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Multiple positive solutions for a bi-nonlocal Kirchhoff-Schr¨odinger-Poisson system with critical growth


  • In this article, we study the following bi-nonlocal Kirchhoff-Schr¨odinger-Poisson system with critical growth:

    {(Ω|u|2dx)rΔu+ϕu=u5+λ(ΩF(x,u)dx)sf(x,u),in  Ω,Δϕ=u2,u>0,in  Ω,u=ϕ=0,on  Ω,

    where ΩR3 is a smooth bounded domain, λ>0, 0r<1, 0<s<1r3(r+1) and f(x,u) satisfies some suitable assumptions. By using the concentration compactness principle, the multiplicity of positive solutions for the above system is established.

    Citation: Guaiqi Tian, Hongmin Suo, Yucheng An. Multiple positive solutions for a bi-nonlocal Kirchhoff-Schr¨odinger-Poisson system with critical growth[J]. Electronic Research Archive, 2022, 30(12): 4493-4506. doi: 10.3934/era.2022228

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  • In this article, we study the following bi-nonlocal Kirchhoff-Schr¨odinger-Poisson system with critical growth:

    {(Ω|u|2dx)rΔu+ϕu=u5+λ(ΩF(x,u)dx)sf(x,u),in  Ω,Δϕ=u2,u>0,in  Ω,u=ϕ=0,on  Ω,

    where ΩR3 is a smooth bounded domain, λ>0, 0r<1, 0<s<1r3(r+1) and f(x,u) satisfies some suitable assumptions. By using the concentration compactness principle, the multiplicity of positive solutions for the above system is established.



    This paper is concerned with the following Kirchhoff-Schr¨odinger-Poisson system:

    {(Ω|u|2dx)rΔu+ϕu=u5+λ(ΩF(x,u)dx)sf(x,u),in  Ω,Δϕ=u2,u>0,in  Ω,u=ϕ=0,on  Ω, (1.1)

    where ΩR3 is a smooth bounded domain, λ>0, 0r<1, 0<s<1r3(r+1) and F(x,u)=u0f(x,ξ)dξ. We assume that fC1(¯Ω×R,R) and there exist constants a1,a2>0 and 6(r+1)r+2<q<4s+1,0<s<1r3(r+1), such that

    a1tq1f(x,t)a2tq1forany(x,t)¯Ω×R. (1.2)

    When (Ω|u|2dx)r=1 and s=0, the system (1.1) reduces to the boundary value problem

    {Δu+Vu+ϕu=f(x,u),in  Ω,Δϕ=u2,u>0,in  Ω,u=ϕ=0,on  Ω. (1.3)

    Problem (1.3) has been extensively studied, by using variational methods and critical point theory under suitable assumptions on V, f; see [1,2,3,4,5,6,7] and the references therein.

    On the other hand, considering just the first equation in (1.1) with the potential equal to zero, we have the problem

    {(a+bΩ|u|2dx)Δu=f(x,u),in  Ω,u=0,on  Ω, (1.4)

    where a,b>0, which was proposed by Kirchhoff in [8] as an extension of the classical D'Alembert's wave equation for free vibrations of elastic strings. The appearance of the nonlocal term Ω|u|2dx in the equations makes them important in many physical applications. We have to point out that such nonlocal problems appear in other fields like biological systems, such as population density, where u describes a process which depends on the average of itself (see [9]). The Kirchhoff type problem (1.4) with critical growth began to call the attention of researchers; we can see [10,11,12,13,14,15,16,17,18] and the references therein.

    In particular, Che et al. in [19] considered the following Kirchhoff-Schr¨odinger-Poisson system with critical growth:

    {(a+bR3|u|2dx)Δu+V(x)u+ϕu=λg(x)|u|q1+h(x)u5,inR3,Δϕ=u2,u>0,inR3,

    where a>0,b0,q[4,6), and λ>0 is a parameter. Under some suitable conditions on V(x), g(x) and h(x), by using the Nehari manifold technique and Ljusternik-Schnirelmann category theory, they established the number of positive solutions with the topology of the global maximum set of h when λ is small enough. Furthermore, with the aid of the mountain pass theorem, they obtained an existence result for λ sufficiently large.

    In [20], Chabrowski investigated the bi-nonlocal problem for the nonlinear elliptic equation of the form

    {(Ω|u|2dx)sΔu=Q(x)|u|p2u+(Ω|u|qdx)r|u|q2u,in  Ω,uν=0,on  Ω,

    where 2<p2,2<q<2,s>0,r>0, and 2=2NN2(N3) denotes the critical Sobolev exponent. The existence of solutions in critical and subcritical cases is obtained by using the variational method. A similar problem with Dirichlet boundary conditions has been considered in [21]. Motivated by the above references, we study the existence of multiple positive solutions for system (1.1). Our main difficulties are as follows: The critical growth of system (1.1) leads to the lack of compactness of the embedding H10(Ω)L6(Ω), and it is difficult to prove the energy functional belongs to the range where the (PS) condition holds. We overcome this difficulty by using the concentration compactness principle.

    Throughout this paper, we make use of the following notations:

    ● The space H10(Ω) is equipped with the norm u=(Ω|u|2dx)12, and the norm in Lp(Ω) is denoted by |u|p=(Ω|u|pdx)1p;

    C,C1,C2, denote various positive constants, which may vary from line to line;

    ● We denote by Sρ (respectively, Bρ) the sphere (respectively, the closed ball) of center zero and radius ρ, i.e., Sρ={uH10(Ω):u=ρ}, Bρ={uH10(Ω):uρ};

    (respectively, ) denotes strong (respectively, weak) convergence;

    ● Let S be the best Sobolev constant, namely,

    S=infuH10(Ω){0}Ω|u|2dx(Ω|u|6dx)1/3. (1.5)

    Our main result is the following:

    Theorem 1.1. Assume that q(s+1)<42(r+1)<6, λ>0, and 0<s<1r3(r+1). Then, there exists Λ>0 such that for any λ(0,Λ), system (1.1) has at least two positive solutions.

    Remark 1.1. As we shall see, in the system (1.1), when f(x,u)=|u|q2u, F(x,u)=|u|q, Chabrowski established the existence of solutions. In this paper, due to the nonlocal term ϕu in (1.1), new treatments are needed for our problem. Therefore, in this article, we extend the relevant results of [20].

    First, by using the Lax-Milgram theorem, for each uH10(Ω), there exists a unique solution ϕuH10(Ω) which satisfies the second equation of system (1.1). We substitute ϕu into the first equation of system (1.1), and then system (1.1) is transformed into the following problem:

    {(Ω|u|2dx)rΔu+ϕuu=u5+λ(ΩF(x,u)dx)sf(x,u),in  Ω,u>0,in  Ω,u=0,on  Ω. (2.1)

    We define the energy functional corresponding to problem (2.1) by

    Iλ(u)=12(r+1)(Ω|u|2dx)r+1+14Ωϕuu2dx16Ω|u|6dxλs+1(ΩF(x,u)dx)s+1.

    We say that a function uH10(Ω) is called a weak solution of problem (2.1) if for every φH10(Ω), there holds

    (Ω|u|2dx)rΩuφdx+ΩϕuuφdxΩ|u|4uφdxλ(ΩF(x,u)dx)sΩf(x,u)φdx=0.

    Before proving Theorem 1.1, we give the following important Lemma.

    Lemma 2.1. (See [22,23]) For all uH10(Ω), there exists a unique solution ϕuH10(Ω) of

    {Δϕ=u2,in  Ω,ϕ=0,on  Ω,

    and

    (1) ϕu2=Ωϕuu2dx;

    (2) ϕu0. Moreover, ϕu>0 when u0;

    (3) For each t0, ϕtu=t2ϕu;

    (4)

    Ω|ϕu|2dx=Ωϕuu2dxS1|u|412/5Cu4;

    (5) If unu in H10(Ω), then ϕunϕu in H10(Ω), and

    ΩϕununvdxΩϕuuvdx,vH10(Ω).

    Lemma 2.2. There exist constants δ,ρ,Λ0>0, for all λ(0,Λ0) such that the functional Iλ satisfies the following conditions:

    (i) Iλ|uSρδ>0, infuBρIλ(u)<0;

    (ii) There exists eH10(Ω) with e>ρ such that Iλ(e)<0.

    Proof. (i) According to H¨older's inequality and (1.5), one has

    (Ω|u|qdx)s+1[(Ω|u|6dx)q6(Ω166qdx)6q6]s+1|Ω|(6q)(s+1)6Sq(s+1)2uq(s+1). (2.2)

    By (1.2), we have

    a1|t|qf(x,t)ta2|t|qforany(x,t)¯Ω×R, (2.3)

    and

    a1q|t|qF(x,t)a2q|t|qforany(x,t)¯Ω×R. (2.4)

    Therefore, it follows from (1.5), (2.2) and (2.4) that

    Iλ(u)12(r+1)u2(r+1)16S3u6λs+1(a2q)s+1(Ω|u|qdx)s+112(r+1)u2(r+1)16S3u6λs+1(a2q|Ω|6q6Sq2uq)s+1=uq(s+1)[u2(r+1)q(s+1)2(r+1)S3u6q(s+1)6λs+1(a2q|Ω|6q6Sq2)s+1].

    Let H(t)=12(r+1)t2(r+1)q(s+1)16S3t6q(s+1) for t>0, and then there exists

    ρ=[3S3[2(r+1)q(s+1)](r+1)[6q(s+1)]]162(r+1)>0,

    such that maxt>0H(t)=H(ρ)>0. Setting

    Λ0=(s+1)qs+1Sq(s+1)2as+12|Ω|(6q)(s+1)6H(ρ),

    there exists a constant δ>0, such that Iλ|uSρδ for each λ(0,Λ0). Moreover, for every uH10(Ω){0}, we get

    limτ0+Iλ(τu)τq(s+1)=limτ0+λs+1(ΩF(x,τu)dx)s+1λs+1(a1q)s+1(Ω|u|qdx)s+1<0.

    So, we obtain Iλ(τu)<0 for all u0 and τ small enough. Hence, for u small enough, we have

    m=infuBρIλ(u)<0.

    (ii) Set uH10(Ω), and we get

    Iλ(τu)τ2(r+1)2(r+1)(Ω|u|2dx)r+1+τ44Ωϕuu2dxτ66Ω|u|6dxλτq(s+1)s+1(a1q)s+1(Ω|u|qdx)s+1,

    as τ+, which implies that Iλ(τu)<0 for τ>0 large enough. Consequently, we can find eH10(Ω) with e>ρ such that Iλ(e)<0. The proof is complete.

    Lemma 2.3. Assume that λ>0, q(s+1)<42(r+1)<6, and 0<s<1r3(r+1). Then, the functional Iλ satisfies the (PS)cλ condition for each

    cλ<c=2r6(r+1)S3(r+1)2rDλ2(r+1)2(r+1)q(s+1),

    where

    D=(2r)[2(r+1)q(s+1)]6q(r+1)(s+1)[6as+12q(s+1)as+112qs(2r)|Ω|(6q)(s+1)6Sq(s+1)2]2(r+1)2(r+1)q(s+1).

    Proof. Let {un}H10(Ω) be a (PS) sequence for Iλ at the level cλ, that is,

    Iλ(un)cλ,andIλ(un)0asn. (2.5)

    Combining with (2.2)–(2.4), we get

    cλ+1+o(un)Iλ(un)16Iλ(un),un=(12(r+1)16)un2(r+1)+(1416)Ωϕunu2ndxλ1s+1(ΩF(x,un)dx)s+1+λ6(ΩF(x,un)dx)sΩf(x,un)undx(12(r+1)16)un2(r+1)+λ[q6(a1q)s+11s+1(a2q)s+1](Ω|un|qdx)s+1(12(r+1)16)un2(r+1)+λ[q6(a1q)s+11s+1(a2q)s+1]|Ω|(6q)(s+1)6Sq(s+1)2unq(s+1).

    Then, this implies that {un} is bounded in H10(Ω) for all q(s+1)<42(r+1)<6. Thus, we may assume up to a subsequence, still denoted by {un}, there exists uH10(Ω) such that

    {unu,weaklyinH10(Ω),unu,stronglyinLp(Ω)(1p<6),un(x)u(x),a.e.inΩ, (2.6)

    as n. Next, we prove that unu strongly in H10(Ω). By using the concentration compactness principle (see [24]), there exist an at most countable set J, a family of points {xj}jJˉΩ, and positive numbers {νj}jJ, {μj}jJ such that

    |un|6dν=|u|6+jJνjδxj,
    |un|2dμ|u|2+jJμjδxj.

    Moreover, we have

    μj,νj0,μjSν13j. (2.7)

    Let φε,j(x) be a smooth cut-off function centered at xj such that 0φε,j1, |φε,j|2ε, ε>0, and

    φε,j(x)={1, in B(xj,ε2),0, in ΩB(xj,ε). (2.8)

    Noting that {unφε,j} is bounded in H10(Ω) uniformly for n, combining with (2.6) and (2.8), we have

    limε0limnΩ|un|qφε,jdxlimε0limnB(xj,ε)|un|qdx=0. (2.9)

    Similarly, we can obtain

    limε0limnΩϕunu2nφε,jdxlimε0B(xj,ε)ϕuu2φε,jdx=0. (2.10)

    By using the H¨older inequality and |φε,j|2ε, there exists C2>0, and we have

    limε0limnΩun,φε,jundxlimε0limn(Ω|un|2dx)12(Ω|un|2|φε,j|2dx)12C1limε0(B(xj,ε)|u|6dx)16(B(xj,ε)|φε,j|3dx)13C1limε0(B(xj,ε)|u|6dx)16[B(xj,ε)(2ε)3dx]13C2limε0(B(xj,ε)|u|6dx)16=0. (2.11)

    We also derive that

    limε0limnΩ|un|2φε,jdxlimε0Ω|u|2φε,jdx+μj=μj, (2.12)

    and

    limε0limnΩ|un|6φε,jdx=limε0Ω|u|6φε,jdx+νj=νj. (2.13)

    By (2.5) and (2.9)–(2.13), we get

    0=limε0limnIλ(un),unφε,j=limε0limn{(Ω|un|2dx)rΩun,(unφε,j)dx+Ωϕunu2nφε,jdxΩ|un|6φε,jdxλ(ΩF(x,un)dx)sΩf(x,un)unφε,jdx}=limε0limn{(Ω|un|2dx)rΩ|un|2φε,jdx+(Ω|un|2dx)rΩun,φε,jundxΩ|un|6φε,jdx}limε0{(Ω|u|2dx+μj)r(Ω|u|2φε,jdx+μj)νj}μr+1jνj,

    that is, νjμr+1j. If νj>0, by (2.7), we obtain

    νjS3(r+1)2r,μjS32r. (2.14)

    Now, we show that (2.14) is impossible. Assume that there exists j0J, such that μj0S32r and xj0Ω. It follows from (2.2)–(2.5) that

    cλ=limn{Iλ(un)16Iλ(un),un}=limn{(12(r+1)16)(Ω|un|2dx)r+1+(1416)Ωϕunu2ndxλs+1(ΩF(x,un)dx)s+1+λ6(ΩF(x,un)dx)sΩf(x,un)undx}2r6(r+1)(Ω|u|2dx+jJμj)r(Ω|u|2dx+jJμj)λ[1s+1(a2q)s+1q6(a1q)s+1](Ω|u|qdx)s+12r6(r+1)μr+1j0+2r6(r+1)u2(r+1)λ6as+12q(s+1)as+116(s+1)qs+1|Ω|(6q)(s+1)6Sq(s+1)2uq(s+1).

    Let

    G(t)=2r6(r+1)t2(r+1)λ6as+12q(s+1)as+116(s+1)qs+1|Ω|(6q)(s+1)6Sq(s+1)2tq(s+1).

    It is clear that limt0G(t)=0, and limt+G(t)=+. Therefore, there exists T>0 such that G(T)=mint0G(t), that is,

    G(t)|T=2r3T2r+1λ6as+12q(s+1)as+116qs|Ω|(6q)(s+1)6Sq(s+1)2Tq(s+1)1=0. (2.15)

    From (2.15) we obtain

    T=(λ(6as+12q(s+1)as+11)2qs(2r)|Ω|(6q)(s+1)6Sq(s+1)2)12(r+1)q(s+1),

    and by simple calculation, we have

    G(T)=2r6(r+1)[λ6as+12q(s+1)as+112qs(2r)|Ω|(6q)(s+1)6Sq(s+1)2]2(r+1)2(r+1)q(s+1)λ6as+12q(s+1)as+116(s+1)qs+1(|Ω|6q6Sq2)s+1[λ6as+12q(s+1)as+112qs(2r)|Ω|(6q)(s+1)6Sq(s+1)2]q(s+1)2(r+1)q(s+1)=(2r)[q(s+1)2(r+1)]6q(r+1)(s+1)[λ6as+12q(s+1)as+112qs(2r)|Ω|(6q)(s+1)6Sq(s+1)2]2(r+1)2(r+1)q(s+1).

    Hence, we can see that

    cλ2r6(r+1)S3(r+1)2rDλ2(r+1)2(r+1)q(s+1),

    where

    D=(2r)[2(r+1)q(s+1)]6q(r+1)(s+1)[6as+12q(s+1)as+112qs(2r)|Ω|(6q)(s+1)6Sq(s+1)2]2(r+1)2(r+1)q(s+1).

    We obtain that cλc. This is a contradiction, which indicates that νj=μj=0 for every jJ, which implies that unu in L6(Ω). We may assume that

    Ω|un|2dxA2,Ω|u|2dxA2.

    Combining with (2.5) and (2.6), we have

    0=limnIλ(un),unu=limn[(Ω|un|2dx)r(Ω|un|2dxΩunudx)+Ωϕunun(unu)dxΩ|un|4un(unu)dxλ(ΩF(x,un)dx)sΩf(x,un)(unu)dx]=A2r(A2Ω|u|2dx).

    Hence, we obtain

    Ω|un|2dxΩ|u|2dxasn,

    which implies unu in H10(Ω). The proof is complete.

    Choose the extremal function

    Uε(x)=(3ε2)14(ε2+|x|2)12,ε>0,

    satisfying

    ΔUε=U5εinR3.

    Let ΨC1(R3) such that Ψ(x)=1 on BR2(0), Ψ(x)=0 on R3BR(0), and 0Ψ(x)1 on R3. Set uε(x)=Ψ(x)Uε(x). From [25], one has

    {Ω|uε|2dx=S32+O(ε),Ω|uε|6dx=S32+O(ε3),

    and

    |uε|αα={O(εα2),α[2,3),O(εα2|lnε|),α=3,O(ε6α2),α(3,6). (2.16)

    Then, we have the following Lemma.

    Lemma 2.4. Suppose that λ>0, q(s+1)<42(r+1)<6, and 0<s<1r3(r+1). Then,

    supt0Iλ(tuε)<2r6(r+1)S3(r+1)2rDλ2(r+1)2(r+1)q(s+1).

    Proof. According to the definition of uε and (2.4), it holds that

    λs+1(ΩF(x,tuε)dx)s+1λs+1tq(s+1)(a1q)s+1(Ω|uε|qdx)s+1C3λ(BR/2(0)εq2(ε2+|x|2)q2dx)s+1=C3λε(6q)(s+1)2(R/2ε0y2(1+y2)q2dy)s+1C3λε(6q)(s+1)2(10y2(1+y2)q2dy)s+1C4λε(6q)(s+1)2. (2.17)

    From Lemma 2.1 and (2.16), we have the following estimate:

    Ωϕuεu2εdxS1|uε|412/5O(ε2). (2.18)

    Since Iλ(tuε) as t, by Lemma 2.2, there exists tε>0 such that

    Iλ(tεuε)=supt>0Iλ(tuε)δ>0.

    Moreover, by the continuity of Iλ, there exist positive constants t1 and t2 such that 0<t1tεt2<+. As a consequence of the above fact, one has

    supt0Iλ(tuε)=supt0{12(r+1)(Ω|tuε|2dx)r+1+14Ωϕtuε|tuε|2dx16Ω|tuε|6dxλs+1(ΩF(x,tuε)dx)s+1}supt0{12(r+1)(Ω|tuε|2dx)r+116Ω|tuε|6dx}+O(ε2)λC4ε(6q)(s+1)22r6(r+1)[(Ω|uε|2dx)r+1Ω|uε|6dx]2(r+1)62(r+1)(Ω|uε|2dx)r+1+O(ε2)λC4ε(6q)(s+1)22r6(r+1)S3(r+1)2r+C5ελC4ε(6q)(s+1)2<2r6(r+1)S3(r+1)2rDλ2(r+1)2(r+1)q(s+1).

    We have used the fact that 6(r+1)r+2<q<4s+1 and let ε=λ2(r+1)2(r+1)q(s+1).

    0<λ<Λ1=min{[2r6(r+1)DS3(r+1)2r]2(r+1)q(s+1)2(r+1),(C5+DC4)2(r+1)q(s+1)(s+1)[(r+1)(6q)q]    },

    and then

    C5εC4λε(6q)(s+1)2=C5λ2(r+1)2(r+1)q(s+1)   C4λ(r+1)(6q)(s+1)2(r+1)q(s+1)    +1=λ2(r+1)2(r+1)q(s+1)   (C5C4λ(s+1)[(r+1)(6q)q]2(r+1)q(s+1)    )<Dλ2(r+1)2(r+1)q(s+1).

    The proof is complete.

    Lemma 2.5. Suppose that 0<λ<Λ0 (Λ0 is as in Lemma 2.2). Then, system (1.1) has a positive solution uλ satisfying Iλ(uλ)<0.

    Proof. It follows from Lemma 2.2 that

    m=infu¯Bρ(0)Iλ(u)<0.

    By the Ekeland variational principle [26], there exists a minimizing sequence {un}¯Bρ(0) such that

    Iλ(un)infu¯Bρ(0)Iλ(u)+1n,Iλ(v)Iλ(un)1nvun,v¯Bρ(0).

    Therefore, we obtain that Iλ(un)m and Iλ(un)0 as n. Since {un} is a bounded sequence, and ¯Bρ(0) is a closed convex set, we may assume up to a subsequence, still denoted by {un}, there exists uλ¯Bρ(0)H10(Ω) such that

    {unuλ,weaklyinH10(Ω),unuλ,stronglyinLq(Ω)(1p<6),un(x)uλ(x),a.e.inΩ.

    By the lower semi-continuity of the norm with respect to weak convergence, we get

    mlim infn[Iλ(un)16Iλ(un),un]=lim infn[(12(r+1)16)(Ω|un|2dx)r+1+(1416)Ωϕunu2ndxλ1s+1(ΩF(x,un)dx)s+1+λ6(ΩF(x,un)dx)sΩf(x,un)undx]2r6(r+1)(Ω|uλ|2dx)r+1+112Ωϕuλu2λdxλ1s+1(ΩF(x,uλ)dx)s+1+λ6(ΩF(x,uλ)dx)sΩf(x,uλ)uλdx=Iλ(uλ)16Iλ(uλ),uλ=Iλ(uλ)m.

    Thus, Iλ(uλ)=m<0, and we can see that uλ0. Iλ(|uλ|)=Iλ(uλ), which suggests that uλ0. Therefore, by the strong maximum principle, we obtain that uλ is a positive solution of system (1.1). The proof is complete.

    Lemma 2.6. Assume that 0<λ<Λ (Λ=min{Λ0,Λ1}). Then, the system (1.1) has a positive solution uH10(Ω) with Iλ(u)>0.

    Proof. From the mountain pass lemma and Lemma 2.2, there exists a sequence {un}H10(Ω) such that

    Iλ(un)cλ>0,andIλ(un)0asn,

    where

    cλ=infγΓmaxt[0,1]Iλ(γ(t)),

    and

    Γ={γC([0,1],H10(Ω)):γ(0)=0,γ(1)=e}.

    According to Lemma 2.3, we know that {un}H10(Ω) has a convergent subsequence, still denoted by {un}, such that unu in H10(Ω) as n.

    Iλ(u)=limnIλ(un)=cλ>δ>0,

    which implies that u0. It is similar to Lemma 2.5 that u>0, that is, u is a positive solution of system (1.1) such that Iλ(u)>0. The proof is complete.

    In this paper, we considered a class of bi-nonlocal Kirchhoff-Schr¨odinger-Poisson system with critical growth. Under some suitable assumptions, by using the concentration compactness principle, we obtained the multiplicity of positive solutions.

    The authors thank the anonymous referees for careful reading and some helpful comments, which greatly improved the manuscript. This work is supported by the National Natural Science Foundation of China (No.11661021; No.11861021).

    The authors declare there are no conflicts of interest.



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