Research article

Existence of three periodic solutions for a quasilinear periodic boundary value problem

  • Received: 02 June 2020 Accepted: 15 July 2020 Published: 24 July 2020
  • MSC : 34B15, 34C25, 70H12

  • In this paper, we prove the existence of at least three periodic solutions for the quasilinear periodic boundary value problem {p(x)x+α(t)x=λf(t,x) a.e. t[0,1],x(1)x(0)=x(1)x(0)=0 under appropriate hypotheses via a three critical points theorem of B. Ricceri. In addition, we give an example to illustrate the validity of our result.

    Citation: Zhongqian Wang, Dan Liu, Mingliang Song. Existence of three periodic solutions for a quasilinear periodic boundary value problem[J]. AIMS Mathematics, 2020, 5(6): 6061-6072. doi: 10.3934/math.2020389

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  • In this paper, we prove the existence of at least three periodic solutions for the quasilinear periodic boundary value problem {p(x)x+α(t)x=λf(t,x) a.e. t[0,1],x(1)x(0)=x(1)x(0)=0 under appropriate hypotheses via a three critical points theorem of B. Ricceri. In addition, we give an example to illustrate the validity of our result.


    In this paper, we consider the quasilinear periodic boundary value problem

    {p(x)x+α(t)x=λf(t,x)a.e.t[0,1],x(1)x(0)=x(1)x(0)=0 (1.1)

    where f(t,x):R×RR is an L1-Carathéodory function which is 1-periodic in t and λ is a positive parameter.

    We need the following assumptions:

    (Q1) p:R(0,+) is a continuous function such that there exist two positive numbers Mm and

    mp(x)M,xR. (1.2)

    (Q2) α(t)C(R) is a 1-periodic positive weight function, that is, there exist α1α0>0 such that

    α0α(t)α1,t[0,1]. (1.3)

    Since the three critical points theorem was obtained by Ricceri [11], it has been one of the most frequently applied abstract multiplicity theorems. After that, Averna-Bonanno [1] and Ricceri [12,13] have given some general three critical points theorems due to Ricceri [11]. These three critical points theorems in [1,11,12,13] are widely used to solve differential equations (see, for example [1,2,3,4,5,6,7,8,9,10,13]). In particular, in [5], using the three critical points theorem of [11], Bonanno and Livrea have studied the existence and multiplicity of solutions for the periodic boundary value problem

    {x+A(t)x=λb(t)G(x),t[0,T],x(T)x(0)=x(T)x(0)=0, (1.4)

    where A(t)=(ai,j(t))n×n is positive definite matrix for all t[0,T],ai,j(t)C([0,T],R),GC1(Rn,R) and b(t)L1([0,T]){0} that is a.e. nonnegative. Noticing that when p(x)1,T=1 and f(t,x)=b(t)g(x), the n-dimensional problem (1.4) from [5] reduces to the one-dimensional problem (1.1) in case n=1. Recently, in [7], using two general three critical points theorems of [1] and [12], Li et al. have studied the existence of three periodic solutions for p-Hamiltonian systems

    {(|x|p2x)+A(t)|x|p2x=λF(t,x)+μG(t,x),t[0,T],x(T)x(0)=x(T)x(0)=0, (1.5)

    where λ,μ[0,+),p>1,A(t)=(ai,j(t))n×n is positive definite matrix for all t[0,T],ai,j(t)C([0,T],R),F:[0,T]×RnR is a function such that F(,x) is continuous in [0,T] for all xRn and F(t,) is a C1-function in Rn for a.e. t[0,T], and G[0,T]×RnR is measurable in [0,T] and C1 in Rn. Noticing that when p=2,T=1,n=1 and μ=0, problem (1.5) becomes problem (1.1) as p(x)1.

    It is well known, the second order Hamiltonian systems satisfying periodic boundary conditions is motivated by celestial mechanics(see [15]). Finding periodic solutions for the system is a classic problem. The authors of [5] and [7] have proved the existence of three periodic solutions for this system. We also want to point out that, in [5] and [7], the nonlinear terms in the differential equations of the problems studied there do not depend on the derivatives of the unknown functions, i.e., p(x)1, in (1.1). So we are interested in problem (1.1).

    On the other hand, in [3], using the three critical points theorem of [11], Afrouzi and Heidarkhani established a three solutions result for the following quasilinear two point boundary value problem

    {x=λh(x)f(t,x),t[a,b],x(a)=x(b)=0, (1.6)

    where f:[a,b]×RR is an L1-Carathéodory function, h:R(0,+) is a continuous function and λ>0,a,bR, and extend the main result of [8] to problem (1.6). Inspired by the ideas of [3,8], we discuss the existence of three periodic solutions for problem (1.1).

    The aim of this paper is to establish some new criteria for problem (1.1) to have at least three periodic solutions by applying the three critical points theorem due to B. Ricceri. In addition, we give an example to illustrate the validity of our result.

    Next we state our results.

    Theorem 1.1. Let g(y)=y0(τ0p(ξ)dξ)dτ(yR) and F(t,x)=x0f(t,ξ)dξ. Assume that f:R×RR is an L1-Carathéodory function with 1-periodic in t, p:R(0,+) satisfies (Q1) and α(t)C(R) satisfies (Q2). Assume that there exist three positive constants c,d and s with s<2, g(2d)+g(2d)+α0d22>c2min{m,α0}, and a function γ(t)L1([0,1]) such that

    (ⅰ) f(t,x)0 for each (t,x)[0,1]×[0,3d2];

    (ⅱ) F(t,x)γ(t)(1+|x|s) for all xR and a.e. t[0,1];

    (ⅲ)

    sup(t,x)[0,1]×[c,c]F(t,x)<c2min{m,α0}g(2d)+g(2d)+4α1d23414F(t,d)dt. (1.7)

    Then, there exist an open set Λ[0,+) and a positive number r0 such that for every λΛ, problem (1.1) has at least three periodic solutions whose norms in Z are less than r0, where

    Z={x:[0,1]R|xisabsolutelycontinuous,x(1)=x(0),xL2([0,1])}.

    When f(t,x)=f1(t)f2(x), we have the following result by using Theorem 1.1.

    Theorem 1.2. Assume that f1:RR+ is a 1-periodic continuous function, f2:RR is a continuous function, p:R(0,+) satisfies (Q1) and α(t)C(R) satisfies (Q2). Assume that there exist three positive constants c,d and s with s<2, g(2d)+g(2d)+α0d22>c2min{m,α0}, and a positive constant γ such that

    (ⅰ) f2(x)0 for each x[0,3d2];

    (ⅱ) x0f2(ξ)dξγ(1+|x|s) for all xR and a.e. t[0,1];

    (ⅲ)

    maxt[0,1]f1(t)maxx[c,c]x0f2(ξ)dξ<c2min{m,α0}g(2d)+g(2d)+4α1d23414f1(t)dtd0f2(ξ)dξ. (1.8)

    Then, there exist an open set Λ[0,+) and a positive number r0 such that for every λΛ, problem

    {p(x)x+α(t)x=λf1(t)f2(x)a.e.t[0,1],x(1)x(0)=x(1)x(0)=0 (1.9)

    has at least three periodic solutions whose norms in Z are less than r0.

    Remark 1.. If we take p(x)1 in Theorem 1.2, then we can get the corresponding results of problem (1.4) as T=1,n=1. Inspecting the conditions of Theorem 3.1 in [5], it is not difficult to find that its hypothesis is different from that of our Theorem 1.2, so this is a new result.

    Furthermore, when α(t) and f(t,x) don't depend on t, we have the following autonomous version of Theorem 1.1.

    Theorem 1.3. Assume that f:RR is a continuous function and p:R(0,+) satisfies (Q1). Assume that there exist three positive constants c,d and s with s<2, g(2d)+g(2d)+α0d22>c2min{m,α0}, and a positive constant γ such that

    (ⅰ) f(x)0 for each x[0,3d2];

    (ⅱ) x0f(ξ)dξγ(1+|x|s) for all xR;

    (ⅲ)

    supx[c,c]x0f(ξ)dξ<c2min{m,α0}2g(2d)+2g(2d)+8α1d2d0f(ξ)dξ. (1.10)

    Then, there exist an open set Λ[0,+) and a positive number r0 such that for every λΛ, problem

    {p(x)x+αx=λf(x),x(1)x(0)=x(1)x(0)=0 (1.11)

    has at least three periodic solutions whose norms in Z are less than r0, where α>0.

    We postpone the proofs to the next section and turn to give an example to illustrate the validity of Theorem 1.1.

    Example 1. Let p(y)=2cosy, α(t)1 and

    f(t,x)={|sin(πt)|ex,if(t,x)[0,1]×(,16],|sin(πt)|(x+e164),if(t,x)[0,1]×(16,+).

    Then, we have m=1,M=3,α0=α1=1,

    g(y)=y2+cosy1,

    and

    F(t,x)=x0f(t,ξ)dξ={|sin(πt)|(ex1),if(t,x)[0,1]×(,16],|sin(πt)|(23x3+xe164x+A),if(t,x)[0,1]×(16,+),

    where A=15e16+613. If d=16 and c=4min{m,α0}=2, then

    g(2d)+g(2d)+α0d22=2174+2cos32>c2min{m,α0}=4,

    and

    sup(t,x)[0,1]×[2,2]F(t,x)(e21)<2(e161)3070+2cos3243070+2cos323414|sin(πt)|(e161)dt=c2min{m,α0}g(2d)+g(2d)+4α1d23414F(t,d)dt.

    This shows that (1.7) of Theorem 1.1 holds. Further, if γ(t)e16 and s=23, then all the assumptions of Theorem 1.1 are satisfied. Hence, there exist an open interval Λ[0,+) and a positive number r0 such that for every λΛ, problem (1.1) has at least three periodic solutions whose norms in Z are less than r0.

    Remark 2 If we take p(x)1 in Theorem 1.1, then we can get the corresponding results of problem (1.5) as T=1,p=2,n=1 and μ=0. Let F(t,x) and α(t) be the functions in Example 1 respectively, and then after a simple calculation, it is not difficult to verify that

    {x+α(t)x=λF(t,x),t[0,1],x(1)x(0)=x(1)x(0)=0,

    satisfy the conditions of Theorem 1 of [7]. Moreover, Example 1 does not satisfy the assumptions of Theorem 1 of [7], so Theorem 1.1 represents a development of Theorem 1 of [7] in some sense.

    For the reader's convenience, we first recall here the three critical points theorem of [12] and Proposition 3.1 of [13].

    Lemma 2.1. ([12], Theorem 1) Let Z be a separable and reflexive real Banach space, Φ:ZR a continuously Gˆateaux differentiable and sequentially weakly lower semi-continuous functional whose Gˆateaux derivative admits a continuous inverse on Z and Φ is bounded on each bounded subset of Z;Ψ:ZR a continuously Gˆateaux differentiable functional whose Gˆateaux derivative is compact. Assume that

    limx+(Φ(x)λΨ(x))=+ (2.1)

    for all λ[0,+), and that there exists βR such that

    supλ0infxZ(Φ(x)+λ(βΨ(x)))<infxZsupλ0(Φ(x)+λ(βΨ(x))). (2.2)

    Then, there exist an open set Λ[0,+) and a positive number r0 such that for every λΛ, the equation

    Φ(x)λΨ(x)=0

    has at least three solutions whose norms in Z are less than r0.

    Proposition 2.2. ([13], Proposition 3.1) Let Z be a nonempty set, and Φ,Ψ two real functions on Z. Assume that there are r>0 and x0,x1Z such that

    Φ(x1)=Ψ(x1)=0,Φ(x0)>r,andsupxΦ1((,r])Ψ(x)<rΨ(x0)Φ(x0).

    Then, for each βR satisfying

    supxΦ1((,r])Ψ(x)<β<rΨ(x0)Φ(x0),

    one has

    supη0infxZ(Φ(x)+λ(βΨ(x)))<infxZsupλ0(Φ(x)+λ(βΨ(x))).

    Remark 3 We recall that a L1-Carathéodory function f:[0,T]×RR is defined by

    (C1) tf(t,x) is measurable for every xR;

    (C2) xf(t,x) is continuous for almost every t[0,T];

    (C3) for every ρ>0 there exists a function lρL1([0,T]) such that

    sup|x|ρ|f(t,x)|lρ(t)foralmosteveryt[0,T].

    Next, we establish the variational setting for problem (1.1).

    Throughout the sequel, the Sobolev space Z is defined by

    Z={x:[0,1]R|xisabsolutelycontinuous,x(1)=x(0),xL2([0,1])}

    with the norm

    x=(10(|x|2+|x|2)dt)12.

    Clearly, Z is a Hilbert space and Z=Z, where Z is the dual space of Z.

    Setting

    g(y)=y0(τ0p(ξ)dξ)dτ,foreveryyR, (2.3)

    we have

    g(y)=y0p(ξ)dξ,andg(y)=p(y),foreveryyR. (2.4)

    Proposition 2.3. Assume that p() satisfies (Q1), then g is strongly monotone.

    Proof. By (Q1) and (2.4), we have

    g(y)g(z)={zyp(ξ)dξm(yz)<0,ify<z;yzp(ξ)dξm(yz)>0,ify>z.

    It follows that

    (g(y)g(z))(yz)m(yz)2

    for all y,zR. This shows that g is strongly monotone.

    Put

    Φ(x)=10g(x(t))dt+1210α(t)|x(t)|2dt,foreveryxZ.

    Proposition 2.4. Assume that p() and α(t) satisfy (1.2) and (1.3) respectively, then

    (1) Φ is well-defined in Z;

    (2) Φ is Gˆateaux differentiable in Z;

    (3) Φ is a Lipschitzian operator;

    (4) Φ is convex in Z.

    Proof. From (1.2) and (1.3), we have

    10g(x(t))dt=10(x(t)0(τ0p(ξ)dξ)dτ)dtM10(x(t)0τdτ)dt=M210|x(t)|2dt

    and

    Φ(x)M210|x(t)|2dt+α1210|x(t)|2dt12max{M,α1}x2,

    for every xZ, which implies that Φ is well-defined in Z.

    Taken that x,yZ and {an}R{0} with limn+an=0. By the mean value theorem of differential calculus, for a.e. t[0,1], we can see that there exist un(t) and vn(t) such that

    limn+un(t)=x(t),limn+vn(t)=x(t)

    and

    g(x(t)+any(t))g(x(t))an=g(un(t))y(t),
    12α(t)(x(t)+any(t))212α(t)(x(t))2an=α(t)vn(t)y(t).

    Then, by (1.2) and (1.3), if n is large enough, we have

    |g(un(t))y(t)|M(|x(t)|+|y(t)|)|y(t)|,|α(t)vn(t)y(t)|α1(|x(t)|+|y(t)|)|y(t)| (2.5)

    for almost every t[0,1]. Again using the Lebesgue's theorem, from the continuity of g and the arbitrariness of {an}, we know that Φ is Gˆateaux differentiable in Z with

    Φ(x)(y)=10g(x(t))y(t)dt+10α(t)x(t)y(t)dt. (2.6)

    If we fix x,y,zZ with z1. Let u(t) be such that

    |g(x(t))g(y(t))||g(u(t))||x(t)y(t)|

    for every t[0,1]. Thus, by the Hölder inequality, (1.2), (1.3) and (2.4), we have

    |10(g(x(t))g(y(t)))z(t)dt+10α(t)(x(t)y(t))z(t)dt|10|g(u(t))||x(t)y(t)||z(t)|dt+α110|x(t)y(t)||z(t)|dtMxyL2zL2+α1xyL2zL2max{M,α1}xyz,

    which shows that

    Φ(x)Φ(y)Zmax{M,α1}xy,

    for every x,yZ and Φ is Lipschitzian.

    Finally, from (1.2) and (2.4), we know that g is convex. Noticing that α(t)x2 is convex in x, we have Φ is convex in Z.

    For every xZ, put

    F(t,x)=x0f(t,ξ)dξ,andΨ(x)=10(x(t)0f(t,ξ)dξ)dt.

    Since f:[0,1]×RR is an L1-Carathéodory function, we know that Ψ is a well-defined and Gˆateaux differentiable functional with

    Ψ(x)(y)=10f(t,x(t))y(t)dt

    for every x,yZ. Since the embeddings ZLq(q1) and ZL are compact (See R. A. Adams[16]), we have Ψ:ZZ is a continuous and compact operator.

    Next, we consider the functional I:ZR defined by

    I(x)=Φ(x)λΨ(x) (2.7)

    for every xZ, where λ>0. Clearly, I is Gˆateaux differentiable. If xZ is a critical point for I, we have

    10g(x(t))y(t)dt+10α(t)x(t)y(t)dt=λ10f(t,x(t))y(t)dt

    for each yZ. This implies that gx has a weak derivative which equals α(t)x(t)λf(t,x(t)) and is thus continuous, so gx is C1([0,1]). Since g is an invertible C1-function, it follows that x is also in C1([0,1]), hence x is in C2([0,1]).

    Set

    e(t)=g(x(t))+t0α(τ)x(τ)dτλt0f(τ,x(τ))dτC

    such that 10e(t)dt=0. Let y(t)=t0e(τ)dτ. Then y(t)Z and 10|e(t)|2dt=0, that is, e(t)=0 for a.e. t[0,1]. This shows that

    (gx)(t)+α(t)x(t)=g(x)x(t)+α(t)x(t)=p(x(t))x(t)+α(t)x(t)=λf(t,x(t))

    for all t[0,1]. Hence we conclude that x is a solution of problem (1.1) belongs to C2([0,1]).

    Proof of Theorem 1.1 Consider the functional I(x)=Φ(x)λΨ(x) for every xZ and λ>0. From Proposition 2.4, we know that Φ is well-defined, Gˆateaux differentiable and convex functional in Z, and Φ is a Lipschitzian operator, which implies that Φ is a sequentially weakly lower semicontinuous via Theorem 1.2 of [15]. Further, we claim that Φ admits a continuous inverse on Z. In fact, by (1.2), (1.3), Proposition 2.3 and (2.6), we have

    Φ(x)Φ(y),xy=10(g(x(t))g(y(t)),x(t)y(t))dt+10α(t)|x(t)y(t)|2dt10m|x(t)y(t)|2dt+10α0|x(t)y(t)|2dtmin{m,α0}xy2,

    for all x,yZ, which shows that Φ is uniformly monotone in Z. Put y=0, then we have

    min{m,α0}x2|Φ(x),x|Φ(x)Zxmin{m,α0}xΦ(x)Z,

    which shows that Φ is coercive in Z. Since Φ is a Lipschitzian operator, Φ is hemicontinuous in Z. By Theorem 26. A of [14] we can see that Φ admits a continuous inverse on Z. From the estimation formula in the proof of Proposition 2.4 Φ(x)12max{M,α1}x2, we see that Φ is bounded on each bounded subset of Z.

    On the other hand, as we saw in above, Ψ:ZR a continuously Gˆateaux differentiable functional whose Gˆateaux derivative is compact. Based on the previous discussion of I, we know that the critical points of I=ΦλΨ in Z are the solutions of problem (1.1). Therefore, we only need to verify that both assumptions of Lemma 2.1 are valid.

    From (1.2) and (1.3), it follows that

    Φ(x)12min{m,α0}x2,

    for all xZ. By assumption (ⅱ), we see that

    limx+I(x)=limx+Φ(x)λΨ(x)=+,

    and so (2.1) of Lemma 2.1 holds.

    Next, We want to prove the validity of (2.2) in Lemma 2.1 by using Proposition 2.2. For xZ, taking into account

    |x(t)||tt1x(τ)dτ|+|x(t1)|10|x(τ)|dτ+|x(t1)|

    and

    |x(t)|10|x(τ)|dτ+10|x(t1)|dt1(10|x(τ)|2dτ)12+(10|x(τ)|2dτ)12,

    we have

    maxt[0,1]|x(t)|2x.

    Thus, for each r>0, we can obtain

    Φ1((,r]){xZ|maxt[0,1]|x(t)|4rmin{m,α0}},

    which shows that

    supxΦ1((,r])Ψ(x)sup(t,x)[0,1]×[c,c]F(t,x), (2.8)

    where c=4rmin{m,α0}.

    Put x1=0 and

    x0(t)={d,ift[0,14],2dt+d2,ift[14,12],2dt+5d2,ift[12,34],d,ift[34,1].

    Then we have Φ(x1)=Ψ(x1)=0, x0Z,

    Φ(x0)=14g(2d)+14g(2d)+1210α(t)|x0(t)|2dt14g(2d)+14g(2d)+α1210|x0(t)|2dt=14g(2d)+14g(2d)+31α1d22×24<14g(2d)+14g(2d)+α1d2, (2.9)

    and

    Φ(x0)14g(2d)+14g(2d)+α0210|x0(t)|2dt=14g(2d)+14g(2d)+31α0d22×24>14g(2d)+14g(2d)+α0d28. (2.10)

    From mint[14,34]{x0(t)}=d, maxt[14,34]{x0(t)}=3d2 and the assumptions (ⅰ), we have

    Ψ(x0)=140d0f(t,ξ)dξdt+3414x0(t)0f(t,ξ)dξdt+134d0f(t,ξ)dξdt3414d0f(t,ξ)dξdt=3414F(t,d)dt. (2.11)

    Moreover, choose r=c2min{m,α0}4, and recall from the assumptions of Theorem 1.1 that

    g(2d)+g(2d)+α0d22>c2min{m,α0}.

    From (1.7), (2.8), (2.9), (2.10) and (2.11) we obtain that

    Φ(x0)>g(2d)4+g(2d)4+α0d28>r>0

    and

    supxΦ1((,r])Ψ(x)<c2min{m,α0}g(2d)+g(2d)+4α1d23414F(t,d)dtrΨ(x0)Φ(x0).

    By Proposition 2.2, we know that (2.2) of Lemma 2.1 holds. So, the proof is complete.

    Proof of Theorem 1.2 Let f(t,x)=f1(t)f2(x). Noting that

    sup(t,x)[0,1]×[c,c]F(t,x)=maxt[0,1]f1(t)maxx[c,c]x0f2(ξ)dξ,

    it is easy to verify that the assumptions of Theorem 1.1 hold. So, the proof is complete.

    Proof of Theorem 1.3 Let f(t,x)=f(x). Noting that

    sup(t,x)[0,1]×[c,c]F(t,x)=maxx[c,c]x0f(ξ)dξ,

    it is easy to see that the assumptions of Theorem 1.1 hold. So, the proof is complete.

    Periodic solutions of Hamiltonian systems are important in applications. For second order Hamiltonian systems or p-Hamiltonian systems subject to periodic boundary conditions, there are many works reported on the existence of three periodic solutions. But the results on the multiplicity of periodic solutions of quasilinear periodic boundary value problem are very rare. In this paper, we study a quasilinear second order differential equation involving periodic boundary condition. Using a three critical points theorem obtained by B. Ricceri, we establish some new existence theorem of at least three periodic solutions for the quasilinear periodic boundary value problem (1.1) under appropriate hypotheses. In addition, we give an example to illustrate the validity of our results.

    This research was supported by the National Natural Science Foundation of China (No. 11901248) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No.18KJB110006). Moreover, the authors express their sincere gratitude to the referee for reading this paper very carefully and specially for valuable suggestions concerning improvement of the manuscript.

    All authors declare no conflicts of interest in this paper.



    [1] D. Averna, G. Bonanno, A three critical point theorem and its applications to the ordinary Dirichlet problem, Topol. Methods Nonlinear Anal., 22 (2003), 93-103. doi: 10.12775/TMNA.2003.029
    [2] G. A. Afrouzi, S. Heidarkhani, Three solutions for a Dirichlet boundary value problem involving the p-Laplacian, Nonlinear Anal., 66 (2007), 2281-2288. doi: 10.1016/j.na.2006.03.019
    [3] G. A. Afrouzi, S. Heidarkhani, Three solutions for a quasilinear boundary value problem, Nonlinear Anal., 69 (2008), 3330-3336. doi: 10.1016/j.na.2007.09.022
    [4] G. A. Afrouzi, A. Hadjian, V. D. Rǎdulescu, A variational approach of Sturm-Liouville problems with the nonlinearity depending on the derivative, Boundary Value Problems, 81 (2015), 1-17.
    [5] G. Bonanno, R. Livrea, Periodic solutions for a class of second order hamiltonian systems, Electron. J. Differential Equations, 115 (2005), 13.
    [6] S. Heidarkhani, Multiple solutions for a quasilinear second order differential equation depending on a parameter, Acta Mathematicae Applicatae Sinica, English Series, 32 (2016), 199-208.
    [7] C. Li, Z. Q. Ou, C. L. Tang, Three periodic solutions for p-Hamiltonian systems, Nonlinear Anal., 74 (2011), 1596-1606. doi: 10.1016/j.na.2010.10.030
    [8] R. Livrea, Existence of three solutions for a quasilinear two point boundary value problem, Arch. Math. 79 (2002), 288-298. doi: 10.1007/s00013-002-8315-0
    [9] J. R. Graef, S. Heidarkhani, L. Kong, A critical points approach for the existence of multiple solutions of a Dirichlet quasilinear system, J. Math. Anal. Appl., 388 (2012), 1268-1278. doi: 10.1016/j.jmaa.2011.11.019
    [10] Q. Meng, Three periodic solutions for a class of ordinary p-Hamiltonian systems, Boundary Value Problems, 150 (2014), 1-6.
    [11] B. Ricceri, On a three critical points theorem, Arch. Math. (Basel), 75 (2000), 220-226. doi: 10.1007/s000130050496
    [12] B. Ricceri, A three critical points theorem revisited, Nonlinear Anal., 70 (2009), 3084-3089. doi: 10.1016/j.na.2008.04.010
    [13] B. Ricceri, Existence of three solutions for a class of elliptic eigenvalue problems, Math. Comput. Modelling, 32 (2000), 1485-1494. doi: 10.1016/S0895-7177(00)00220-X
    [14] E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. II/B, Springer, Berlin, Heidelberg, New York, 1985.
    [15] J. Mawhin, M. Willem, Critical point theory and Hamiltonian systems, Springer-Verlag, New York, 1989.
    [16] R. A. Adams, J. J. F. Fournier, Sobolev spaces, Elsevier, 2003.
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