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Theory article

Existence of infinitely many solutions for critical sub-elliptic systems via genus theory

  • Received: 15 October 2023 Revised: 24 January 2024 Accepted: 02 February 2024 Published: 25 March 2024
  • 35R03, 35J70, 35B33

  • We are devoted to the study of the following sub-Laplacian system with Hardy-type potentials and critical nonlinearities

    {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,

    where ΔG is the sub-Laplacian on Carnot group G, μ1, μ2[0,μG), α,γ(0,2), λ1, λ2, β, p1, p2>0 with 1<p1+p2<2, d(z) is the ΔG-gauge, ψ=|Gd(z)|, 2(α):=2(Qα)Q2 is the critical Sobolev-Hardy exponents, and μG=(Q22)2 is the best Hardy constant on G. By combining a variant of the symmetric mountain pass theorem with the genus theory, we prove the existence of infinitely many weak solutions whose energy tends to zero when β or λ1, λ2 belong to a suitable range.

    Citation: Hongying Jiao, Shuhai Zhu, Jinguo Zhang. Existence of infinitely many solutions for critical sub-elliptic systems via genus theory[J]. Communications in Analysis and Mechanics, 2024, 16(2): 237-261. doi: 10.3934/cam.2024011

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  • We are devoted to the study of the following sub-Laplacian system with Hardy-type potentials and critical nonlinearities

    {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,

    where ΔG is the sub-Laplacian on Carnot group G, μ1, μ2[0,μG), α,γ(0,2), λ1, λ2, β, p1, p2>0 with 1<p1+p2<2, d(z) is the ΔG-gauge, ψ=|Gd(z)|, 2(α):=2(Qα)Q2 is the critical Sobolev-Hardy exponents, and μG=(Q22)2 is the best Hardy constant on G. By combining a variant of the symmetric mountain pass theorem with the genus theory, we prove the existence of infinitely many weak solutions whose energy tends to zero when β or λ1, λ2 belong to a suitable range.



    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

    0μGˆρ0ˆμ0,0(1μ1μG)ˆμ0ˆμ0μ1ˆρ0, (3.23)

    and

    0μGˉρ0ˉμ0,0(1μ2μG)ˉμ0ˉμ0μ2ˉρ0. (3.24)

    If ˆν0=0, from (3.18) and (3.23), it follows that ˆμ0=ˆρ0=0. Similarly, if ˉν0=0, by (3.18) and (3.24), we conclude ˉμ0=ˉρ0=0.

    To analyze the concentration at infinity, for R>0, we choose the function ϕC1(G) such that 0ϕ1, ϕ(z)=0 on Bd(0,1), ϕ(z)=1 on GBd(0,2) and |Gϕ|cR. Set ϕR(z)=ϕ(δ1R(z)), then {ϕRun}S10(G) is bounded. Testing Iλ1,λ2,β(un,vn) with (ϕRun,0) we obtain limnIλ1,λ2,β(un,vn),(ϕRun,0)=0, i.e.,

    limnGGun,GϕRundz=limn[G(|Gun|2ϕRμ1ψ2|un|2d(z)2ϕR)dzλ1Gψα|un|2(α)d(z)αϕRdzβp1Gf(z)ψγ|un|p1|vn|p2d(z)γϕRdz]. (3.25)

    Since

    Sα,μ1(Gψα|unϕR|2(α)d(z)αdz)22(α)G(|G(unϕR)|2μ1ψ2|unϕR|2d(z)2)dz,

    we conclude that

    μ1Gψ2|unϕR|2d(z)2dz+Sα,μ1(Gψα|unϕR|2(α)d(z)αdz)22(α)G|G(unϕR)|2dzG|Gun|2|ϕR|2dz+G|GϕR|2|un|2dz+2G|GunϕRunGϕR|dz. (3.26)

    By Hölder inequality, it is easy to get that

    limRlim supnG|ϕRGun||unGϕR|dzlimRlim supn(Bd(0,2R)Bd(0,R)|Gun|2dz)12(Bd(0,2R)Bd(0,R)|unGϕR|2dz)12ClimR(Bd(0,2R)Bd(0,R)|GϕR|2|u0|2dz)12ClimR(Bd(0,2R)Bd(0,R)|Gϕε|Qdz)1Q(Bd(0,2)Bd(0,R)|u0|2dz)12ClimR(Bd(0,2R)Bd(0,R)|u0|2dz)12=0. (3.27)

    Similarly,

    limRlim supnG|GϕR|2|un|2dz=0. (3.28)

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

    μμ1ρSα,μ1ν22(α). (3.29)

    On the other hand, from Hölder inequality and the definition of ϕR we have

    |Gf(z)ψγ|un|p1|vn|p2d(z)γϕRdz||GBd(0,R)f(z)ψγ|un|pd(z)γϕRdz|+|GBd(0,R)f(z)ψγ|vn|pd(z)γϕRdz|(GBd(0,R)ψγ|f(z)|2(γ)2(γ)pd(z)γdz)2(γ)p2(γ)[(GBd(0,R)ψγ|un|2(γ)d(z)γϕRdz)p2(γ)+(GBd(0,R)ψγ|vn|2(γ)d(z)γϕRdz)p2(γ)](GBd(0,R)ψγd(z)γ|f(z)|2(γ)2(γ)pdz)2(γ)p2(γ)[Sp2γ,μ1unpμ2+Sp2γ,μ1unpμ2].

    Since fLp(G,ψγd(z)γdz), it follows that

    limRlim supn|Gf(z)ψγ|un|p1|vn|p2d(z)γϕRdz|limRC(GBd(0,R)ψγ|f(z)|2(γ)2(γ)pd(z)γdz)2(γ)p2(γ)=0.

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

    μμ1ρλ1ν. (3.30)

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

    either(5)ν=0,or(6)ν(Sα,μ1λ1)Qα2α.

    In contrast, the Hardy inequality implies that

    0μGρμ,0(1μ1μG)μμμ1ρ. (3.31)

    If ν=0, by combining (3.30) and (3.31), we get μ=ρ=0.

    From above argument the same conclusion holds for ˉν, namely,

    ˉμμ2ˉρSα,μ2ˉν22(α),
    ˉμμ1ˉρλ2ˉν,

    and

    either(5)ˉν=0,or(6)ˉν(Sα,μ2λ2)Qα2α.

    If ˉν=0, we have that ˉμ=ˉρ=0.

    Now we claim that (2), (2), (4), (4) and (6), (6) cannot occur if λ1, λ2 and β are chosen properly. In fact, applying (f) and Hölder inequality, we have

    0>c=limn(Iλ1,λ2,β(un,vn)12(α)Iλ1,λ2,β(un,vn),(un,vn))=limn((1212(α))(un,vn)2Hβ(1p2(α))Gf(z)ψγ|un|p1|vn|p2d(z)γdz)2(α)222(α)(u0,v0)2Hβ(2(α)p)2(α)fLp(G,ψγd(z)γdz)(u0pL2(γ)(G,ψγd(z)γdz)+v0pL2(γ)(G,ψγd(z)γdz))2(α)222(α)(Sγ,μ1u02L2(γ)(G,ψγd(z)γdz)+Sγ,μ2v02L2(γ)(G,ψγd(z)γdz))β(2(α)p)2(α)fLp(G,ψγd(z)γdz)(u0pL2(γ)(G,ψγd(z)γdz)+v0pL2(γ)(G,ψγd(z)γdz)). (3.32)

    Since

    u0pL2(γ)(G,ψγd(z)γdz)+v0pL2(γ)(G,ψγd(z)γdz)2(u0L2(γ)(G,ψγd(z)γdz)+v0L2(γ)(G,ψγd(z)γdz))p,
    u02L2(γ)(G,ψγd(z)γdz)+v02L2(γ)(G,ψγd(z)γdz)12(u0L2(γ)(G,ψγd(z)γdz)+v0L2(γ)(G,ψγd(z)γdz))2,

    which and (3.32) yield that

    2β(2(α)p)2(α)fLp(G,ψγd(z)γdz)(u0L2(γ)(G,ψγd(z)γdz)+v0L2(γ)(G,ψγd(z)γdz))p2(α)242(α)min{Sγ,μ1,Sγ,μ2}(u0L2(γ)(G,ψγd(z)γdz)+v0L2(γ)(G,ψγd(z)γdz))2,

    namely,

    u0L2(γ)(G,ψγd(z)γdz)+v0L2(γ)(G,ψγd(z)γdz)(8(2(α)p)fLp(G,ψγd(z)γdz)(2(α)2)min{Sγ,μ2,Sγ,μ2})12pβ12p. (3.33)

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

    0>c=limn(Iλ1,λ2,β(un,vn)12(α)Iλ1,λ2,β(un,vn),(un,vn))2(α)222(α)(μμ1ρ+ˉμμ2ˉρ)22(α)(8(2(α)2)min{Sγ,μ1,Sγ,μ2})p2p((2(α)p)fLp(G,ψγd(z)γdz))22pβ22p2(α)222(α)(Sα,μ1ν22(α)+Sα,μ2ˉν22(α))22(α)(8(2(α)2)min{Sγ,μ1,Sγ,μ2})p2p((2(α)p)fLp(G,ψγd(z)γdz))22pβ22p2(α)222(α)(Sα,μ1[(Sα,μ1λ1)Qα2α]22(α)+Sα,μ2[(Sα,μ2λ2)Qα2α]22(α))22(α)(8(2(α)2)min{Sγ,μ1,Sγ,μ2})p2p((2(α)p)fLp(G,ψγd(z)γdz))22pβ22p=2(α)222(α)((Sα,μ1)Qα2αλQ22α1+(Sα,μ2)Qα2αλQ22α2)22(α)(8(2(α)2)min{Sγ,μ1,Sγ,μ2})p2p((2(α)p)fLp(G,ψγd(z)γdz))22pβ22p,

    that is,

    0>2(α)222(α)((Sα,μ1)Qα2αλQ22α1+(Sα,μ2)Qα2αλQ22α2)22(α)(8(2(α)2)min{Sγ,μ1,Sγ,μ2})p2p((2(α)p)fLpp(G,ψγd(z)γdz))22pβ22p. (3.34)

    From the above inequality, we can find that if β>0 is given, there exists λ>0 small enough such that for λ1,λ2(0,λ), the right-hand side of (3.34) is greater than 0, which is a contradiction. Similarly, if λ1,λ2>0 is given, we can take β>0 so small that for β(0,β), right-hand side of (3.34) is greater than 0.

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

    limnGψα|un|2(α)d(z)αdz=Gψα|u0|2(α)d(z)αdz

    and

    limnGψα|vn|2(α)d(z)αdz=Gψα|v0|2(α)d(z)αdz.

    In view of (un,vn)(u0,v0) weakly in H and the Brezis-Lieb lemma [38], we have

    limnGψα|unu0|2(α)d(z)αdz=0,limnGψα|vnv0|2(α)d(z)αdz=0.

    We are now going to prove that (un,vn)(u0,v0) strongly in H. First, we have

    (unu0,vnv0)2H=(Iλ1,λ2,β(un,vn)Iλ1,λ2,β(u0,v0)),(unu0,vnv0)+λ1Gψα(|un|2(α)2un|u0|2(α)2u0)(unu0)d(z)αdz+λ2Gψα(|vn|2(α)2vn|v0|2(α)2v0)(vnv0)d(z)αdz+βp1Gf(z)ψγ[|un|p12un|vn|p2|u0|p12u0|v0|p2](unu0)d(z)γdz+βp2Gf(z)ψγ[|un|p1|vn|p22vn|u0|p1|v0|p22v0](vnv0)d(z)γdz. (3.35)

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

    |Gψα(|un|2(α)2un|u0|2(α)2u0)(unu0)d(z)αdz|Gψα|un|2(α)1|unu0|d(z)αdz+Gψα|u0|2(α)1|unu0|d(z)αdz(Gψα|un|2(α)d(z)αdz)2(α)12(α)(Gψα|unu0|2(α)d(z)αdz)12(α)+(Gψα|u0|2(α)d(z)αdz)2(α)12(α)(Gψα|unu0|2(α)d(z)αdz)12(α)0asn. (3.36)

    Similarly,

    |Gψα(|vn|2(α)2vn|v0|2(α)2v0)(vnv0)d(z)αdz|0asn. (3.37)

    On the other hand, using the Hölder inequality and (un,vn)(u0,v0) weakly in H, we get that

    |Gf(z)ψγ[|un|p12un|vn|p2|u0|p12u0|v0|p2](unu0)d(z)γdz|Gψγ|f(z)||un|p1|unu0|d(z)γdz+G|f(z)|ψγ|u0|p1|unu0|d(z)γdz(Gψγ|f(z)|2(γ)2(γ)pd(z)γdz)2(γ)p2(γ)(Gψγ|un|2(γ)d(z)γdz)p12(γ)(Gψγ|unu0|2(γ)d(z)γdz)12(γ)+(Gψγ|f(z)|2(γ)2(γ)pd(z)γdz)2(γ)p2(γ)(Gψγ|u0|2(γ)d(z)γdz)p12(γ)(Gψγ|unu0|2(γ)d(z)γdz)12(γ)0asn, (3.38)

    and

    |Gf(z)ψγ[|un|p1|vn|p22vn|u0|p1|v0|p22v0](vnv0)d(z)γdz|0asn, (3.39)

    Combining (3.36), (3.37), (3.38), (3.39), (3.35) with limnIλ1,λ2,β(un,vn),(unu0,vnv0)=0 and limnIλ1,λ2,β(u0,v0),(unu0,vnv0)=0, we deduce that

    limn(unu0,vnv0)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

    Gψα|u|2(α)d(z)αdz+Gψα|v|2(α)d(z)αdzS2(α)2α,μ1u2(α)μ1+S2(α)2α,μ2v2(α)μ2(S2(α)2α,μ1+S2(α)2α,μ2)(u,v)2(α)H,

    and

    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)pψγd(z)γdz)2(γ)p2(γ)(Gψγ|un|2(γ)d(z)γdz)p2(γ)+p2p(G|f(z)|2(γ)2(γ)pψγd(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. (3.40)

    Then,

    Iλ1,λ2,β(u,v)=12(u,v)2Hλ12(α)Gψα|u|2(α)d(z)αdzλ22(α)Gψα|v|2(α)d(z)αdzβGf(z)ψγ|u|p1|v|p2d(z)γdz12(u,v)2H(λ1+λ2)(S2(α)2α,μ1+S2(α)2α,μ2)2(α)(u,v)2(α)HβfLp(G,ψγd(z)γdz)(Sp2γ,μ1+Sp2γ,μ2)(u,v)pH.

    Define the function

    g(t)=12t2C1(λ1+λ2)t2(α)C2βtp,t>0,

    where

    C1:=(S2(α)2α,μ1+S2(α)2α,μ2)2(α),C2:=fLp(G,ψγd(z)γdz)(Sp2γ,μ1+Sp2γ,μ2)>0.

    Because 1<p<2<2(α), for the given β>0, there exists λ>0 so small that for λ1+λ2(0,λ), there exist t1, t2>0 with t1<t2 such that g(t1)=g(t2)=0, and g(t)<0 for t(0,t1), g(t)>0 for t(t1,t2), g(t)<0 for t(t2,+). Similarly, given λ1, λ2>0, we can choose β>0 small enough such that for all β(0,β), there exist ˆt1, ˆt2>0 with ˆt1<ˆt2 such that g(ˆt1)=g(ˆt2)=0 and g(t)<0 for t(0,ˆt1), g(t)>0 for t(ˆt1,ˆt2), g(t)<0 for t(ˆt2,+).

    Let us define a function ϕC0([0,),R) such that 0ϕ(t)1, ϕ(t)=ϕ(t) for all t[0,+), ϕ(t)=1 if t[0,t1] and ϕ(t)=0 if t[t2,). So we consider the equation

    {ΔGuμ1ψ2ud(z)2=λ1ϕ((u,v)H)ψα|u|2(α)2ud(z)α+βp1f(z)ψγ|u|p12u|v|p2d(z)γin G,ΔGvμ2ψ2vd(z)2=λ2ϕ((u,v)H)ψα|v|2(α)2vd(z)α+βp2f(z)ψγ|u|p1|v|p22vd(z)γin G, (3.41)

    and we observe that if (u,v) is a weak solution of (3.41) such that (u,v)H<t1, then (u,v) is also a solution of (1.1). For this reason we look for critical points of the following functional Jλ1,λ2,β:HR defined as

    Jλ1,λ2,β(u,v)=12(u,v)|2H12(α)Gϕ((u,v)H)(λ1ψα|u|2(α)d(z)α+λ2ψα|v|2(α)d(z)α)dzβGf(z)ψγ|u|p1|v|p2d(z)γdz,(u,v)H.

    In view of the definition of ϕ and p<2 we can see that Jλ1,λ2,β(u,v) as (u,v)H, Jλ1,λ2,β(u,v)=Jλ1,λ2,β(u,v) and Jλ1,λ2,β(u,v) is bounded from below. Moreover, Iλ1,λ2,β(u,v)Jλ1,λ2,β(u,v) for all (u,v)H.

    Next, we show that Jλ1,λ2,β satisfies the assumptions of Theorem 2.1.

    Lemma 3.2. (i) If Jλ1,λ2,β(u,v)<0, then (u,v)H<t1 and Jλ1,λ2,β(˜u,˜v)=Iλ1,λ2,β(˜u,˜v) for all (˜u,˜v)N(u,v), where N(u,v) denotes the enough neighborhood of (u,v).

    (ii) For λ1, λ2>0, there exists ˜β=min{β,β} such that if β(0,˜β) and c(,0), then Jλ1,λ2,β satisfies (PS)c-condition;

    (iii) For β>0, there exists ˜λ=min{λ,λ} such that if λ1,λ2(0,˜λ) and c(,0), then Jλ1,λ2,β satisfies (PS)c-condition.

    Proof. We prove (i) by contradiction, assume Jλ1,λ2,β(u),v0 and (u,v)Ht1. If (u,v)Ht2, then we have

    Jλ1,λ2,β(u,v)12(u,v)2HβfLp(G,ψγd(z)γdz)(Sp2α,μ1+Sp2α,μ2)(u,v)pH>0.

    This contradicts Jλ1,λ2,β(u,v)<0.

    If t1(u,v)H<t2, since 0ϕ(t)1, we get

    Jλ1,λ2,β(u,v)Iλ1,λ2,β(u,v)g((u,v)H)>0,

    which again contradicts Jλ1,λ2,β(u,v)<0. Hence, (u,v)H<t1. Furthermore, by continuity of Jλ1,λ2,β, applying Iλ1,λ2,β(u,v)=Jλ1,λ2,β(u,v) for all (u,v)H<t1 there exists a small neighborhood B(u,v)Bd((0,0),R) of (u,v) such that Iλ1,λ2,β(˜u,˜v)=Jλ1,λ2,β(˜u,˜v) for any (˜u,˜v)B(u,v), we conclude the proof of (i).

    Now we prove (ii), let ˜β=min{β,β}, and let {(un,vn)}H be a (PS)c-sequence for Jλ1,λ2,β with the level c<0, then Jλ1,λ2,β(un,vn)c and Jλ1,λ2,β(un,vn)0 in H1. By (i), we have (un,vn)H<t1, hence Jλ1,λ2,β(un,vn)=Iλ1,λ2,β(un,vn). By Proposition 3.1, Iλ1,λ2,β satisfies the (PS)c-condition for c<0. Thus, Jλ1,λ2,β 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

    Jελ1,λ2,β={(u,v)H:Jλ1,λ2,β(u,v)ε}.

    Lemma 3.3. Given kN, there exists ε=ε(k)>0 such that γ(Jελ1,λ2,β)k for any λ1, λ2, β>0.

    Proof. Fix λ1, λ2>0, kN and let Ek be a k-dimensional vectorial subspace of H. Taking (u,v)Ek{(0,0)} with (u,v)=rk(ω1,ω2), where (ω1,ω2)Ek and (ω1,ω2)H=1. Then, by (3.40) there is a constant C>0 such that

    |Gf(z)ψγ|ω1|p1|ω2|p2d(z)γdz|C(ω1,ω2)pH=C<,

    which implies that there exists ck(,+) such that

    Gf(z)ψγ|ω1|p1|ω2|p2d(z)γdzck>.

    Thus, for each (u,v)=rk(ω1,ω2) with rk(0,t1), we have

    Jλ1,λ2,β(u,v)=Jλ,β(rk(ω1,ω2))=r2k2r2(α)k2(α)ϕ(rk)G(λ1ψα|ω1|2(α)d(z)α+λ2ψα|ω2|2(α)d(z)α)dzβrpkGf(z)ψγ|ω1|p1|ω2|p2d(z)γdz12r2kβckrpk.

    For any ε:=ε(k)>0, there exists rk(0,t1) small enough such that Jλ1,λ2,β(u,v)ε for any (u,v)H with (u,v)H=rk.

    Denote Sk={(u,v)H:(u,v)H=rk}. Clearly, Sk is homeomorphic to the k1 dimensional sphere Sk1 and SkEkJελ1,λ2,β. By Proposition 2.1 (2) and (4) it follows that

    γ(Jελ1,λ2,β)γ(SkEk)=k,

    concluding the proof.

    Let us set the number

    ck=infAΓksup(u,v)AJλ1,λ2,β(u,v),

    with

    Γk={AH:Ais closed,A=Aandγ(A)k}.

    Clearly, ckck+1 for each kN. Before proving our main result, we state the following technical results.

    Lemma 3.4. ck<0 for all kN.

    Proof. Fix kN. By Lemma 3.3, there exists ε>0 such that γ(Jελ1,λ2,β)k. This and Jλ1,λ2,β is a continuous even functional imply that Jελ1,λ2,βΓk. Then

    (0,0)Jελ1,λ2,βandsup(u,v)Jελ1,λ2,βJλ1,λ2,β(u,v)ε<0.

    Therefore, taking into account that Jλ1,λ2,β is bounded from below, we get

    <ck=infAΓksup(u,v)AJλ1,λ2,β(u,v)sup(u,v)Jελ1,λ2,βJλ1,λ2,β(u,v)ε<0.

    Let

    Kc={(u,v)H:Jλ1,λ2,β(u,v)=0andJλ1,λ2,β(u,v)=c}.

    Lemma 3.5. For any λ1, λ2, β>0, the critical values {ck}kN of Jλ1,λ2,β satisfy ck0 as k.

    Proof. Fix μ1, μ2[0,μG) and λ1, λ2, β>0. By Lemma 3.4 it follows that ck<0. Since ckck+1 we can assume that limkckc00. Moreover, by Lemma 3.2, it is easy to see that the functional Jλ1,λ2,β satisfies the (PS)ck-condition at level ck.

    Now we prove that c0=0. We argue by contradiction and we suppose that c0<0. In view of Lemma 3.2, Kc0 is compact. Furthermore, it is easy to see that

    Kc0E:={AH{(0,0)}:Ais closed and A=A},

    which and Proposition 2.1 (6) imply that γ(Kc0)=k0< and there exists δ>0 such that Nδ(Kc0)E and

    γ(Kc0)=γ(Nδ(Kc0))=k0<, (3.42)

    where Nδ(Kc0)={(u,v)H:dist((u,v),Kc0)δ}. Moreover, By [38, Theorem A.4], there exists an odd homeomorphism η:HH such that

    η(Jc0+ελ1,λ2,βNδ(Kc0))Jc0ελ1,λ2,β,for someε(0,c0) (3.43)

    Taking into account that ck+1ck and ckc0 as k, we can find kN such that ck>c0ε and ck+k0c0, where k0 given in (3.42). Take AΓk+k0 such that sup(u,v)AJλ1,λ2,β(u,v)ck+k0<c0+ε, by using Properties 2.1 (4), we have

    γ(¯ANδ(Kc0))γ(A)γ(Nδ(Kc0))kandγ(η(¯ANδ(Kc0)))k,

    from which we have η(¯ANδ(Kc0))Γk. Hence

    sup(u,v)η(¯ANδ(Kc0))Jλ1,λ2,β(u,v)ck>c0ε. (3.44)

    On the other hand, in view of (3.43) and AJc0+ελ1,λ2,β, we see that

    η(ANδ(Kc0))η(Jc0+ελ1,λ2,βNδ(Kc0))Jc0ελ1,λ2,β,

    which gives a contradiction in virtue of (3.44). Hence, c0=0 and limkck=0 hold.

    Lemma 3.6. Let λ1, λ2, β be as in (ii) or (iii) of Lemma 3.2. If k,lN such that c=ck=ck+1==ck+l, then

    γ(Kc)l+1.

    Proof. From Lemma 3.4 we have that c=ck=ck+1==ck+l<0. By Lemma 3.2, Jλ1,λ2,β satisfies the (PS)c-condition on the compact set Kc.

    Suppose the result is not true, that is, γ(Kc)l. Then, by Proposition 2.1 (6) there is a neighborhood of Kc, say Nδ(Kc), such that γ(Nδ(Kc))=γ(Kc)l. By [38, Theorem A.4], there exists an odd homeomorphism η:HH such that

    η(Jc+ελ1,λ2,βNδ(Kc))Jcελ1,λ2,βfor someε(0,c). (3.45)

    From the definition of c=cn+l, we know there exists AΓn+l such that

    sup(u,v)AJλ1,λ2,β(u,v)<c+ε,

    that is, AJc+ελ1,λ2,β, and so by (3.45) we get

    η(ANδ(Kc))η(Jc+ελ1,λ2,βNδ(Kc))Jcελ1,λ2,β.

    This yields

    supuη(¯ANδ(Kc))Jλ1,λ2,β(u,v)cε, (3.46)

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

    γ(η(¯ANδ(Kc)))γ(¯ANδ(Kc))γ(A)γ(Nδ(Kc))n.

    Hence, we conclude that η(¯ANδ(Kc))Γn and so

    supuη(¯ANδ(Kc))Jλ1,λ2,β(u,v)cn=c,

    which contradicts (3.46). Thus, we conclude γ(Kc)l+1.

    Proof of Theorem 1.1 Let λ1, λ2, β 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 Jλ1,λ2,β satisfies the (PS)ck-condition with cn<0. That is, ck is a critical value of Jλ1,λ2,β.

    We consider two situations.

    If all ck's are distinct, that is, <c1<c2<<ck<ck+1<, then γ(Kck)1 since Kck is a compact set. Thus, in this case Jλ1,λ2,β admits infinitely many critical values. By Lemma 3.2 (1) we can see that Iλ1,λ2,β has infinitely many critical points, i.e., (1.1) has infinitely many solutions.

    If for some kN there exists lN such that ck=ck+1==ck+l=c, then γ(Kc)l+12 by Lemma 3.6. Thus, the set Kc has infinitely many distinct elements, (see [38, Remark 7.3]), i.e., Iλ1,λ2,β 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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