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Research article

Local controllability of complex networks

  • Received: 19 April 2021 Accepted: 20 June 2021 Published: 23 June 2021
  • The control of complex networks has been studied extensively in the last decade, with different control models been introduced. In this paper, we propose a new network control framework, called local controllability. Local controllability extends the entire network control onto a local scale, and it concerns about the minimum number of inputs required to control a subset of nodes in a directed network. We develop a mathematical formulation for local controllability as an optimization problem and show that it can be solved by a cubic-time algorithm. Moreover, applications to both real networks and model networks are presented and results of these numerical studies are then discussed.

    Citation: Chang Luo. Local controllability of complex networks[J]. Mathematical Modelling and Control, 2021, 1(2): 121-133. doi: 10.3934/mmc.2021010

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  • The control of complex networks has been studied extensively in the last decade, with different control models been introduced. In this paper, we propose a new network control framework, called local controllability. Local controllability extends the entire network control onto a local scale, and it concerns about the minimum number of inputs required to control a subset of nodes in a directed network. We develop a mathematical formulation for local controllability as an optimization problem and show that it can be solved by a cubic-time algorithm. Moreover, applications to both real networks and model networks are presented and results of these numerical studies are then discussed.



    In this paper, we are concerned with the following sub-Laplacian system with Sobolev-Hardy critical nonlinearities on Carnot group G:

    {ΔGuμ1ψ2ud(z)2=λ1ψα|u|2(α)2ud(z)α+βp1f(z)ψγ|u|p12u|v|p2d(z)γin G,ΔGvμ2ψ2vd(z)2=λ2ψα|v|2(α)2vd(z)α+βp2f(z)ψγ|u|p1|v|p22vd(z)γin G, (1.1)

    where ΔG stands for the sub-Laplacian operator on Carnot group G, μ1, μ2[0,μG), α,γ(0,2), λ1, λ2, β are positive parameters, p1, p2>0 with 1<p1+p2<2, ψ=|Gd(z)|, G denotes the horizontal gradient and d is the natural gauge associated with the fundamental solution of ΔG on G. Here, μG=(Q22)2 is the best Hardy constant and 2(α):=2(Qα)Q2 is the Sobolev-Hardy critical exponents, Q3 being the homogeneous dimension of the space G with respect to the dilation δγ. Moreover, the function f(z) satisfies the following assumption:

    (f)f(z)Lp(G,ψγd(z)γdz) and the Lebesgue measure of set {zG:f(z)>0} is positive, where p=2(γ)2(γ)(p1+p2), 0<γ<2.

    Our goal is to prove, by means of variational methods, the existence of weak solutions to (1.1). We define the energy functional Iλ1,λ2,β associated to (1.1) as follows

    Iλ1,λ2,β(u,v)=12G(|Gu|2+|Gv|2μ1ψ2|u|2d(z)2μ2ψ2|v|2d(z)2)dzλ12(α)Gψα|u|2(α)d(z)αdzλ22(α)Gψα|v|2(α)d(z)αdzβGf(z)ψγ|u|p1|v|p2d(z)γdz

    defined on the product space H:=S10(G)×S10(G), where the Folland-Stein space S10(G)={uL2(G):G|Gu|2dz<+} is the closure of C0(G) with respect to the norm

    uS10(G)=(G|Gu|2dz)12.

    Here, 2=2QQ2 is the Sobolev critical exponent. Further, we endow the product space H with the following norm

    (u,v)H=(u2μ1+v2μ2)12,

    where

    u2μi=G(|Gu|2μiψ2|u|2d(z)2)dz,i=1,2.

    The above norm is well-defined due to the following Hardy-type inequality on Carnot group

    μGGψ2|u|2d(z)2dzG|Gu|2dz,uC0(G), (1.2)

    where μG=(Q22)2 is the optimal constant for (1.2). We can note that the norms μi and S10(G) for any μi<μG with i=1,2 are equivalent due to the Hardy's inequality (1.2).

    The inequality (1.2) was first proved by Garofalo and Lanconelli in [1] for the Heisenberg group (see also [2]), and extended it to Carnot groups by D' Ambrosio, see [3]. In the Euclidean space setting, the weight function ψ appearing in the l.h.s. of (1.2) is constant, i.e., ψ1. So, (1.2) becomes the well-known Hardy inequality:

    ˉμRN|u|2|x|2dxRN|u|2dx,uC0(RN),

    where ˉμ=(N22)2 is the best constant and it is never attained. In the Euclidean space, the existence and non-existence, as well as qualitative properties, of nontrivial weak solutions for p-Laplacian equations with singular potentials and critical exponents were recently studied by several authors, we refer, e.g., in bounded domains and for p=2 to [4,5,6,7,8], and for general p>1 to [9,10,11,12]; while in Rn and for p=2 to [13,14,15], and for general p>1 to [16,17,18], and for fractional (p,q)-Laplacian to [19], and the references therein. Moreover, a more interesting result can be found in [20], which studies the critical p-Laplace equation on the Heisenberg group with a Hardy-type term.

    In recent years, people have paid much attention to the following singular sub-elliptic problem:

    {ΔGuμψ2ud(z)2=f(z,u) in Ω,u=0 on Ω, (1.3)

    where Ω is a smooth bounded domain in Carnot group G, 0Ω. It should be mentioned that [21], by using Moser-type iteration, the author studied the asymptotic behavior of weak solutions to (1.3) when the function f satisfies the following condition:

    |f(z,t)|C(|t|+|t|21)for all(z,t)Ω×R,

    and obtained the following asymptotic behavior at origin:

    u(z)d(z)(μGμGμ) as d(z)0.

    Subsequently, in [22] also the behavior at infinity has been determined for the purely critical problem

    ΔGuμψ2ud(z)2=|u|22uonG

    for which the asymptotic estimates at the origin and at infinity are then, respectively:

    u(z)1d(z)a(μ) as d(z)0,u(z)1d(z)b(μ) as d(z),

    where a(μ)=μGμGμ, b(μ)=μG+μGμ and the notation fg means that there exists a constant C>0 such that 1Cg(z)f(z)Cg(z). From a technical point of view, these asymptotic estimates have a fundamental role in the study of the associated Brezis-Nirenberg type sub-elliptic problems on Carnot group. For more details on this topic, please refer to [22], which provides a detailed analysis of the Brezis-Nirenberg problem on Carnot group.

    Motivated by the aforementioned articles and their results, we are interested in finding existence and multiplicity results for a system with critical Sobolev-Hardy critical terms. While dealing with the system (1.1), if we suppose μ1=μ2=μ, λ1=λ2=1 and β=0, problem (1.1) reduces to a sub-elliptic critical problem

    ΔGuμψ2ud(z)2=ψα|u|2(α)2ud(z)αinG. (1.4)

    In 2015, Loiudice in the paper [23] proved the existence of ground state solutions of (1.4) using variational approach for μ=0 and 0<α<2, and obtained the asymptotic behavior of this solution at infinity. Recently, Zhang [24] proved the existence of ground state solutions of (1.4) 0<μ<μG and 0<α<2 and considered the following sub-elliptic system with critical Sobolev-Hardy nonlinearities on Carnot group

    {ΔGuμψ2ud(z)2=ψα|u|2(α)2ud(z)α+ληη+θψα|u|η2u|v|θd(z)αinG,ΔGvμψ2vd(z)2=ψα|v|2(α)2vd(z)α+λθη+θψα|u|η|v|θ2vd(z)αinG,

    where α(0,2), λ>0 and η, θ>1. The existence of nontrivial solutions of the above sub-Laplacian system through variational methods was obtained for the critical case, i.e., η+θ=2(α). Other subelliptic problems with multiple critical exponents can be found in [25] and the references therein.

    Let us recall that solutions of (1.4) arise as minimizers uS10(G) of the following Rayleigh quotient:

    Sα,μ=infuS10(G){0}G|Gu|2dzμGψ2|u|2d(z)2dz(Gψα|u|2(α)d(z)αdz)22(α).

    Actually, up to a normalization, it holds that

    G|Gu|2dzμGψ2|u|2d(z)2dz=Gψα|u|2(α)d(z)αdz=(Sα,μ)Qα2α. (1.5)

    Moreover, for any ε>0, rescaled functions uε(z)=εQ22u(δ1ε(z)) are solutions, up to multiplicative constants, of the equation (1.4) and satisfy (1.5) too. However, the explicit form of ground state solutions is unknown, which is also the focus of our future work.

    As a natural extension of the above papers, we are mainly interested in searching infinitely many solutions of singular sub-elliptic problem (1.1). Our point is here a combination of sub-Laplace operator and critical Sobolev-Hardy terms on the Carnot group. In the Euclidean elliptic setting, i.e., when G is the ordinary Euclidean space (RN,+), starting with the pioneering work of Kajikiya [26], established a critical point theorem related to the symmetric mountain pass lemma and applied it to find the existence of infinitely many solutions to elliptic equation. A large number of scholars have investigated the application of this method and achieved rich results, such as He-Zou [27], Baldelli-Filippucci [28], Liang-Zhang [29,30], Ambrosio-Isernia [19] and Liang-Shi [31] in this direction.

    Motivated by the above results, our aim of this paper is to show the existence of infinitely many solutions of sub-elliptic problem (1.1), and that there exists a sequence of infinitely many arbitrarily small solutions converging to zero using the symmetric mountain-pass lemma due to Kajikiya [26]. To the best of our knowledge, there are only some results that deal with the sub-Laplacian problem with Sobolev-Hardy critical exponents and Hardy-type terms on the Carnot group.

    Before stating our main result, let us recall the definition of weak solutions to (1.1).

    Definition 1.1. We say that (u,v)H is a weak solutions of (1.1), if (u,v) satisfies

    GGuGϕ1dz+GGvGϕ2dzμ1Gψ2uϕ1d(z)2dzμ2Gψ2vϕ2d(z)2dzλ1Gψα|u|2(α)2uϕ1d(z)αdzλ2Gψα|v|2(α)2vϕ2d(z)αdzβp1Gf(z)ψγ|u|p12|v|p2uϕ1d(z)γdzβp2Gf(z)ψγ|u|p1|v|p22vϕ2d(z)γdz=0

    for all (ϕ1,ϕ2)H.

    By Hardy-Sobolev inequality, it is clear that Iλ1,λ2,β is well-defined on H and belongs to C1(H,R). Then, from Definition 1.1 we see that any weak solution of (1.1) is just a critical point of Iλ1,λ2,β. Therefore, we are now in position to state our main result as follows.

    Theorem 1.1. Assume that (f) holds, and 1<p1+p2<2, 0α<2, 0γ<2. Then

    (i) for any β>0, there exists ˜λ>0 such that if 0<λ1<˜λ, 0<λ2<˜λ, problem (1.1) has a sequence of solutions {(un,vn)}H with Iλ1,λ2,β(un,vn)<0 and Iλ1,λ2,β(un,vn)0 as n.

    (ii) for any λ1, λ2>0, there exists ˜β>0 such that if 0<β<˜β, problem (1.1) has a sequence of solutions {(un,vn)}H with Iλ1,λ2,β(un,vn)<0 and Iλ1,λ2,β(un,vn)0 as n.

    Remark 1.1 Using the symmetric mountain pass lemma (see Theorem 2.1) we can conclude that the solutions obtained from Theorem 1.1 satisfy (un,vn)(0,0) as n.

    The main idea to prove Theorem 1.1 is based on concentration-compactness result on the Carnot group and the symmetric mountain pass lemma [26]. One of the main difficulties to prove the existence and multiplicity of solutions of equation (1.1) using variational methods is that the energy functional does not satisfy the Palais-Smale condition for large energy levels, since the embedding S10(G)L2(α)(G,ψαd(z)αdz) is not compact. Another difficulty is that every nontrivial solution of (1.1) is singular at {z=0} due to the presence of the Hardy terms. Thus, different techniques are needed to deal with the singular case.

    The rest of this paper is organized as follows. In Section 2, the variational setting and some preliminary are recalled. Finally, Section 3 contains several preliminary lemmas, including the crucial concentration-compactness lemma, as well as the proof of Theorem 1.1.

    We devote this section to state some useful facts on the Carnot groups. For more details, we refer the reader to [32,33,34,35,36] and references therein.

    A Carnot group (or Stratified group) (G,) is a connected, simply connected nilpotent Lie group, whose Lie algebra g admits a stratification, namely a decomposition g=rk=1Vk with

    [V1,Vk]=Vk+1for 1kr1and[V1,Vr]={0}.

    Here, the integer r is called the step of G, dim(Vk)=Nk and the symbol [V1,Vk] denotes the subspace of g generated by the commutators [X,Y], where XV1 and YVk.

    By means of the natural identification of G with its Lie algebra via the exponential map, it is not restrictive to suppose that G is a homogeneous group, i.e., Lie group equipped with a family {δγ}γ>0 of dilations, acting on zRN as follows

    δγ(z(1),,z(r))=(γ1z(1),γ2z(2),,γrz(r)),

    where z(k)RNk for every k{1,,r} and N=rk=1Nk. Then, the structure G:=(RN,,{δγ}γ>0) is called a homogeneous group with homogeneous dimension Q:=rk=1kNk. Note that the number Q is naturally associated to the family {δγ}γ>0 since, for every γ>0, the Jacobian of the map zδγ(z) equals γQ. Moreover, the number N:=rk=1Nk is called the topological dimension of G.

    Now, let {X1,,XN1} be any basis of V1, the sub-Laplacian on G is define as the second order differential operator

    ΔG:=X21+X22++X2N1.

    The horizontal gradient on G is define as

    G:=(X1,X2,XN1).

    The horizontal divergence on G is define by

    divGu=Gu.

    It is easy to check that G and ΔG are left-translation invariant with respect to the group action τz and δγ-homogeneous, respectively, of degree one and two, that is, G(uτz)=Guτz, G(uδγ)=γGuδγ; ΔG(uτz)=ΔGuτz and ΔG(uδγ)=γ2ΔGuδγ, where the left translation τz:GG is defined by

    τz(z)=zz,z,zG.

    Let us now define the homogeneous norm Carnot group G.

    Definition 2.1 A continuous function d:G[0,+) is said to be a homogeneous norm on G if it satisfies the following condition:

    (i) d(z)=0 if and only if z=0;

    (ii) d(z1)=d(z) for all zG;

    (iii) d(δγ(z))=γd(z) for every γ>0 and zG.

    Throughout this paper, we almost exclusively work with the homogeneous norm, which is related to the fundamental solution of the sub-Laplace operator ΔG, that is the function d such that

    Γ(z)=Cd(z)Q2,zG

    is the fundamental solution of ΔG with pole at 0, for a suitable constant C>0, see [22,33]. Moreover, if we define d(z1,z2):=d(z12z1), then d is a pseudo-distence on G. In particular, d satisfies the pseudo-triangular inequality:

    d(z1,z2)c(d(z1,z3)+d(z3,z2)),z1,z2,z3G

    for a suitable positive constant c. The ball of radius R>0 centered at zG with respect to the norm d, calling them d-balls, defined as

    Bd(z,R)={yG:d(z,y)<R}.

    In fact, the norm on G can be induced by the Euclidean distance || on g through the exponential mapping, which also induces the homogeneous pseudo-norm ||g on g, namely, for ξg with ξ=ξ1++ξk, where ξiVi, define a pseudo-norm on g as follows

    |ξ|g=|(ξ1,,ξk)|g:=(ki=1|ξi|2k!i)12k!.

    The induced norm on G has the form

    |g|G=|exp1G(g)|g,gG.

    The function ||G is usually known as the non-isotropic gauge. It defines a pseudo-distence on G given by

    d(g,h):=|h1g|G,g,hG.

    The simplest example of a stratified Lie group is the Heisenberg group HN:=(R2N+1,) with the composition law as

    (x,y,t)(x,y,t):=(x1+x1,,xn+xn,y1+y1,,yn+yn,t+t+2(x,yx,y)),

    where (x,y,t),(x,y,t)RN×RN×R1 and , represents the inner product on RN. The sub-Laplacian on HN is given by

    ΔHN=Ni=1(X2i+Y2i),

    where

    Xi=xi+2yit,Yi=yi2xitfori=1,2,,N.

    In order to prove Theorem 1.1, we will recall some basic facts involved in the so-called Krasnoselskii genus, which can be found in [37,38].

    For a symmetric group Z2={id,id} and let E be a Banach space we set

    Σ={AE{0}:Ais closed and A=A}.

    For any AΣ, the Krasnoselskii's genus of A is defined by

    γ(A)=inf{k:ϕC(A,Rk) ϕis oddandϕ(z)0}.

    If k does not exist, we set γ(A)=. By above definition, it is obvious that γ()=0.

    Let Σk denote the family of closed symmetric subsets A of E such that 0E and γ(A)k, that is,

    Σk={A:AEis closed symmetric,0Eandγ(A)k}.

    Then we have the following result, see [26,37].

    Proposition 2.1. Let A and B be closed symmetric subsets of E which do not contain the origin. Then the following statements hold:

    (1) If there exists an odd continuous mapping from A to B, then γ(A)γ(B).

    (2) If AB, then γ(A)γ(B).

    (3) If there is an odd homeomorphism from A to B, then γ(A)=γ(B).

    (4) If γ(B)<, then γ¯(AB)γ(A)γ(B).

    (5) If Sn is a n-dimensional sphere, then γ(Sn)=n+1.

    (6) If A is compact, then γ(A)<+ and there exists a δ-closed symmetric neighborhood of A, i.e., Nδ(A)={uE:dist(u,A)δ} such that Nδ(A)Σk and γ(Nδ(A))=γ(A).

    Now, we state the following variant of symmetric mountain-pass lemma due to Kajikiya [26].

    Theorem 2.1. Let E be an infinite-dimensional Banach space, and let JC1(E,R) be a functional satisfying the conditions below:

    (1) J(u) is even, bounded from below, J(0)=0 and J(u) satisfies the local Palais-Smale condition, i.e. for some ˉc>0, every sequence {un} in E satisfying limnJ(un)=c<ˉc and limnJ(un)E=0 has a convergent subsequence;

    (2) For each kN, there exists AkΣk such that supuAkJ(u)<0.

    Then either (i) or (ii) below holds.

    (i) There exists a sequence {un} such that J(un)=0, J(un)<0 and {un} converges to zero as n.

    (ii) There exist two sequences {un} and {vn} such that J(un)=0, J(un)=0, un0, limnun=0; J(vn)=0, J(vn)<0,limnJ(vn)=0, and {vn} converges to a non-zero limit.

    In this section, we first discuss a compactness property for the energy functional Iλ1,λ2,β, given by the Palais-Smale condition.

    Let cR, H be a Banach space and Iλ1,λ2,βC1(H,R). {(un,vn)}H is a Palais-Smale sequence for Iλ1,λ2,β in H at level c, (PS)c-sequence for short, if

    Iλ1,λ2,β(un,vn)candIλ1,λ2,β(un,vn)0inH1as n.

    We say that Iλ1,λ2,β satisfies (PS)c-condition at level c if for any (PS)c-sequence {(un,vn)}H for Iλ1,λ2,β has a convergent subsequence in H.

    In order to apply Theorem 2.1, we need the following preliminary results for (PS)c-sequence of Iλ1,λ2,β.

    Lemma 3.1. Suppose that 1<p:=p1+p2<2 and α,γ(0,2). Let {(un,vn)}H be a (PS)c-sequence for Iλ1,λ2,β. Then, {(un,vn)} is bounded in H.

    Proof. Let {(un,vn)}H be a (PS)c-sequence for Iλ1,λ2,β, then

    Iλ1,λ2,β(un,vn)=c+on(1)andIλ1,λ2,β(un,vn)=on(1) in H1asn.

    By Young inequality and Hölder inequality, we have

    Gf(z)ψγ|un|p1|vn|p2d(z)γdzp1pGf(z)ψγ|un|pd(z)γdz+p2pGf(z)ψγ|vn|pd(z)γdzp1p(G|f(z)|2(γ)2(γ)pψ(z)γd(z)γdz)2(γ)p2(γ)(Gψγ|un|2(γ)d(z)γdz)p2(γ)+p2p(G|f(z)|2(γ)2(γ)pψ(z)γd(z)γdz)2(γ)p2(γ)(Gψγ|vn|2(γ)d(z)γdz)p2(γ)fLp(G,ψγd(z)γdz)(p1pSp2γ,μ1unpμ1+p2pSp2γ,μ2vnpμ2)fLp(G,ψγd(z)γdz)(Sp2γ,μ1+Sp2γ,μ2)(un,vn)pH.

    Then,

    on(1)+|c|+on((un,vn)H)Iλ1,λ2,β(un)12(α)Iλ1,λ2,β(un,vn),(un,vn)=(1212(α))(un,vn)2Hβ(1p2(α))Gf(z)ψγ|un|p1|vn|p2d(z)γdz2α2(Qα)(un,vn)2Hβ2(α)p2(α)fLp(G,ψγd(z)γdz)(Sp2γ,μ1+Sp2γ,μ2)un,vnpH,

    which implies that {(un,vn)} is bounded in H since p<2<2(α) and β>0.

    Proposition 3.1. Let 1<p<2, α,γ(0,2) and let {(un,vn)}H be a (PS)c-sequence of Iλ1,λ2,β with c<0. Then,

    (i) for any λ1, λ2>0, there exists β>0 such that if 0<β<β, Iλ1,λ2,β satisfies (PS)c condition, where β is independent on the sequence {(un,vn)};

    (ii) for any β>0, there exists λ>0 such that is 0<λ1<λ, 0<λ2<λ, Iλ1,λ2,β satisfies (PS)c condition, where λ is independent on the sequence {(un,vn)}.

    Proof. Since the sequence {(un,vn)} is bounded in H, thanks to Lemma 3.1, then there exists (u0,v0)H such that, up to a subsequence, it follows that

    (un,vn)(u0,v0)weakly in H,(un,vn)(u0,v0)weakly in [L2(α)(G,ψαd(z)αdz)]2,(un,vn)(u0,v0)strongly in [Ltloc(G,ψγd(z)γdz)]2,t[1,2(γ)),(un(z),vn(z))(u0(z),v0(z))a.e. in G.

    Then, by the concentration-compactness principle [39,40,41] and up to a subsequence if necessary, there exist positive finite Radon measure ˆμ, ˆν, ˆρ, ˉμ, ˉν, ˉρR(G{}); at most countable set J and ˉJ; real numbers ˆμj, ˆνj(jJ), ˉμk, ˉνk(kˉJ), ˆμ0, ˆν0, ˆρ0, ˉμ0, ˉν0, ˉρ0 and different points zjG{0} (jJ), ˉzkG{0} (kˉJ) such that

    |Gun|2dzˆμ|Gu0|2dz+jJδzjˆμj+δ0ˆμ0, (3.1)
    |Gvn|2dzˉμ|Gv0|2dz+kˉJδˉzkˉμk+δ0ˉμ0, (3.2)
    ψα|un|2(α)d(z)αdzˆν=ψα|u0|2(α)d(z)αdz+jJδzjˆνj+δ0ˆν0, (3.3)
    ψα|vn|2(α)d(z)αdzˉν=ψα|v0|2(α)d(z)αdz+kˉJδˉzkˉνk+δ0ˉν0, (3.4)
    ψ2|un|2d(z)2dzˆρ=ψ2|u0|2d(z)2dz+δ0ˆρ0, (3.5)
    ψ2|vn|2d(z)2dzˉρ=ψ2|v0|2d(z)2dz+δ0ˉρ0, (3.6)

    where δz is the Dirac mass at z. Moreover, by the Sobolev-Hardy and the Hardy inequalities, we get

    ˆμjS(α,G)ˆν22(α)j for all jJ{0}, and ˆμ0μGˆρ0, (3.7)
    ˉμkS(α,G)ˉν22(α)k for all kˉJ{0}, and ˉμ0μGˉρ0, (3.8)

    where S(α,G) is the best Hardy-Sobolev constant, i.e.,

    S(α,G)=infuS10(G){0}G|Gu|2dz(Gψα|u|2(α)d(z)αdz)22(α).

    In order to study the concentration at infinity of {un} and {vn}, we use a method of concentration-compactness principle at infinity, which was first established by Chabrowski [42]. We set

    μ:=limRlim supnG{d(z)>R}|Gun|2dz, (3.9)
    ν:=limRlim supnG{d(z)>R}ψα|un|2(α)d(z)αdz, (3.10)
    ρ:=limRlim supnG{d(z)>R}ψ2|un|2d(z)2dz, (3.11)

    and

    ˉμ:=limRlim supnG{d(z)>R}|Gvn|2dz,ˉν:=limRlim supnG{d(z)>R}ψα|vn|2(α)d(z)αdz,ˉρ:=limRlim supnG{d(z)>R}ψ2|vn|2d(z)2dz.

    For the sequence {un}, let ϕj(z)C0(G,[0,1]) be a cut-off function centered at zjG{0} with ϕj=1 on Bd(zj,1), ϕj=0 on GBd(zj,2). Let ϕj,ε(z)=ϕj(δ1ε(z)). Then |Gϕj,ε|Cε and {ϕj,εun} is bounded in S10(G). Testing Iλ1,λ2,β(un,vn) with (ϕj,εun,0), we obtain limnIλ1,λ2,β(un,vn),(ϕj,εun,0)=0, that is,

    limn(G|Gun|2ϕj,εdzμ1Gψ2|un|2ϕj,εd(z)2dzλ1Gψα|un|2(α)ϕj,εd(z)αdzβp1Gf(z)ψγ|un|p1ϕj,ε|vn|p2d(z)γdz)=limnGunGunGϕj,εdz. (3.12)

    Now, we estimate each term in (3.12). From (3.1)–(3.6), we get

    limnG|Gun|2ϕj,εdz=Gϕj,εdˆμG|Gu0|2ϕj,εdz+ˆμj, (3.13)
    limnGψα|un|2(α)ϕj,εd(z)αdz=Gϕj,εdˆν=Gψα|u0|2(α)ϕj,εd(z)αdz+ˆνj, (3.14)
    limε0limn|Gψ2|un|2ϕj,εd(z)2dz|limε0limnBd(zj,2ε)ψ2|un|2d(z)2dz=0, (3.15)

    and

    limε0limnGf(z)ψγ|un|p1ϕj,ε|vn|p2d(z)γdzlimε0limnBd(zj,2ε)f(z)ψγ|un|p1ϕj,ε|vn|p2d(z)γdzlimε0limnfLp(Bd(zj,2ε),ψγd(z)γdz)[(Bd(zj,2ε)ψγ|un|2(γ)d(z)γdz)p2(γ)+(Bd(zj,2ε)ψγ|vn|2(γ)d(z)γdz)p2(γ)]=0. (3.16)

    From Hölder inequality, it follows that

    0limε0limn|GunGunGϕj,εdz|limε0limn(G|Gun|2dz)12(G|Gϕj,ε|2|un|2dz)12Climε0(G|Gϕj,ε|2|u0|2dz)12Climε0(Bd(zj,2ε)|Gϕj,ε|Qdz)1Q(Bd(zj,2ε)|u0|2dz)12Climε0(Bd(zj,2ε)|u0|2dz)12=0. (3.17)

    Consequently, from the above arguments (3.13)–(3.17), we get

    0=limε0Iλ1,λ2,β(un,vn),(ϕεun,0)ˆμjλ1ˆνj,jJ.

    Combining with (3.7), we have

    either(1)ˆνj=0,or(2)ˆνj(S(α,G)λ1)Qα2α,

    which implies that the set J is finite.

    Similarly, for ˉνk and ˉJ, the following conclusion holds:

    ˉJis finite, and either(1)ˉνk=0,or(2)ˉνk(S(α,G)λ2)Qα2αforkˉJ.

    On the other hand, choosing a suitable cutoff function centered at the origin, by the analogous argument we can prove that

    ˆμ0μ1ˆρ0λ1ˆν0 and ˉμ0μ1ˉρ0λ1ˉν0. (3.18)

    It follows from the definition of Sα,μ1 and Sα,μ2 that

    ˆμ0μ1ˆρ0Sα,μ1ˆν22(α)0 (3.19)
    ˉμ0μ2ˉρ0Sα,μ2ˉν22(α)0. (3.20)

    Thus, by combining (3.18) and (3.19), (3.20) we get

    either(3)ˆν0=0,or(4)ˆν0(Sα,μ1λ1)Qα2α (3.21)

    and

    either(3)ˉν0=0,or(4)ˉν0(Sα,μ2λ2)Qα2α. (3.22)

    Furthermore, the Hardy inequality (1.2) implies that

    \begin{equation} 0\leq \mu_{\mathbb{G}}\hat{\rho}_{0}\leq \hat{\mu}_{0},\qquad 0\leq \Big(1-\frac{\mu_{1}}{\mu_{\mathbb{G}}}\Big)\hat{\mu}_{0}\leq \hat{\mu}_{0}-\mu_{1}\hat{\rho}_{0}, \end{equation} (3.23)

    and

    \begin{equation} 0\leq \mu_{\mathbb{G}}\bar{\rho}_{0}\leq \bar{\mu}_{0},\qquad 0\leq \Big(1-\frac{\mu_{2}}{\mu_{\mathbb{G}}}\big)\bar{\mu}_{0}\leq \bar{\mu}_{0}-\mu_{2}\bar{\rho}_{0}. \end{equation} (3.24)

    If \hat{\nu}_{0} = 0 , from (3.18) and (3.23), it follows that \hat{\mu}_{0} = \hat{\rho}_{0} = 0 . Similarly, if \bar{\nu}_{0} = 0 , by (3.18) and (3.24), we conclude \bar{\mu}_{0} = \bar{\rho}_{0} = 0 .

    To analyze the concentration at infinity, for R > 0 , we choose the function \phi\in C^{\infty}_{1}(\mathbb{G}) such that 0\leq \phi\leq 1 , \phi(z) = 0 on B_{d}(0, 1) , \phi(z) = 1 on \mathbb{G}\backslash B_{d}(0, 2) and |\nabla_{\mathbb{G}}\phi|\leq \frac{c}{R} . Set \phi_{R}(z) = \phi(\delta_{\frac{1}{R}}(z)) , then \{\phi_{R}u_{n}\}\subset S^{1}_{0}(\mathbb{G}) is bounded. Testing I'_{\lambda_{1}, \lambda_{2}, \beta}(u_{n}, v_{n}) with (\phi_{R}u_{n}, 0) we obtain \lim\limits_{n\to \infty}\langle I'_{\lambda_{1}, \lambda_{2}, \beta}(u_{n}, v_{n})\, ,\, (\phi_{R}u_{n}, 0)\rangle = 0, i.e.,

    \begin{equation} \begin{aligned} -\lim\limits_{n\to \infty}&\int_{\mathbb{G}}\langle\nabla_{\mathbb{G}} u_{n}, \nabla_{\mathbb{G}}\phi_{R}\rangle u_{n}dz = \lim\limits_{n\to \infty}\left[ \int_{\mathbb{G}}\Big(|\nabla_{\mathbb{G}}u_{n}|^{2}\phi_{R} - \mu_{1}\frac{\psi^{2}|u_{n}|^{2}}{\text{d}(z)^{2}}\phi_{R}\Big)dz\right.\\ &\qquad\qquad\qquad\left.-\lambda_{1}\int_{\mathbb{G}} \frac{\psi^{\alpha}|u_{n}|^{2^*(\alpha)}}{\text{d}(z)^{\alpha}}\phi_{R}dz -\beta p_{1}\int_{\mathbb{G}}f(z) \frac{\psi^{\gamma}|u_{n}|^{p_{1}}|v_{n}|^{p_{2}}}{\text{d}(z)^{\gamma}}\phi_{R}dz\right]. \end{aligned} \end{equation} (3.25)

    Since

    \begin{equation*} S_{\alpha,\mu_{1}}\left(\int_{\mathbb{G}} \frac{\psi^{\alpha}|u_{n}\phi_{R}|^{2^*(\alpha)}}{\text{d}(z)^{\alpha}}dz\right)^{\frac{2}{2^*(\alpha)}}\leq \int_{\mathbb{G}}\Big(|\nabla_{\mathbb{G}}(u_{n}\phi_{R})|^{2} - \mu_{1}\frac{\psi^{2}|u_{n}\phi_{R}|^{2}}{\text{d}(z)^{2}}\Big)dz, \end{equation*}

    we conclude that

    \begin{equation} \begin{aligned} &\mu_{1}\int_{\mathbb{G}} \frac{\psi^{2}|u_{n}\phi_{R}|^{2}}{\text{d}(z)^{2}}dz + S_{\alpha,\mu_{1}}\left(\int_{\mathbb{G}} \frac{\psi^{\alpha}|u_{n}\phi_{R}|^{2^*(\alpha)}}{\text{d}(z)^{\alpha}}dz\right)^{\frac{2}{2^*(\alpha)}}\\ &\leq \int_{\mathbb{G}}|\nabla_{\mathbb{G}}(u_{n}\phi_{R})|^{2}dz\\ &\leq\int_{\mathbb{G}}|\nabla_{\mathbb{G}}u_{n}|^{2}|\phi_{R}|^{2}dz +\int_{\mathbb{G}}|\nabla_{\mathbb{G}}\phi_{R}|^{2}|u_{n}|^{2}dz +2\int_{\mathbb{G}}|\nabla_{\mathbb{G}}u_{n}\phi_{R}u_{n}\nabla_{\mathbb{G}}\phi_{R}|dz. \end{aligned} \end{equation} (3.26)

    By Hölder inequality, it is easy to get that

    \begin{equation} \begin{aligned} \lim\limits_{R\to \infty}&\limsup\limits_{n\to \infty} \int_{\mathbb{G}}|\phi_{R}\nabla_{\mathbb{G}} u_{n}| |u_{n} \nabla_{\mathbb{G}}\phi_{R}|dz\\ &\leq \lim\limits_{R\to \infty}\limsup\limits_{n\to \infty} \Big(\int_{B_{\text{d}}(0,2R)\backslash B_{\text{d}}(0,R)} |\nabla_{\mathbb{G}} u_{n}|^{2}dz\Big)^{\frac{1}{2}} \Big(\int_{B_{\text{d}}(0,2R)\backslash B_{\text{d}}(0,R)}|u_{n}\nabla_{\mathbb{G}}\phi_{R}|^{2}dz\Big)^{\frac{1}{2}}\\ &\leq C \lim\limits_{R\to \infty} \Big(\int_{B_{\text{d}}(0,2R)\backslash B_{\text{d}}(0,R)} |\nabla_{\mathbb{G}}\phi_{R}|^{2} |u_{0}|^{2}dz\Big)^{\frac{1}{2}}\\ &\leq C\lim\limits_{R\to \infty} \Big(\int_{B_{\text{d}}(0,2R)\backslash B_{d}(0,R)} |\nabla_{\mathbb{G}}\phi_{\varepsilon}|^{Q}dz\Big)^{\frac{1}{Q}} \Big(\int_{B_{\text{d}}(0,2)\backslash B_{\text{d}}(0,R)} |u_{0}|^{2^*}dz\Big)^{\frac{1}{2^*}}\\ &\leq C\lim\limits_{R\to \infty} \Big(\int_{B_{\text{d}}(0,2R)\backslash B_{\text{d}}(0,R)} |u_{0}|^{2^*}dz\Big)^{\frac{1}{2^*}} = 0. \end{aligned} \end{equation} (3.27)

    Similarly,

    \begin{equation} \lim\limits_{R\to \infty}\limsup\limits_{n\to \infty} \int_{\mathbb{G}}|\nabla_{\mathbb{G}}\phi_{R}|^{2}| u_{n}|^{2}dz = 0. \end{equation} (3.28)

    Thus, we see from(3.27), (3.28) and (3.26), we have

    \begin{equation} \mu_{\infty}-\mu_{1}\rho_{\infty}\geq S_{\alpha,\mu_{1}}\cdot \nu_{\infty}^{\frac{2}{2^*(\alpha)}}. \end{equation} (3.29)

    On the other hand, from Hölder inequality and the definition of \phi_{R} we have

    \begin{align*} \label{eq3-31} &\left|\int_{\mathbb{G}}f(z) \frac{\psi^{\gamma}|u_{n}|^{p_{1}}|v_{n}|^{p_{2}}}{\text{d}(z)^{\gamma}}\phi_{R}dz\right|\nonumber\\ &\leq\left|\int_{\mathbb{G}\backslash B_{\text{d}}(0,R)}f(z) \frac{\psi^{\gamma}|u_{n}|^{p}}{\text{d}(z)^{\gamma}}\phi_{R}dz\right| + \left|\int_{\mathbb{G}\backslash B_{\text{d}}(0,R)}f(z) \frac{\psi^{\gamma}|v_{n}|^{p}}{\text{d}(z)^{\gamma}}\phi_{R}dz\right|\nonumber\\ &\leq \left(\int_{\mathbb{G}\backslash B_{\text{d}}(0,R)}\frac{\psi^{\gamma}|f(z)|^{\frac{2^*(\gamma)}{2^*(\gamma)-p}}}{\text{d}(z)^{\gamma}}dz\right)^{\frac{2^*(\gamma)-p}{2^*(\gamma)}}\left[ \left(\int_{\mathbb{G}\backslash B_{\text{d}}(0,R)} \frac{\psi^{\gamma}|u_{n}|^{2^*(\gamma)}}{\text{d}(z)^{\gamma}}\phi_{R}dz\right)^{\frac{p}{2^*(\gamma)}}\right.\nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.+ \left(\int_{\mathbb{G}\backslash B_{\text{d}}(0,R)} \frac{\psi^{\gamma}|v_{n}|^{2^*(\gamma)}}{\text{d}(z)^{\gamma}}\phi_{R}dz\right)^{\frac{p}{2^*(\gamma)}}\right]\nonumber\\ &\leq \left(\int_{\mathbb{G}\backslash B_{\text{d}}(0,R)}\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}|f(z)|^{\frac{2^*(\gamma)}{2^*(\gamma)-p}}dz\right)^{\frac{2^*(\gamma)-p}{2^*(\gamma)}}\left[ S_{\gamma,\mu_{1}}^{-\frac{p}{2}}\|u_{n}\|^{p}_{\mu_{2}}+ S_{\gamma,\mu_{1}}^{-\frac{p}{2}}\|u_{n}\|^{p}_{\mu_{2}}\right]. \end{align*}

    Since f\in L^{p_{*}}(\mathbb{G}, \frac{\psi^{\gamma}}{d(z)^{\gamma}}dz) , it follows that

    \begin{equation*} \label{eq3-32} \begin{aligned} \lim\limits_{R\to \infty}&\limsup\limits_{n\to \infty}\left|\int_{\mathbb{G}}f(z) \frac{\psi^{\gamma}|u_{n}|^{p_{1}}|v_{n}|^{p_{2}}} {\text{d}(z)^{\gamma}}\phi_{R}dz\right| \leq \lim\limits_{R\to \infty}C \left(\int_{\mathbb{G}\backslash B_{\text{d}}(0,R)} \frac{\psi^{\gamma}|f(z)|^{\frac{2^*(\gamma)}{2^*(\gamma)-p}}} {\text{d}(z)^{\gamma}}dz\right)^{\frac{2^*(\gamma)-p}{2^*(\gamma)}} = 0. \end{aligned} \end{equation*}

    Thus, taking limits by letting n\to \infty in (3.25), we have

    \begin{equation} \mu_{\infty}-\mu_{1}\rho_{\infty}\leq \lambda_{1} \nu_{\infty}. \end{equation} (3.30)

    Hence, it follows from (3.29) and (3.30) that

    \begin{equation*} \label{eq3-34} \text{either}\,\,\, (5)\,\,\nu_{\infty} = 0,\,\,\,\text{or} \,\,\, (6)\,\, \nu_{\infty}\geq \Big(\frac{S_{\alpha,\mu_{1}}}{\lambda_{1}}\Big)^{\frac{Q-\alpha}{2-\alpha}}. \end{equation*}

    In contrast, the Hardy inequality implies that

    \begin{equation} 0\leq \mu_{\mathbb{G}}\rho_{\infty}\leq \mu_{\infty},\qquad 0\leq \Big(1-\frac{\mu_{1}}{\mu_{\mathbb{G}}}\Big)\mu_{\infty}\leq \mu_{\infty}-\mu_{1}\rho_{\infty}. \end{equation} (3.31)

    If \nu_{\infty} = 0 , by combining (3.30) and (3.31), we get \mu_{\infty} = \rho_{\infty} = 0 .

    From above argument the same conclusion holds for \bar{\nu}_{\infty} , namely,

    \begin{equation*} \label{eq3-36} \bar{\mu}_{\infty}-\mu_{2}\bar{\rho}_{\infty}\geq S_{\alpha,\mu_{2}}\cdot \bar{\nu}_{\infty}^{\frac{2}{2^*(\alpha)}}, \end{equation*}
    \begin{equation*} \label{eq3-37} \bar{\mu}_{\infty}-\mu_{1}\bar{\rho}_{\infty}\leq \lambda_{2} \bar{\nu}_{\infty}, \end{equation*}

    and

    \begin{equation*} \label{eq3-38} \text{either}\,\,\, (5)'\,\,\bar{\nu}_{\infty} = 0,\,\,\,\text{or} \,\,\, (6)'\,\, \bar{\nu}_{\infty}\geq \Big(\frac{S_{\alpha,\mu_{2}}}{\lambda_{2}}\Big)^{\frac{Q-\alpha}{2-\alpha}}. \end{equation*}

    If \bar{\nu}_{\infty} = 0 , we have that \bar{\mu}_{\infty} = \bar{\rho}_{\infty} = 0 .

    Now we claim that (2) , (2)' , (4) , (4)' and (6) , (6)' cannot occur if \lambda_{1} , \lambda_{2} and \beta are chosen properly. In fact, applying (f) and Hölder inequality, we have

    \begin{equation} \begin{aligned} 0 > c & = \lim\limits_{n\to \infty}(I_{\lambda_{1},\lambda_{2},\beta}(u_n,v_{n})- \frac{1}{2^*(\alpha)}\langle I'_{\lambda_{1},\lambda_{2},\beta}(u_n,v_{n}),(u_n,v_{n})\rangle)\\ & = \lim\limits_{n\to \infty}\left(\Big(\frac{1}{2}-\frac{1}{2^*(\alpha)}\Big) \|(u_n,v_{n})\|^{2}_{\mathcal{H}}-\beta\Big(1-\frac{p}{2^*(\alpha)}\Big) \int_{\mathbb{G}}f(z)\frac{\psi^{\gamma}|u_{n}|^{p_{1}}|v_{n}|^{p_{2}}}{\text{d}(z)^{\gamma}}dz\right)\\ &\geq \frac{2^*(\alpha)-2}{2 \cdot2^*(\alpha)}\|(u_{0},v_{0})\|^{2}_{\mathcal{H}}\\ &\quad- \frac{\beta(2^*(\alpha)-p)}{2^*(\alpha)}\|f\|_{L^{p_{*}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)} \Big(\|u_{0}\|^{p}_{L^{2^*{(\gamma)}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)}+\|v_{0}\|^{p}_{L^{2^*{(\gamma)}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)}\Big)\\ &\geq \frac{2^*(\alpha)-2}{2\cdot 2^*(\alpha)} \Big(S_{\gamma,\mu_{1}}\|u_{0}\|_{L^{2^*{(\gamma)}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)}^{2} +S_{\gamma,\mu_{2}}\|v_{0}\|_{L^{2^*{(\gamma)}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)}^{2}\Big)\\ &\quad- \frac{\beta(2^*(\alpha)-p)}{2^*(\alpha)} \|f\|_{L^{p_{*}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)} \Big(\|u_{0}\|^{p}_{L^{2^*{(\gamma)}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)} +\|v_{0}\|_{L^{2^*{(\gamma)}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)}^{p}\Big).\\ \end{aligned} \end{equation} (3.32)

    Since

    \|u_{0}\|^{p}_{L^{2^*{(\gamma)}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)} +\|v_{0}\|_{L^{2^*{(\gamma)}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)}^{p}\leq 2\left(\|u_{0}\|_{L^{2^*{(\gamma)}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)} +\|v_{0}\|_{L^{2^*{(\gamma)}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)}\right)^{p},
    \|u_{0}\|^{2}_{L^{2^*{(\gamma)}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)} +\|v_{0}\|_{L^{2^*{(\gamma)}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)}^{2}\geq \frac{1}{2} \left(\|u_{0}\|_{L^{2^*{(\gamma)}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)} +\|v_{0}\|_{L^{2^*{(\gamma)}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)}\right)^{2},

    which and (3.32) yield that

    \begin{equation*} \label{eq3-40} \begin{aligned} &\frac{2\beta(2^*(\alpha)-p)}{2^*(\alpha)} \|f\|_{L^{p_{*}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)} \Big(\|u_{0}\|_{L^{2^*{(\gamma)}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)}+\|v_{0}\|_{L^{2^*{(\gamma)}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)}\Big)^{p}\\ &\quad \geq \frac{2^*(\alpha)-2}{4 \cdot 2^*(\alpha)} \min\{S_{\gamma,\mu_{1}},S_{\gamma,\mu_{2}}\} \Big(\|u_{0}\|_{L^{2^*{(\gamma)}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)} +\|v_{0}\|_{L^{2^*{(\gamma)}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)}\Big)^{2}, \end{aligned} \end{equation*}

    namely,

    \begin{equation} \|u_{0}\|_{L^{2^*{(\gamma)}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)} +\|v_{0}\|_{L^{2^*{(\gamma)}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)} \leq \left(\frac{8(2^*(\alpha)-p)\|f\|_{L^{p_{*}}(\mathbb{G}, \frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)}} {(2^*(\alpha)-2)\min\{S_{\gamma,\mu_{2}},S_{\gamma,\mu_{2}}\}} \right)^{\frac{1}{2-p}}\,\beta^{\frac{1}{2-p}}. \end{equation} (3.33)

    If (6) or (6)' occurs, we obtain by (3.32) and (3.33) that

    \begin{align*} 0& > c = \lim\limits_{n\to \infty}\Big(I_{\lambda_{1},\lambda_{2},\beta}(u_{n},v_{n})- \frac{1}{2^*(\alpha)} \langle I'_{\lambda_{1},\lambda_{2},\beta}(u_{n},v_{n}),(u_{n},v_{n})\rangle\Big)\\ &\geq \frac{2^*(\alpha)-2}{2\cdot 2^*(\alpha)} \Big(\mu_{\infty}-\mu_{1}\rho_{\infty} +\bar{\mu}_{\infty}-\mu_{2}\bar{\rho}_{\infty}\Big)\\ &-\frac{2}{2^*(\alpha)} \left(\frac{8}{(2^*(\alpha)-2)\min\{S_{\gamma,\mu_{1}},S_{\gamma,\mu_{2}}\}}\right)^{\frac{p}{2-p}} \left((2^*(\alpha)-p)\|f\|_{L^{p_{*}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)} \right)^{\frac{2}{2-p}}\cdot\beta^{\frac{2}{2-p}}\\ &\geq \frac{2^*(\alpha)-2}{2\cdot 2^*(\alpha)} \Big(S_{\alpha,\mu_{1}}\nu_{\infty}^{\frac{2}{2^*(\alpha)}} +S_{\alpha,\mu_{2}}\bar{\nu}_{\infty}^{\frac{2}{2^*(\alpha)}}\Big)\\ &-\frac{2}{2^*(\alpha)} \left(\frac{8}{(2^*(\alpha)-2)\min\{S_{\gamma,\mu_{1}},S_{\gamma,\mu_{2}}\}}\right)^{\frac{p}{2-p}} \left((2^*(\alpha)-p)\|f\|_{L^{p_{*}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)} \right)^{\frac{2}{2-p}}\cdot\beta^{\frac{2}{2-p}}\\ &\geq \frac{2^*(\alpha)-2}{2\cdot 2^*(\alpha)} \left(S_{\alpha,\mu_{1}} \Big[\Big(\frac{S_{\alpha,\mu_{1}}}{\lambda_{1}}\Big)^{\frac{Q-\alpha}{2-\alpha}}\Big]^{\frac{2}{2^*(\alpha)}}+ S_{\alpha,\mu_{2}} \Big[\Big(\frac{S_{\alpha,\mu_{2}}}{\lambda_{2}}\Big)^{\frac{Q-\alpha}{2-\alpha}}\Big]^{\frac{2}{2^*(\alpha)}}\right)\\ &-\frac{2}{2^*(\alpha)} \left(\frac{8}{(2^*(\alpha)-2)\min\{S_{\gamma,\mu_{1}},S_{\gamma,\mu_{2}}\}}\right)^{\frac{p}{2-p}} \left((2^*(\alpha)-p)\|f\|_{L^{p_{*}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)} \right)^{\frac{2}{2-p}}\cdot\beta^{\frac{2}{2-p}}\\ & = \frac{2^*(\alpha)-2}{2\cdot 2^*(\alpha)} \left((S_{\alpha,\mu_{1}})^{\frac{Q-\alpha}{2-\alpha}} \lambda_{1}^{-\frac{Q-2}{2-\alpha}}+ (S_{\alpha,\mu_{2}})^{\frac{Q-\alpha}{2-\alpha}} \lambda_{2}^{-\frac{Q-2}{2-\alpha}}\right)\\ &-\frac{2}{2^*(\alpha)} \left(\frac{8}{(2^*(\alpha)-2)\min\{S_{\gamma,\mu_{1}},S_{\gamma,\mu_{2}}\}}\right)^{\frac{p}{2-p}} \left((2^*(\alpha)-p)\|f\|_{L^{p_{*}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)} \right)^{\frac{2}{2-p}}\cdot\beta^{\frac{2}{2-p}}, \end{align*}

    that is,

    \begin{equation} \begin{aligned} 0 > &\frac{2^*(\alpha)-2}{2\cdot 2^*(\alpha)} \left((S_{\alpha,\mu_{1}})^{\frac{Q-\alpha}{2-\alpha}} \lambda_{1}^{-\frac{Q-2}{2-\alpha}}+ (S_{\alpha,\mu_{2}})^{\frac{Q-\alpha}{2-\alpha}} \lambda_{2}^{-\frac{Q-2}{2-\alpha}}\right)\\ &-\frac{2}{2^*(\alpha)} \left(\frac{8}{(2^*(\alpha)-2)\min\{S_{\gamma,\mu_{1}},S_{\gamma,\mu_{2}}\}}\right)^{\frac{p}{2-p}} \left((2^*(\alpha)-p)\|f\|_{L^{p_{*}}{p_{*}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)} \right)^{\frac{2}{2-p}}\cdot\beta^{\frac{2}{2-p}}. \end{aligned} \end{equation} (3.34)

    From the above inequality, we can find that if \beta > 0 is given, there exists \lambda_{*} > 0 small enough such that for \lambda_{1}, \, \lambda_{2}\in (0, \lambda_{*}) , the right-hand side of (3.34) is greater than 0 , which is a contradiction. Similarly, if \lambda_{1}, \, \lambda_{2} > 0 is given, we can take \beta_{*} > 0 so small that for \beta\in (0, \beta_{*}) , right-hand side of (3.34) is greater than 0 .

    Similarly we can prove that (2) , (2)' and (4) , (4)' cannot occur. So

    \begin{equation*} \lim\limits_{n\to \infty}\int_{\mathbb{G}}\frac{\psi^{\alpha}|u_{n}|^{2^*(\alpha)}}{\text{d}(z)^{\alpha}}dz = \int_{\mathbb{G}}\frac{\psi^{\alpha}|u_{0}|^{2^*(\alpha)}}{\text{d}(z)^{\alpha}}dz \end{equation*}

    and

    \begin{equation*} \lim\limits_{n\to \infty}\int_{\mathbb{G}}\frac{\psi^{\alpha}|v_{n}|^{2^*(\alpha)}}{\text{d}(z)^{\alpha}}dz = \int_{\mathbb{G}}\frac{\psi^{\alpha}|v_{0}|^{2^*(\alpha)}}{\text{d}(z)^{\alpha}}dz. \end{equation*}

    In view of (u_{n}, v_{n})\rightharpoonup (u_{0}, v_{0 }) weakly in \mathcal{H} and the Brezis-Lieb lemma [38], we have

    \begin{equation*} \label{eq3-43} \lim\limits_{n\to \infty}\int_{\mathbb{G}} \frac{\psi^{\alpha}|u_{n}-u_{0}|^{2^*(\alpha)}}{\text{d}(z)^{\alpha}}dz = 0, \qquad \lim\limits_{n\to \infty}\int_{\mathbb{G}} \frac{\psi^{\alpha}|v_{n}-v_{0}|^{2^*(\alpha)}}{\text{d}(z)^{\alpha}}dz = 0. \end{equation*}

    We are now going to prove that (u_{n}, v_{n})\to (u_{0}, v_{0}) strongly in \mathcal{H} . First, we have

    \begin{equation} \begin{aligned} \|(u_{n}-u_{0},v_{n}-v_{0})\|_{\mathcal{H}}^{2} & = \langle (I'_{\lambda_{1},\lambda_{2},\beta}(u_{n},v_{n}) -I'_{\lambda_{1},\lambda_{2},\beta}(u_{0},v_{0})),(u_{n}-u_{0},v_{n}-v_{0})\rangle\\ &+\lambda_{1} \int_{\mathbb{G}} \frac{\psi^{\alpha}(|u_{n}|^{2^*(\alpha)-2}u_{n}-|u_{0}|^{2^*(\alpha)-2}u_{0})(u_{n}-u_{0})} {\text{d}(z)^{\alpha}}dz\\ &+\lambda_{2} \int_{\mathbb{G}} \frac{\psi^{\alpha}(|v_{n}|^{2^*(\alpha)-2}v_{n}-|v_{0}|^{2^*(\alpha)-2}v_{0})(v_{n}-v_{0})} {\text{d}(z)^{\alpha}}dz\\ &+\beta p_{1}\int_{\mathbb{G}}f(z) \frac{\psi^{\gamma}[|u_{n}|^{p_{1}-2}u_{n}|v_{n}|^{p_{2}}-|u_{0}|^{p_{1}-2}u_{0}|v_{0}|^{p_{2}}](u_{n}-u_{0})}{\text{d}(z)^{\gamma}}dz\\ &+\beta p_{2}\int_{\mathbb{G}}f(z) \frac{\psi^{\gamma}[|u_{n}|^{p_{1}}|v_{n}|^{p_{2}-2}v_{n}-|u_{0}|^{p_{1}}|v_{0}|^{p_{2}-2}v_{0}](v_{n}-v_{0})}{\text{d}(z)^{\gamma}}dz. \end{aligned} \end{equation} (3.35)

    For the first term in (3.35), by using Hölder inequality, we get that

    \begin{equation} \begin{aligned} &\Big|\int_{\mathbb{G}} \frac{\psi^{\alpha}(|u_{n}|^{2^*(\alpha)-2}u_{n}-|u_{0}|^{2^*(\alpha)-2}u_{0})(u_{n}-u_{0})} {\text{d}(z)^{\alpha}}dz\Big|\\ &\leq\int_{\mathbb{G}} \frac{\psi^{\alpha}|u_{n}|^{2^*(\alpha)-1}|u_{n}-u_{0}|}{\text{d}(z)^{\alpha}}dz+ \int_{\mathbb{G}} \frac{\psi^{\alpha}|u_{0}|^{2^*(\alpha)-1}|u_{n}-u_{0}|}{\text{d}(z)^{\alpha}}dz \\ &\leq\left(\int_{\mathbb{G}} \frac{\psi^{\alpha}|u_{n}|^{2^*(\alpha)}}{\text{d}(z)^{\alpha}}dz\right)^{\frac{2^*(\alpha)-1}{2^*(\alpha)}} \left(\int_{\mathbb{G}} \frac{\psi^{\alpha}|u_{n}-u_{0}|^{2^*(\alpha)}}{\text{d}(z)^{\alpha}}dz\right)^{\frac{1}{2^*(\alpha)}}\\ &\quad+ \left(\int_{\mathbb{G}} \frac{\psi^{\alpha}|u_{0}|^{2^*(\alpha)}}{\text{d}(z)^{\alpha}}dz\right)^{\frac{2^*(\alpha)-1}{2^*(\alpha)}} \left(\int_{\mathbb{G}} \frac{\psi^{\alpha}|u_{n}-u_{0}|^{2^*(\alpha)}}{\text{d}(z)^{\alpha}}dz\right)^{\frac{1}{2^*(\alpha)}}\\ &\to 0\,\,\,\text{as}\,\,n\to \infty. \end{aligned} \end{equation} (3.36)

    Similarly,

    \begin{equation} \Big|\int_{\mathbb{G}} \frac{\psi^{\alpha}(|v_{n}|^{2^*(\alpha)-2}v_{n}-|v_{0}|^{2^*(\alpha)-2}v_{0})(v_{n}-v_{0})} {\text{d}(z)^{\alpha}}dz\Big| \to 0\,\,\text{as}\,\,n\to \infty. \end{equation} (3.37)

    On the other hand, using the Hölder inequality and (u_{n}, v_{n})\rightharpoonup (u_{0}, v_{0}) weakly in \mathcal{H} , we get that

    \begin{equation} \begin{aligned} &\Big|\int_{\mathbb{G}}f(z) \frac{\psi^{\gamma}[|u_{n}|^{p_{1}-2}u_{n}|v_{n}|^{p_{2}}-|u_{0}|^{p_{1}-2}u_{0}|v_{0}|^{p_2}](u_{n}-u_{0})}{\text{d}(z)^{\gamma}}dz\Big|\\ &\leq \int_{\mathbb{G}} \frac{\psi^{\gamma}|f(z)||u_{n}|^{p-1}|u_{n}-u_{0}|} {\text{d}(z)^{\gamma}}dz +\int_{\mathbb{G}}|f(z)| \frac{\psi^{\gamma}|u_{0}|^{p-1}|u_{n}-u_{0}|} {\text{d}(z)^{\gamma}}dz\\ &\leq\left(\int_{\mathbb{G}} \frac{\psi^{\gamma}|f(z)|^{\frac{2^*(\gamma)}{2^*(\gamma)-p}}} {\text{d}(z)^{\gamma}}dz\right)^{\frac{2^*(\gamma)-p}{2^*(\gamma)}} \left(\int_{\mathbb{G}}\frac{\psi^{\gamma}|u_{n}|^{2^*(\gamma)}} {\text{d}(z)^{\gamma}}dz\right)^{\frac{p-1}{2^*(\gamma)}} \left(\int_{\mathbb{G}}\frac{\psi^{\gamma}|u_{n}-u_{0}|^{2^*(\gamma)}} {\text{d}(z)^{\gamma}}dz\right)^{\frac{1}{2^*(\gamma)}}\\ &+\left(\int_{\mathbb{G}}\frac{\psi^{\gamma}|f(z)|^{\frac{2^*(\gamma)}{2^*(\gamma)-p}}} {\text{d}(z)^{\gamma}}dz\right)^{\frac{2^*(\gamma)-p}{2^*(\gamma)}} \left(\int_{\mathbb{G}}\frac{\psi^{\gamma}|u_{0}|^{2^*(\gamma)}}{\text{d}(z)^{\gamma}}dz\right)^{\frac{p-1}{2^*(\gamma)}} \left(\int_{\mathbb{G}}\frac{\psi^{\gamma}|u_{n}-u_{0}|^{2^*(\gamma)}}{\text{d}(z)^{\gamma}}dz\right)^{\frac{1}{2^*(\gamma)}}\\ &\to 0\quad\text{as}\,\,\,n\to \infty, \end{aligned} \end{equation} (3.38)

    and

    \begin{equation} \Big|\int_{\mathbb{G}}f(z) \frac{\psi^{\gamma}[|u_{n}|^{p_{1}}|v_{n}|^{p_{2}-2}v_{n} -|u_{0}|^{p_{1}}|v_{0}|^{p_{2}-2}v_{0}](v_{n}-v_{0})}{\text{d}(z)^{\gamma}}dz\Big| \to 0\,\,\,\text{as}\,\,n\to \infty, \end{equation} (3.39)

    Combining (3.36), (3.37), (3.38), (3.39), (3.35) with \lim\limits_{n\to \infty}\langle I'_{\lambda_{1}, \lambda_{2}, \beta}(u_{n}, v_{n}), (u_{n}-u_{0}, v_{n}-v_{0})\rangle = 0 and \lim\limits_{n\to \infty}\langle I'_{\lambda_{1}, \lambda_{2}, \beta}(u_{0}, v_{0}), (u_{n}-u_{0}, v_{n}-v_{0})\rangle = 0 , we deduce that

    \lim\limits_{n\to \infty}\|(u_{n}-u_{0},v_{n}-v_{0})\|_{\mathcal{H}} = 0.

    The proof is completed.

    In the end of this section, we will prove the existence of infinitely many weak solutions of (1.1) which tend to zero. First, by using Hölder's inequality and Young's inequality, we get

    \begin{equation*} \label{eq4-1} \begin{aligned} \int_{\mathbb{G}}\frac{\psi^{\alpha}|u|^{2^*(\alpha)}}{\text{d}(z)^{\alpha}}dz +\int_{\mathbb{G}}\frac{\psi^{\alpha}|v|^{2^*(\alpha)}}{\text{d}(z)^{\alpha}}dz &\leq S_{\alpha,\mu_{1}}^{-\frac{2^*(\alpha)}{2}}\|u\|_{\mu_{1}}^{2^*(\alpha)} +S_{\alpha,\mu_{2}}^{-\frac{2^*(\alpha)}{2}}\|v\|_{\mu_{2}}^{2^*(\alpha)}\\ &\leq (S_{\alpha,\mu_{1}}^{-\frac{2^*(\alpha)}{2}} +S_{\alpha,\mu_{2}}^{-\frac{2^*(\alpha)}{2}})\|(u,v)\|_{\mathcal{H}}^{2^*(\alpha)}, \end{aligned} \end{equation*}

    and

    \begin{equation} \begin{aligned} \int_{\mathbb{G}}f(z) &\frac{\psi^{\gamma}|u_{n}|^{p_{1}}|v_{n}|^{p_{2}}}{\text{d}(z)^{\gamma}}dz \leq \frac{p_{1}}{p}\int_{\mathbb{G}}f(z)\frac{\psi^{\gamma}|u_{n}|^{p}}{\text{d}(z)^{\gamma}}dz +\frac{p_{2}}{p}\int_{\mathbb{G}}f(z)\frac{\psi^{\gamma}|v_{n}|^{p}}{\text{d}(z)^{\gamma}}dz\\ &\leq \frac{p_{1}}{p} \left(\int_{\mathbb{G}}|f(z)|^{\frac{2^*(\gamma)}{2^*(\gamma)-p}}\frac{\psi^{\gamma}} {\text{d}(z)^{\gamma}}dz\right)^{\frac{2^*(\gamma)-p}{2^*(\gamma)}} \left(\int_{\mathbb{G}}\frac{\psi^{\gamma}|u_{n}|^{2^*(\gamma)}} {\text{d}(z)^{\gamma}}dz\right)^{\frac{p}{2^*(\gamma)}}\\ &+ \frac{p_{2}}{p} \left(\int_{\mathbb{G}}|f(z)|^{\frac{2^*(\gamma)}{2^*(\gamma)-p}}\frac{\psi^{\gamma}} {\text{d}(z)^{\gamma}}dz\right)^{\frac{2^*(\gamma)-p}{2^*(\gamma)}} \left(\int_{\mathbb{G}}\frac{\psi^{\gamma}|v_{n}|^{2^*(\gamma)}} {\text{d}(z)^{\gamma}}dz\right)^{\frac{p}{2^*(\gamma)}}\\ &\leq \|f\|_{L^{p_{*}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)} \Big(\frac{p_{1}}{p} S_{\gamma,\mu_{1}}^{-\frac{p}{2}}\|u_{n}\|_{\mu_{1}}^{p} + \frac{p_{2}}{p}S_{\gamma,\mu_{2}}^{-\frac{p}{2}}\|v_{n}\|_{\mu_{2}}^{p}\Big)\\ &\leq \|f\|_{L^{p_{*}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)} (S_{\gamma,\mu_{1}}^{-\frac{p}{2}} + S_{\gamma,\mu_{2}}^{-\frac{p}{2}})\|(u_{n},v_{n})\|_{\mathcal{H}}^{p}. \end{aligned} \end{equation} (3.40)

    Then,

    \begin{equation*} \begin{aligned} I_{\lambda_{1},\lambda_{2},\beta}(u,v) & = \frac{1}{2}\|(u,v)\|^{2}_{\mathcal{H}} -\frac{\lambda_{1}}{2^*(\alpha)}\int_{\mathbb{G}}\frac{\psi^{\alpha}|u|^{2^*(\alpha)}}{\text{d}(z)^{\alpha}}dz -\frac{\lambda_{2}}{2^*(\alpha)}\int_{\mathbb{G}}\frac{\psi^{\alpha}|v|^{2^*(\alpha)}}{\text{d}(z)^{\alpha}}dz \\ &\quad-\beta \int_{\mathbb{G}}f(z)\frac{\psi^{\gamma}|u|^{p_{1}}|v|^{p_{2}}}{\text{d}(z)^{\gamma}}dz\\ &\geq \frac{1}{2}\|(u,v)\|_{\mathcal{H}}^{2} -(\lambda_{1}+\lambda_{2})\frac{ (S_{\alpha,\mu_{1}}^{-\frac{2^*(\alpha)}{2}} +S_{\alpha,\mu_{2}}^{-\frac{2^*(\alpha)}{2}})}{2^*(\alpha)}\|(u,v)\|_{\mathcal{H}}^{2^{*}(\alpha)}\\ &\quad-\beta\|f\|_{L^{p_{*}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)} (S_{\gamma,\mu_{1}}^{-\frac{p}{2}} + S_{\gamma,\mu_{2}}^{-\frac{p}{2}})\|(u,v)\|^{p}_{\mathcal{H}}. \end{aligned} \end{equation*}

    Define the function

    g(t) = \frac{1}{2}t^{2}-C_{1}(\lambda_{1}+\lambda_{2})t^{2^*(\alpha)}-C_{2}\beta t^{p},\quad \forall t > 0,

    where

    C_{1}: = \frac{ (S_{\alpha,\mu_{1}}^{-\frac{2^*(\alpha)}{2}} +S_{\alpha,\mu_{2}}^{-\frac{2^*(\alpha)}{2}})}{2^*(\alpha)}, \quad C_{2}: = \|f\|_{L^{p_{*}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)} (S_{\gamma,\mu_{1}}^{-\frac{p}{2}} + S_{\gamma,\mu_{2}}^{-\frac{p}{2}}) > 0.

    Because 1 < p < 2 < 2^*(\alpha) , for the given \beta > 0 , there exists \lambda_{**} > 0 so small that for \lambda_{1}+\lambda_{2} \in (0, \lambda_{**}) , there exist t_{1} , t_{2} > 0 with t_1 < t_2 such that g(t_1) = g(t_2) = 0 , and g(t) < 0 for t \in (0, t_1) , g(t) > 0 for t \in (t_{1}, t_2) , g(t) < 0 for t \in (t_{2}, +\infty) . Similarly, given \lambda_{1} , \lambda_{2} > 0 , we can choose \beta_{**} > 0 small enough such that for all \beta \in(0, \beta_{**}) , there exist \hat{t}_{1} , \hat{t}_{2} > 0 with \hat{t}_1 < \hat{t}_2 such that g(\hat{t}_1) = g(\hat{t}_2) = 0 and g(t) < 0 for t\in (0, \hat{t}_1) , g(t) > 0 for t\in (\hat{t}_1, \hat{t}_2) , g(t) < 0 for t\in (\hat{t}_{2}, +\infty) .

    Let us define a function \phi\in C^{\infty}_{0}([0\, ,\, \infty), \mathbb{R}) such that 0\leq \phi(t)\leq 1 , \phi(-t) = \phi(t) for all t\in [0, +\infty) , \phi(t) = 1 if t\in [0, t_{1}] and \phi(t) = 0 if t\in [t_{2}, \infty) . So we consider the equation

    \begin{equation} \left\{\begin{aligned} &-\Delta_{\mathbb{G}}u-\mu_{1}\frac{\psi^{2}u}{\text{d}(z)^{2}} = \lambda_{1}\phi(\|(u,v)\|_{\mathcal{H}})\frac{\psi^{\alpha}|u|^{2^*(\alpha)-2}u}{\text{d}(z)^{\alpha}}+\beta p_{1}f(z)\frac{\psi^{\gamma}|u|^{p_{1}-2}u|v|^{p_{2}}}{\text{d}(z)^{\gamma}}\,\,\, &\text{in } \,\,\mathbb{G},\\ &-\Delta_{\mathbb{G}}v-\mu_{2}\frac{\psi^{2}v}{\text{d}(z)^{2}} = \lambda_{2}\phi(\|(u,v)\|_{\mathcal{H}})\frac{\psi^{\alpha}|v|^{2^*(\alpha)-2}v}{\text{d}(z)^{\alpha}}+\beta p_{2}f(z)\frac{\psi^{\gamma}|u|^{p_{1}}|v|^{p_{2}-2}v}{\text{d}(z)^{\gamma}}\,\,\, &\text{in } \,\,\mathbb{G}, \end{aligned}\right. \end{equation} (3.41)

    and we observe that if (u, v) is a weak solution of (3.41) such that \|(u, v)\|_{\mathcal{H}} < t_{1} , then (u, v) is also a solution of (1.1). For this reason we look for critical points of the following functional \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}: \mathcal{H}\to \mathbb{R} defined as

    \begin{equation*} \begin{aligned} \mathcal{J}_{\lambda_{1},\lambda_{2},\beta}(u,v) & = \frac{1}{2}\|(u,v)|^{2}_{\mathcal{H}} -\frac{1}{2^*(\alpha)}\int_{\mathbb{G}}\phi(\|(u,v)\|_{\mathcal{H}}) \Big(\lambda_{1}\frac{\psi^{\alpha}|u|^{2^*(\alpha)}}{\text{d}(z)^{\alpha}} +\lambda_{2} \frac{\psi^{\alpha}|v|^{2^*(\alpha)}}{\text{d}(z)^{\alpha}}\Big)dz\\ &-\beta \int_{\mathbb{G}}f(z)\frac{\psi^{\gamma}|u|^{p_1}|v|^{p_{2}}}{\text{d}(z)^{\gamma}}dz, \quad \forall (u,v)\in \mathcal{H}.\\ \end{aligned} \end{equation*}

    In view of the definition of \phi and p < 2 we can see that \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u, v)\to \infty as \|(u, v)\|_{\mathcal{H}}\to \infty , \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(-u, -v) = \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u, v) and \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u, v) is bounded from below. Moreover, I_{\lambda_{1}, \lambda_{2}, \beta}(u, v)\leq \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u, v) for all (u, v)\in \mathcal{H} .

    Next, we show that \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta} satisfies the assumptions of Theorem 2.1.

    Lemma 3.2. (i) If \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u, v) < 0 , then \|(u, v)\|_{\mathcal{H}} < t_1 and \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(\tilde{u}, \tilde{v}) = I_{\lambda_{1}, \lambda_{2}, \beta}(\tilde{u}, \tilde{v}) for all (\tilde{u}, \tilde{v})\in N_{(u, v)} , where N_{(u, v)} denotes the enough neighborhood of (u, v) .

    (ii) For \lambda_{1} , \lambda_{2} > 0 , there exists \widetilde{\beta } = \min\{\beta_{*}, \beta_{**}\} such that if \beta \in (0, \widetilde{\beta}) and c \in (-\infty, 0) , then \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta} satisfies (PS)_c -condition;

    (iii) For \beta > 0 , there exists \widetilde{\lambda} = \min\{\lambda_{*}, \lambda_{**}\} such that if \lambda_{1}, \, \lambda_{2} \in(0, \widetilde{\lambda}) and c \in(-\infty, 0) , then \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta} satisfies (PS)_c -condition.

    Proof. We prove (i) by contradiction, assume \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u), v\leq0 and \|(u, v)\|_{\mathcal{H}}\geq t_1 . If \|(u, v)\|_{\mathcal{H}}\geq t_2 , then we have

    \mathcal{J}_{\lambda_{1},\lambda_{2},\beta}(u,v) \geq \frac{1}{2}\|(u,v)\|_{\mathcal{H}}^{2} -\beta\|f\|_{L^{p_{*}}(\mathbb{G},\frac{\psi^{\gamma}}{\text{d}(z)^{\gamma}}dz)} (S_{\alpha,\mu_{1}}^{-\frac{p}{2}} + S_{\alpha,\mu_{2}}^{-\frac{p}{2}})\|(u,v)\|^{p}_{\mathcal{H}} > 0.

    This contradicts \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u, v) < 0 .

    If t_{1}\leq \|(u, v)\|_{\mathcal{H}} < t_2 , since 0\leq\phi(t)\leq 1 , we get

    \mathcal{J}_{\lambda_{1},\lambda_{2},\beta}(u,v)\geq I_{\lambda_{1},\lambda_{2},\beta}(u,v) \geq g(\|(u,v)\|_{\mathcal{H}}) > 0,

    which again contradicts \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u, v) < 0 . Hence, \|(u, v)\|_{\mathcal{H}} < t_1 . Furthermore, by continuity of \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta} , applying I_{\lambda_{1}, \lambda_{2}, \beta}(u, v) = \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u, v) for all \|(u, v)\|_{\mathcal{H}} < t_1 there exists a small neighborhood \mathcal{B}_{(u, v)}\subset B_{\text{d}}((0, 0), R) of (u, v) such that I_{\lambda_{1}, \lambda_{2}, \beta}(\tilde{u}, \tilde{v}) = \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(\tilde{u}, \tilde{v}) for any (\tilde{u}, \tilde{v})\in \mathcal{B}_{(u, v)} , we conclude the proof of (i).

    Now we prove (ii), let \widetilde{\beta} = \min\{\beta_{*}, \beta_{**}\} , and let \{(u_{n}, v_{n})\}\subset \mathcal{H} be a (PS)_{c} -sequence for \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta} with the level c < 0 , then \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u_{n}, v_{n})\to c and \mathcal{J}'_{\lambda_{1}, \lambda_{2}, \beta}(u_{n}, v_{n})\to 0 in \mathcal{H}^{-1} . By (i), we have \|(u_{n}, v_{n})\|_{\mathcal{H}} < t_1 , hence \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u_{n}, v_{n}) = I_{\lambda_{1}, \lambda_{2}, \beta}(u_{n}, v_{n}) . By Proposition 3.1, I_{\lambda_{1}, \lambda_{2}, \beta} satisfies the (PS)_{c} -condition for c < 0 . Thus, \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta} satisfies the (PS)_{c} -condition for c < 0 , (ii) holds.

    The proof of (iii) goes exactly as (ii) with only minor modification, we omit it here.

    Let

    \mathcal{J}^{-\varepsilon}_{\lambda_{1},\lambda_{2},\beta} = \{(u,v)\in \mathcal{H}:\,\mathcal{J}_{\lambda_{1},\lambda_{2},\beta}(u,v)\leq -\varepsilon\}.

    Lemma 3.3. Given k\in \mathbb{N} , there exists \varepsilon = \varepsilon(k) > 0 such that \gamma(\mathcal{J}^{-\varepsilon}_{\lambda_{1}, \lambda_{2}, \beta})\geq k for any \lambda_{1} , \lambda_{2} , \beta > 0 .

    Proof. Fix \lambda_{1} , \lambda_{2} > 0 , k\in \mathbb{N} and let E_{k} be a k -dimensional vectorial subspace of \mathcal{H} . Taking (u, v)\in E_{k}\backslash\{(0, 0)\} with (u, v) = r_{k}(\omega_{1}, \omega_{2}) , where (\omega_{1}, \omega_{2})\in E_{k} and \|(\omega_{1}, \omega_{2})\|_{\mathcal{H}} = 1 . Then, by (3.40) there is a constant C > 0 such that

    \Big|\int_{\mathbb{G}}f(z) \frac{\psi^{\gamma}|\omega_{1}|^{p_{1}}|\omega_{2}|^{p_{2}}}{\text{d}(z)^{\gamma}}dz\Big| \leq C\|(\omega_{1},\omega_{2})\|^{p}_{\mathcal{H}} = C < \infty,

    which implies that there exists c_{k}\in (-\infty, +\infty) such that

    \int_{\mathbb{G}}f(z)\frac{\psi^{\gamma}|\omega_{1}|^{p_{1}}|\omega_{2}|^{p_{2}}}{\text{d}(z)^{\gamma}}dz\geq c_{k} > -\infty.

    Thus, for each (u, v) = r_{k}(\omega_{1}, \omega_{2}) with r_{k}\in (0, t_{1}) , we have

    \begin{equation*} \begin{aligned} \mathcal{J}_{\lambda_{1},\lambda_{2},\beta}(u,v) & = \mathcal{J}_{\lambda,\beta}(r_{k}(\omega_{1},\omega_{2}))\\ & = \frac{r_{k}^{2}}{2} -\frac{ r_{k}^{2^*(\alpha)}}{2^*(\alpha)}\phi(r_{k}) \int_{\mathbb{G}}(\lambda_{1}\frac{\psi^{\alpha}|\omega_{1}|^{2^*(\alpha)}}{\text{d}(z)^{\alpha}} +\lambda_{2}\frac{\psi^{\alpha}|\omega_{2}|^{2^*(\alpha)}}{\text{d}(z)^{\alpha}})dz\\ &-\beta r_{k}^{p} \int_{\mathbb{G}}f(z)\frac{\psi^{\gamma}|\omega_{1}|^{p_{1}}|\omega_{2}|^{p_2}}{\text{d}(z)^{\gamma}}dz\\ &\leq \frac{1}{2}r_{k}^{2} -\beta c_{k}r_{k}^{p}. \end{aligned} \end{equation*}

    For any \varepsilon: = \varepsilon(k) > 0 , there exists r_{k}\in (0, t_{1}) small enough such that \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u, v)\leq -\varepsilon for any (u, v)\in \mathcal{H} with \|(u, v)\|_{\mathcal{H}} = r_{k} .

    Denote \mathcal{S}_{k} = \{(u, v)\in \mathcal{H}:\, \|(u, v)\|_{\mathcal{H}} = r_{k}\}. Clearly, \mathcal{S}_{k} is homeomorphic to the k-1 dimensional sphere \mathbb{S}^{k-1} and \mathcal{S}_{k}\cap E_{k}\subset \mathcal{J}^{-\varepsilon}_{\lambda_{1}, \lambda_{2}, \beta} . By Proposition 2.1 (2) and (4) it follows that

    \gamma(\mathcal{J}^{-\varepsilon}_{\lambda_{1},\lambda_{2},\beta}) \geq \gamma(\mathcal{S}_{k}\cap E_{k}) = k,

    concluding the proof.

    Let us set the number

    \begin{equation*} \label{eq4-3} c_{k} = \inf\limits_{A\in \Gamma_{k}}\sup\limits_{(u,v)\in A} \mathcal{J}_{\lambda_{1},\lambda_{2},\beta}(u,v), \end{equation*}

    with

    \Gamma_{k} = \{A\subset \mathcal{H}: A\,\,\text{is closed}, A = -A\,\,\text{and}\,\, \gamma(A)\geq k\}.

    Clearly, c_{k}\leq c_{k+1} for each k\in \mathbb{N} . Before proving our main result, we state the following technical results.

    Lemma 3.4. c_{k} < 0 for all k\in \mathbb{N} .

    Proof. Fix k\in \mathbb{N} . By Lemma 3.3, there exists \varepsilon > 0 such that \gamma(\mathcal{J}^{-\varepsilon}_{\lambda_{1}, \lambda_{2}, \beta})\geq k . This and \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta} is a continuous even functional imply that \mathcal{J}^{-\varepsilon}_{\lambda_{1}, \lambda_{2}, \beta}\in \Gamma_{k} . Then

    (0,0)\not\in \mathcal{J}^{-\varepsilon}_{\lambda_{1},\lambda_{2},\beta}\,\,\, \text{and}\,\,\, \sup\limits_{(u,v)\in \mathcal{J}^{-\varepsilon}_{\lambda_{1},\lambda_{2},\beta}}\mathcal{J}_{\lambda_{1},\lambda_{2},\beta}(u,v)\leq -\varepsilon < 0.

    Therefore, taking into account that \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta} is bounded from below, we get

    -\infty < c_{k} = \inf\limits_{A\in \Gamma_{k}}\sup\limits_{(u,v)\in A} \mathcal{J}_{\lambda_{1},\lambda_{2},\beta}(u,v)\leq \sup\limits_{(u,v)\in \mathcal{J}^{-\varepsilon}_{\lambda_{1},\lambda_{2},\beta}} \mathcal{J}_{\lambda_{1},\lambda_{2},\beta}(u,v)\leq -\varepsilon < 0.

    Let

    K_{c} = \{(u,v)\in \mathcal{H}:\, \mathcal{J}_{\lambda_{1},\lambda_{2},\beta}'(u,v) = 0\,\,\text{and}\,\, \mathcal{J}_{\lambda_{1},\lambda_{2},\beta}(u,v) = c\}.

    Lemma 3.5. For any \lambda_{1} , \lambda_{2} , \beta > 0 , the critical values \{c_k\}_{k\in \mathbb{N}} of \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta} satisfy c_{k}\to 0 as k\to \infty .

    Proof. Fix \mu_{1} , \mu_{2}\in [0, \mu_{\mathbb{G}}) and \lambda_{1} , \lambda_{2} , \beta > 0 . By Lemma 3.4 it follows that c_{k} < 0 . Since c_{k}\leq c_{k+1} we can assume that \lim\limits_{k\to \infty}c_{k}\to c_{0} \leq 0 . Moreover, by Lemma 3.2, it is easy to see that the functional \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta} satisfies the (PS)_{c_{k}} -condition at level c_{k} .

    Now we prove that c_{0} = 0 . We argue by contradiction and we suppose that c_{0} < 0 . In view of Lemma 3.2, K_{c_{0}} is compact. Furthermore, it is easy to see that

    K_{c_{0}}\subset \mathcal{E} : = \{ A\subset \mathcal{H}\setminus\{(0,0)\}:\,\,A\,\,\text{is closed and }\,\,A = -A \},

    which and Proposition 2.1 (6) imply that \gamma(K_{c_{0}}) = k_{0} < \infty and there exists \delta > 0 such that N_{\delta}(K_{c_0})\subset \mathcal{E} and

    \begin{equation} \gamma(K_{c_{0}}) = \gamma(N_{\delta}(K_{c_{0}})) = k_{0} < \infty, \end{equation} (3.42)

    where N_{\delta}(K_{c_{0}}) = \{(u, v)\in \mathcal{H}:\text{dist}((u, v), K_{c_0})\leq \delta\} . Moreover, By [38, Theorem A.4], there exists an odd homeomorphism \eta:\, \mathcal{H}\to \mathcal{H} such that

    \begin{equation} \eta(\mathcal{J}^{c_{0}+\varepsilon}_{\lambda_{1},\lambda_{2},\beta} \setminus N_{\delta}(K_{c_{0}}))\subset \mathcal{J}^{c_{0}-\varepsilon}_{\lambda_{1},\lambda_{2},\beta},\,\,\, \text{for some}\,\,\, \varepsilon \in (0,-c_{0}) \end{equation} (3.43)

    Taking into account that c_{k+1}\leq c_{k} and c_{k}\to c_{0} as k\to \infty , we can find k\in \mathbb{N} such that c_{k} > c_{0}-\varepsilon and c_{k+k_{0}}\leq c_{0} , where k_{0} given in (3.42). Take A \in \Gamma_{k+k_{0}} such that \sup\limits_{(u, v)\in A}\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u, v) \leq c_{k+k_{0}} < c_{0} +\varepsilon , by using Properties 2.1 (4), we have

    \gamma(\overline{A\setminus N_{\delta}(K_{c_{0}})})\geq \gamma(A)-\gamma( N_{\delta}(K_{c_{0}})) \geq k \,\,\text{and}\,\,\, \gamma( \eta(\overline{A\setminus N_{\delta}(K_{c_{0}})}))\geq k,

    from which we have \eta(\overline{A\setminus N_{\delta}(K_{c_{0}})})\in \Gamma_{k} . Hence

    \begin{equation} \sup\limits_{(u,v)\in\eta(\overline{A\setminus N_{\delta}(K_{c_0})}) } \mathcal{J}_{\lambda_{1},\lambda_{2},\beta}(u,v) \geq c_{k} > c_{0}-\varepsilon. \end{equation} (3.44)

    On the other hand, in view of (3.43) and A\subset \mathcal{J}^{c_{0}+\varepsilon}_{\lambda_{1}, \lambda_{2}, \beta} , we see that

    \eta(A\setminus N_{\delta}(K_{c_{0}}))\subset \eta(\mathcal{J}^{c_{0}+\varepsilon}_{\lambda_{1},\lambda_{2},\beta}\setminus N_{\delta}(K_{c_{0}})) \subset \mathcal{J}^{c_{0}-\varepsilon}_{\lambda_{1},\lambda_{2},\beta},

    which gives a contradiction in virtue of (3.44). Hence, c_{0} = 0 and \lim\limits_{k\to \infty}c_{k} = 0 hold.

    Lemma 3.6. Let \lambda_{1} , \lambda_{2} , \beta be as in (ii) or (iii) of Lemma 3.2. If k, \, l\in\mathbb{N} such that c = c_{k} = c_{k+1} = \cdots = c_{k+l} , then

    \gamma(K_{c})\geq l+1.

    Proof. From Lemma 3.4 we have that c = c_k = c_{k+1} = \ldots = c_{k+l} < 0 . By Lemma 3.2, \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta} satisfies the (PS)_{c} -condition on the compact set K_{c} .

    Suppose the result is not true, that is, \gamma(K_c)\leq l . Then, by Proposition 2.1 (6) there is a neighborhood of K_{c} , say N_\delta(K_c) , such that \gamma(N_\delta(K_c)) = \gamma(K_c)\leq l . By [38, Theorem A.4], there exists an odd homeomorphism \eta: \mathcal{H}\to \mathcal{H} such that

    \begin{equation} \eta(\mathcal{J}_{\lambda_{1},\lambda_{2},\beta}^{c+\varepsilon}\setminus N_\delta(K_c))\subset \mathcal{J}_{\lambda_{1},\lambda_{2},\beta}^{c-\varepsilon}\,\,\, \text{for some}\,\,\, \varepsilon \in (0,-c). \end{equation} (3.45)

    From the definition of c = c_{n+l} , we know there exists A\in\Gamma_{n+l} such that

    \sup\limits_{(u,v)\in A}\mathcal{J}_{\lambda_{1},\lambda_{2},\beta}(u,v) < c+\varepsilon,

    that is, A\subset\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}^{c+\varepsilon} , and so by (3.45) we get

    \begin{equation*} \eta(A\setminus N_\delta(K_c))\subset \eta(\mathcal{J}_{\lambda_{1},\lambda_{2},\beta}^{c+\varepsilon}\setminus N_\delta(K_c)) \subset \mathcal{J}_{\lambda_{1},\lambda_{2},\beta}^{c-\varepsilon}. \end{equation*}

    This yields

    \begin{equation} \sup\limits_{u\in \eta(\overline{A \setminus N_\delta(K_{c})})} \mathcal{J}_{\lambda_{1},\lambda_{2},\beta}(u,v) \leq c-\varepsilon, \end{equation} (3.46)

    On the other hand, by parts (1), (3) of Proposition 2.1 we have

    \begin{equation*} \gamma(\eta(\overline{A \setminus N_\delta(K_{c})})) \geq\gamma(\overline{A \setminus N_\delta(K_{c})}) \geq \gamma(A)-\gamma(N_\delta(K_{c}))\geq n. \end{equation*}

    Hence, we conclude that \eta(\overline{A \setminus N_\delta(K_{c})}) \in \Gamma_n and so

    \begin{equation*} \sup\limits_{u\in \eta(\overline{A \setminus N_\delta(K_{c})})} \mathcal{J}_{\lambda_{1},\lambda_{2},\beta}(u,v) \geq c_n = c, \end{equation*}

    which contradicts (3.46). Thus, we conclude \gamma(K_{c})\geq l+1 .

    Proof of Theorem 1.1 Let \lambda_{1} , \lambda_{2} , \beta be as in (ii) or (iii) of Lemma 3.2. Putting together Lemma 3.4 and Lemma 3.2 (ii) or (iii), we can see that the functional \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta} satisfies the (PS)_{c_{k}} -condition with c_n < 0 . That is, c_k is a critical value of \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta} .

    We consider two situations.

    If all c_{k} 's are distinct, that is, -\infty < c_1 < c_2 < \cdots < c_k < c_{k+1} < \cdots , then \gamma(K_{c_{k}})\geq 1 since K_{c_{k}} is a compact set. Thus, in this case \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta} admits infinitely many critical values. By Lemma 3.2 (1) we can see that I_{\lambda_{1}, \lambda_{2}, \beta} has infinitely many critical points, i.e., (1.1) has infinitely many solutions.

    If for some k\in \mathbb{N} there exists l\in\mathbb{N} such that c_k = c_{k+1} = \cdots = c_{k+l} = c , then \gamma(K_c)\geq l+1\geq2 by Lemma 3.6. Thus, the set K_c has infinitely many distinct elements, (see [38, Remark 7.3]), i.e., I_{\lambda_{1}, \lambda_{2}, \beta} has infinitely many distinct critical point. Thus again, system (1.1) has infinitely many distinct weak solutions. Moreover, Lemma 3.5 implies that the energy of this solutions converges to zero.

    The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors declare that there are no conflicts of interest.



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