x|δ.

    Hence, (˜f1) holds. Note that

    θ˜F(t,x)(˜F(t,x),x)=ρ(|x|)(θF(t,x)(F(t,x),x))+(θp1)(1ρ(|x|))M1|x|p1|x|ρ(|x|)(F(t,x)M1|x|p1).

    It is obvious that the conclusion holds for 0|x|δ/2 and |x|>δ. If δ/2<|x|δ, by using θp1, (f2), (˜f1) and the fact sρ(s)0 for all sR, we can get the conclusion (˜f2). Finally, since ρ(|x|) is even for all xRN, by (f3) and the definition of ˜F(t,x), it is easy to get (˜f3).

    Remark 3.1. From the proof of Lemma 3.1, it is easy to see that (F0), (f0) (or (f0)) and (f1) (or (f1)) independently imply (˜F0), (˜f0) and (˜f1), respectively.

    Consider the modified system

    {ddt(eL(t)p(t)(12 0Dα1t(c0Dαtu(t))12 tDα1T(ctDαTu(t))))+eL(t)q(t)u(t)=λeL(t)˜F(t,u),u(0)=u(T)=0, (3.1)

    for a.e. t[0,T], where α=1ξ2(12,1].

    If the equality

    T0eL(t)[12p(t)((c0Dαtu(t),ctDαTv(t))+(ctDαTu(t),c0Dαtv(t)))+p(t)(u(t),v(t))+(q(t)p(t))(u(t),v(t))λ(˜F(t,u(t)),v(t))]dt=0

    holds for every vEα,20, then we call uEα,20 is a weak solution of system (3.1).

    Define the functional ˜J:Eα,20R by

    ˜J(u)=T0eL(t)[12p(t)((c0Dαtu(t),ctDαTu(t))+|u(t)|2)+12(q(t)p(t))|u(t)|2λ˜F(t,u(t))]dt, for all uEα,20.

    Then (˜F0) and Theorem 6.1 in [9] imply that ˜JC1(Eα,20,R), and for every u,vEα,20, we have

    ˜J(u),v=T0eL(t)[12p(t)((c0Dαtu(t),ctDαTv(t))+(ctDαTu(t),c0Dαtv(t)))+p(t)(u(t),v(t))+(q(t)p(t))(u(t),v(t))λ(˜F(t,u(t)),v(t))]dt.

    Hence, a critical point of ˜J(u) corresponds to a weak solution of problem (3.1).

    Let

    Au,v:=T0eL(t)[12p(t)((c0Dαtu(t),ctDαTv(t))+(ctDαTu(t),c0Dαtv(t)))+p(t)(u(t),v(t))+(q(t)p(t))(u(t),v(t))]dt.

    Lemma 3.2. ([13])

    γ1u2Au,uγ2u2,for all uEα,20, (3.2)

    where γ1=|cos(πα)| and γ2=(max{Mm|cosπα|,1}+maxt[0,T](q(t)p(t))).

    Lemma 3.3. Assume that (F0), (f0) and (f1) (or (f0) and (f1)) hold. Then for each λ>0, ˜J is bounded from below on Eα,20 and satisfies (PS) condition.

    Proof. By (˜f1), (2.9) and (3.2), we have

    ˜J(u)=12Au,uλT0eL(t)˜F(t,u(t))dtγ12u2λmax{M1,M2}T0eL(t)(|u(t)|p1+|u(t)|p2)dtγ12u2λmax{M1,M2}Tmaxt[0,T]eL(t)(up1+up2)γ12u2λmax{M1,M2}Tmaxt[0,T]eL(t)[Bp1up1+Bp2up2].

    It follows from 0<p2<p1<2 that

    ˜J(u)+, as u.

    Hence, ˜J is coercive and then is bounded from below. Now we prove that ˜J satisfies the (PS) condition. Assume that {un} is a (PS) sequence of ˜J, that is,

    ˜J(un) is bounded, ˜J(un)0. (3.3)

    Then by the coercivity of ˜J and (3.3), there exists C0>0 such that unC0 and then by Lemma 2.3, there exists a subsequence (denoted again by {un}) such that

    unu, weakly in Eα,20, (3.4)
    unu, a.e.  in C([0,T],R). (3.5)

    Therefore, the boundness of {un} and (3.3) imply that

    |˜J(un),unu|˜J(un)(Eα,20)unu,˜J(un)(Eα,20)(un+u)0, (3.6)

    where (Eα,20) is the dual space of Eα,20, and (˜F0), (2.9) together with (3.5) imply that

    |λT0(˜F(t,un(t)),un(t)u(t))dt|λT0|˜F(t,un(t))||(un(t)u(t))|dtλunuT0[(1+ρ0)a0b(t)+M1p1|un(t)|p11+ρ0M1|un(t)|p1]dtλunu[(1+ρ0)a0T0b(t)dt+M1p1TBp11Cp110+M1Tρ0Bp1Cp10]0. (3.7)

    Note that

    ˜J(un),unu=Aun,unuλT0(˜F(t,un(t)),un(t)u(t))dt.

    Then (3.6) and (3.7) imply that limnAun,unu=0. Moreover, by (3.4), we have

    limnAu,unu=0.

    Therefore

    limnAunAu,unu=0.

    Since A is of type (S)+ (see [13]), by Definition 2.6, we obtain unu in Eα,20.

    Define a Nehari manifold by

    Nλ={uEα,20/{0}|˜Jλ(u),u=0}.

    Lemma 3.4. Assume that (F0) and (f0) (or (f0)) hold. For each λ>0, ˜Jλ has a nontrivial least energy (ground state) weak solution uλ, that is, uλNλ and ˜Jλ(uλ)=infNλ˜Jλ. Moreover, the least energy can be estimated as follows

    ˜Jλ(uλ)Gλ:=(p1/γ2)p12p1[λM1mint[0,T]eL(t)T0|w0(t)|p1dt]22p1(p12)2.

    where w0=ww, and w=(TπsinπtT,0,,0)Eα,20.

    Proof. By Lemma 3.3 and ˜JC1(Eα,20,R), for each λ>0, Lemma 2.5 implies that there exists some uλEα,20 such that

    ˜J(uλ)=infvEα,20˜J(v)and ˜J(uλ)=0. (3.8)

    By (3.2) and (˜f0), we have

    ˜Jλ(sw0)=12A(sw0),sw0λT0eL(t)˜F(t,sw0(t))dtγ22s2w02λT0eL(t)M1|sw0(t)|p1dtγ22s2λM1mint[0,T]eL(t)sp1T0|w0(t)|p1dt. (3.9)

    for all s[0,). Define g:[0,+)R by

    g(s)=γ22s2λM1mint[0,T]eL(t)sp1T0|w0(t)|p1dt.

    Then g(s) achieves its minimum at

    s0,λ=(p1λM1mint[0,T]eL(t)T0|w0(t)|p1dtγ2)12p1

    and

    g(s0,λ)=(p1/γ2)p12p1[λM1mint[0,T]eL(t)T0|w0(t)|p1dt]22p1(p12)2.

    Note that p1<2. So g(s0,λ)<0. Hence, (3.9) implies that

    ˜Jλ(uλ)=infvEα,20˜Jλ(v)˜Jλ(s0,λw0)g(s0,λ)<0=˜Jλ(0)

    and then uλ0 which together with (3.8) implies that uλNλ and ˜Jλ(uλ)=infNλ˜Jλ.

    Lemma 3.5. Assume that (F0), (f1) and (f2) hold. If 0<λ|cos(πα)|2C, then the following estimates hold

    uλ2p22λC|cos(πα)|,uλ2p22λCB2p2|cos(πα)|.

    Proof. It follows from Lemma 3.1, (2.9) and ˜J(uλ),uλ=0 that

    T0eL(t)[p(t)(c0Dαtuλ(t),ctDαTuλ(t))+p(t)(uλ(t),uλ(t))+(q(t)p(t))(uλ(t),uλ(t))]dt=λT0eL(t)(˜F(t,uλ(t)),uλ(t))dtλθT0eL(t)˜F(t,uλ(t))dtλθmax{M1,M2}maxt[0,T]eL(t)T0(|uλ(t)|p1+|uλ(t)|p2)dtλθmax{M1,M2}Tmaxt[0,T]eL(t)(uλp1+uλp2)λθmax{M1,M2}Tmaxt[0,T]eL(t)[Bp1uλp1+Bp2uλp2]λC(uλp1+uλp2). (3.10)

    We claim that uλ1 uniformly for all 0<λ|cos(πα)|2C. Otherwise, we have a sequence of {λn|cos(πα)|2C} such that uλn>1. Thus uλnp2<uλnp1 since p2<p1<2. By (2.10) and (3.10), we obtain

    T0eL(t)[p(t)(c0Dαtuλn(t),ctDαTuλn(t))+p(t)(uλn(t),uλn(t))+(q(t)p(t))(uλn(t),uλn(t))]dt|cos(πα)|uλn2+T0eL(t)(q(t)p(t))|uλn(t)|2dt. (3.11)

    By (3.10) and (3.11), we obtain

    |cos(πα)|uλn2+T0eL(t)(q(t)p(t))|uλn(t)|2dtλnC(uλnp1+uλnp2).

    Since q(t)p(t)>0,

    |cos(πα)|uλn2λnC(uλnp1+uλnp2)2λnCuλnp1.

    Then

    uλn2p12λnC|cos(πα)|1,

    which contradicts with the assumption uλn>1. Now, from (3.10) we can get

    |cos(πα)|uλ2λC(uλp1+uλp2)2λCuλp2.

    So

    uλ2p22λC|cos(πα)|.

    By (2.9), we can obtain

    uλBuλB(2λC|cos(πα)|)12p2.

    Observe that, in the proof of Lemma 3.5, (˜f2) is used only in (3.10). If we directly use (˜F0) to rescale (˜F(t,uλ(t)),uλ(t)) in (3.10). Then the assumption (f2) is not necessary but we have to pay the price that p(0,1). To be precise, we have the following lemma.

    Lemma 3.6. Assume that (F0) and (f0) hold. If 0<λ|cos(πα)|3C, then the following estimates hold

    uλ2p13λC|cos(πα)|,uλ2p13λCB2p1|cos(πα)|.

    Proof. It follows from (F0), Lemma 3.1, Remark 3.1, (2.9) and ˜J(uλ),uλ=0 that

    T0eL(t)[p(t)(c0Dαtuλ(t),ctDαTuλ(t))+p(t)(uλ(t),uλ(t))+(q(t)p(t))(uλ(t),uλ(t))]dt=λT0eL(t)(˜F(t,uλ(t)),uλ(t))dtλmaxt[0,T]eL(t)T0|˜F(t,uλ(t))||uλ(t)|dtλmaxt[0,T]eL(t)T0[(1+ρ0)a0b(t)|uλ(t)|+M1p1|uλ(t)|p1+ρ0M1|uλ(t)|p1+1]dtλmaxt[0,T]eL(t)[(1+ρ0)a0uλT0b(t)dt+M1p1uλp1+ρ0M1Tuλp1+1]λmaxt[0,T]eL(t)[(1+ρ0)a0BuλT0b(t)dt+M1p1TBp1uλp1+ρ0M1TBp1+1uλp1+1]λC(uλ+uλp1+uλp1+1). (3.12)

    We claim that uλ1 uniformly for all 0<λ|cos(πα)|3C. Otherwise, we have a sequence of {λn|cos(πα)|3C} such that uλn>1. Thus uλnp1<uλn<uλnp1+1 since p1<1. By (2.10) and (3.12), we obtain

    T0eL(t)[p(t)(c0Dαtuλn(t),ctDαTuλn(t))+p(t)(uλn(t),uλn(t))+(q(t)p(t))(uλn(t),uλn(t))]dt|cos(πα)|uλn2+T0eL(t)(q(t)p(t))|uλn(t)|2dt. (3.13)

    By (3.12) and (3.13), we obtain

    |cos(πα)|uλn2+T0eL(t)(q(t)p(t))|uλn|2dtλnC(uλ+uλp1+uλp1+1).

    Since q(t)p(t)>0,

    |cos(πα)|uλn2λnC(uλ+uλp1+uλp1+1)3λnCuλnp1+1.

    Then

    uλn1p13λnC|cos(πα)|1,

    which contradicts with the assumption uλn>1. Now, we can get from (3.12) that

    |cos(πα)|uλ2λC(uλ+uλp1+uλp1+1)3λCuλp1.

    So

    uλ2p13λC|cos(πα)|.

    By (2.9), we can obtain

    uλBuλB(3λC|cos(πα)|)12p1.

    Proof of Theorem 1.1. Since 0<λmin{|cos(πα)|2C,(1B)2p2(δ2)2p2|cos(πα)|2C}, Lemma 3.5 implies that

    uλδ2.

    Therefore, for all 0<λmin{|cos(πα)|2C,(1B)2p2(δ2)2p2|cos(πα)|2C}, we have ˜F(t,u(t))=F(t,u(t)) and then uλ is a nontrivial weak solution of the original problem (1.1). Moreover, Lemma 3.5 implies that limλ0uλ=0 as λ0 and

    uλ2p2min{1,(1B)2p2(δ2)2p2},uλ2p2B2p2{1,(1B)2p2(δ2)2p2}.

    Proof of Theorem 1.2. Note that 0<λmin{|cos(πα)|3C,(1B)2p1(δ2)2p1|cos(πα)|3C}. Similar to the proof of Theorem 1.1, by Lemma 3.6, it is easy to complete the proof.

    Proof of Theorem 1.3. By Lemma 3.1 and Lemma 3.3, we obtain that ˜J satisfies (PS) condition and is even and bounded from below, and ˜J(0)=0. Next, we prove that for any kN, there exists a subspace k-dimensional subspace XkEα,20 and ρk>0 such that

    supuXkSρk˜Jλ(u)<0.

    In fact, for any kN, assume that Xk is any subspace with dimension k in Eα,20. Then by (2.10) and Lemma 3.1, there exist constants C1,C2>0 such that

    ˜J(u)max{Mm|cos(πα)|,1}u2+12T0eL(t)(q(t)p(t))|u(t)|2dtλT0eL(t)˜F(t,u(t))dtmax{Mm|cos(πα)|,1}u2+C12u2λC2T0˜F(t,u(t))dtmax{Mm|cos(πα)|,1}u2+C1B22u2λC2M1T0|u(t)|p1dt[max{Mm|cos(πα)|,1}+C1B22]u2λC2M1up1Lp1.

    Since all norms on Xk are equivalent and p1<2, for each fixed λ>0, we can choose ρk>0 small enough such that

    supuXkSρk˜Jλ(u)<0.

    Thus, by Lemma 2.6 and Remark 2.3. ˜Jλ has a sequence of nonzero critical points {uλn}Eα,20 converging to 0 and ˜Jλ(uλn)0. Hence, for each fixed λ>0, (3.1) has a sequence of weak solutions {uλn}Eα,20 with uλn0, as n. Furthermore, there exists n0 large enough such that uλnδ2B for all nn0 and then (2.9) implies that uλnδ2 for all nn0. Thus, ˜F(t,u(t))=F(t,u(t)) and then {uλn}n0 is a sequence of weak solutions of the original problem (1.1) for each fixed λ>0.

    When the nonlinear term F(t,x) is local subquadratic only near the origin with respect to x, system (1.1) with λ in some given interval has a ground state weak solution uλ. If the nonlinear term F(t,x) is also locally even near the origin with respect to x, system (1.1) with λ>0 has infinitely many weak solutions {uλn}.

    This project is supported by Yunnan Ten Thousand Talents Plan Young & Elite Talents Project and Candidate Talents Training Fund of Yunnan Province (No: 2017HB016).

    The authors declare that they have no conflicts of interest.



    [1] F. Jiao, Y. Zhou, Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl., 62 (2011), 1181-1199. doi: 10.1016/j.camwa.2011.03.086
    [2] H. R. Sun, Q. G. Zhang, Existence of solutions for a fractional boundary value problem via the Mountain Pass method and an iterative technique, Comput. Math. Appl., 64 (2012), 3436-3443. doi: 10.1016/j.camwa.2012.02.023
    [3] Y. N. Li, H. R. Sun, Q. G. Zhang, Existence of solutions to fractional boundary-value problems with a parameter, Electron. J. Differ. Equ., 141 (2013), 1783-1812.
    [4] C. Bai, Existence of three solutions for a nonlinear fractional boundary value problem via a critical points theorem, Abstr. Appl. Anal, 2012 (2012), 1-13.
    [5] C. Bai, Existence of soluition for a nonlinear fractional boundary value problem via a local minmum themorem, Electron. J. Differ. Equ., 176 (2012), 1-9.
    [6] G. Bin, Multiple solutions for a class of fractional boundary value problems, Abstr. Appl. Anal., 2012 (2012), 1-16.
    [7] J. Chen, X. H. Tang, Existence and multiplicity of solutions for some fractional boundary value problem via critical point theory, Abstr. Appl. Anal., 2012 (2012), 1-21.
    [8] K. Teng, H. Jia, H. Zhang, Existence and multiplicity results for fractional differential inclusions with Dirichlet boundary conditions, Appl. Math. Comput., 220 (2013), 792-801.
    [9] Y. Zhou, Basic theory of fractional differential equations, World Scientific Publishing Company, 2014.
    [10] D. Gao, J. Li, Infinitely many solutions for impulsive fractional differential equations through variational methods, Quaest. Math., 2019, 1-17.
    [11] K. B. Ali, A. Ghanmi, K. Kefi, Existence of solutions for fractional differential equations with Dirichlet boundary conditions, Electron. J. Differ. Equ, 2016 (2016), 1-11. doi: 10.1186/s13662-015-0739-5
    [12] J. Chen, X. H. Tang, Infinitely many solutions for boundary value problems arising from the fractional advection dispersion equation, Appl. Math., 60 (2015), 703-724. doi: 10.1007/s10492-015-0118-2
    [13] N. Nyamoradi, Y. Zhou, Multiple solutions for a nonlinear fractional boundary value problems via varitional mathods, Fixed. Point. Theor., 17 (2016), 111-122.
    [14] N. Nyamoradi, Y. Zhou, Existence results to some damped-like fractional differential equations, Int. J. Nonlin. Sci. Num., 18 (2017), 88-103.
    [15] P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, American Mathematical Society, Providence, RI, 1986.
    [16] E. Zeidler, Nonlinear Functional Analysis and its Applications: III: Variational Methods and Optimization, Berlin, Springer-Verlag, 1985.
    [17] Z. L. Liu, Z. Q. Wang, On Clark's theorem and its applications to partially sublinear problems, Ann I Poincare-AN, 32 (2015), 1015-1037. doi: 10.1016/j.anihpc.2014.05.002
    [18] Z. Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin, NoDEA-Nonlinear. Diff., 8 (2001), 15-33. doi: 10.1007/PL00001436
    [19] I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer, 1990.
    [20] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, Yverdon-lesBains, Switzerland: Gordon and Breach Science Publishers, Yverdon, 1993.
    [21] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
    [22] A. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science Limited, 2006.
    [23] D. Kaus, Nonlinear Functional Analysis, Dover Publications, Dover, 2009.
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