We are devoted to the study of the following sub-Laplacian system with Hardy-type potentials and critical nonlinearities
{−ΔGu−μ1ψ2ud(z)2=λ1ψα|u|2∗(α)−2ud(z)α+βp1f(z)ψγ|u|p1−2u|v|p2d(z)γin G,−ΔGv−μ2ψ2vd(z)2=λ2ψα|v|2∗(α)−2vd(z)α+βp2f(z)ψγ|u|p1|v|p2−2vd(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(Q−α)Q−2 is the critical Sobolev-Hardy exponents, and μG=(Q−22)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
{−ΔGu−μ1ψ2ud(z)2=λ1ψα|u|2∗(α)−2ud(z)α+βp1f(z)ψγ|u|p1−2u|v|p2d(z)γin G,−ΔGv−μ2ψ2vd(z)2=λ2ψα|v|2∗(α)−2vd(z)α+βp2f(z)ψγ|u|p1|v|p2−2vd(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(Q−α)Q−2 is the critical Sobolev-Hardy exponents, and μG=(Q−22)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:
{−ΔGu−μ1ψ2ud(z)2=λ1ψα|u|2∗(α)−2ud(z)α+βp1f(z)ψγ|u|p1−2u|v|p2d(z)γin G,−ΔGv−μ2ψ2vd(z)2=λ2ψα|v|2∗(α)−2vd(z)α+βp2f(z)ψγ|u|p1|v|p2−2vd(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=(Q−22)2 is the best Hardy constant and 2∗(α):=2(Q−α)Q−2 is the Sobolev-Hardy critical exponents, Q≥3 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 {z∈G: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)=12∫G(|∇Gu|2+|∇Gv|2−μ1ψ2|u|2d(z)2−μ2ψ2|v|2d(z)2)dz−λ12∗(α)∫Gψα|u|2∗(α)d(z)αdz−λ22∗(α)∫Gψα|v|2∗(α)d(z)αdz−β∫Gf(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)={u∈L2∗(G):∫G|∇Gu|2dz<+∞} is the closure of C∞0(G) with respect to the norm
‖u‖S10(G)=(∫G|∇Gu|2dz)12. |
Here, 2∗=2QQ−2 is the Sobolev critical exponent. Further, we endow the product space H with the following norm
‖(u,v)‖H=(‖u‖2μ1+‖v‖2μ2)12, |
where
‖u‖2μi=∫G(|∇Gu|2−μiψ2|u|2d(z)2)dz,∀i=1,2. |
The above norm is well-defined due to the following Hardy-type inequality on Carnot group
μG∫Gψ2|u|2d(z)2dz≤∫G|∇Gu|2dz,∀u∈C∞0(G), | (1.2) |
where μG=(Q−22)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., ψ≡1. So, (1.2) becomes the well-known Hardy inequality:
ˉμ∫RN|u|2|x|2dx≤∫RN|∇u|2dx,∀u∈C∞0(RN), |
where ˉμ=(N−22)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:
{−ΔGu−μψ2ud(z)2=f(z,u) in Ω,u=0 on ∂Ω, | (1.3) |
where Ω is a smooth bounded domain in Carnot group G, 0∈Ω. 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|2∗−1)for all(z,t)∈Ω×R, |
and obtained the following asymptotic behavior at origin:
u(z)∼d(z)−(√μG−√μG−μ) as d(z)→0. |
Subsequently, in [22] also the behavior at infinity has been determined for the purely critical problem
−ΔGu−μψ2ud(z)2=|u|2∗−2uonG |
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)=√μG−√μG−μ, b(μ)=√μG+√μG−μ and the notation f∼g 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
−ΔGu−μψ2ud(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
{−ΔGu−μψ2ud(z)2=ψα|u|2∗(α)−2ud(z)α+ληη+θψα|u|η−2u|v|θd(z)αinG,−ΔGv−μψ2vd(z)2=ψα|v|2∗(α)−2vd(z)α+λθη+θψα|u|η|v|θ−2vd(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 u∈S10(G) of the following Rayleigh quotient:
Sα,μ=infu∈S10(G)∖{0}∫G|∇Gu|2dz−μ∫Gψ2|u|2d(z)2dz(∫Gψα|u|2∗(α)d(z)αdz)22∗(α). |
Actually, up to a normalization, it holds that
∫G|∇Gu|2dz−μ∫Gψ2|u|2d(z)2dz=∫Gψα|u|2∗(α)d(z)αdz=(Sα,μ)Q−α2−α. | (1.5) |
Moreover, for any ε>0, rescaled functions uε(z)=ε−Q−22u(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
∫G∇Gu⋅∇Gϕ1dz+∫G∇Gv⋅∇Gϕ2dz−μ1∫Gψ2uϕ1d(z)2dz−μ2∫Gψ2vϕ2d(z)2dz−λ1∫Gψα|u|2∗(α)−2uϕ1d(z)αdz−λ2∫Gψα|v|2∗(α)−2vϕ2d(z)αdz−βp1∫Gf(z)ψγ|u|p1−2|v|p2uϕ1d(z)γdz−βp2∫Gf(z)ψγ|u|p1|v|p2−2vϕ2d(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, 0≤α<2, 0≤γ<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 1≤k≤r−1and[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 X∈V1 and Y∈Vk.
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 z∈RN 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=1k⋅Nk. Note that the number Q is naturally associated to the family {δγ}γ>0 since, for every γ>0, the Jacobian of the map z↦δγ(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=∇G⋅u. |
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(u∘τz)=∇Gu∘τz, ∇G(u∘δγ)=γ∇Gu∘δγ; ΔG(u∘τz)=ΔGu∘τz and ΔG(u∘δγ)=γ2ΔGu∘δγ, where the left translation τz:G→G is defined by
τz(z′)=z∘z′,∀z,z′∈G. |
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(z−1)=d(z) for all z∈G;
(iii) d(δγ(z))=γd(z) for every γ>0 and z∈G.
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)Q−2,∀z∈G |
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(z−12∘z1), 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,z3∈G |
for a suitable positive constant c. The ball of radius R>0 centered at z∈G with respect to the norm d, calling them d-balls, defined as
Bd(z,R)={y∈G: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 ξ∈g with ξ=ξ1+⋯+ξk, where ξi∈Vi, define a pseudo-norm on g as follows
|ξ|g=|(ξ1,⋯,ξk)|g:=(k∑i=1|ξi|2k!i)12k!. |
The induced norm on G has the form
|g|G=|exp−1G(g)|g,∀g∈G. |
The function |⋅|G is usually known as the non-isotropic gauge. It defines a pseudo-distence on G given by
d(g,h):=|h−1∘g|G,∀g,h∈G. |
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+x′1,⋯,xn+x′n,y1+y′1,⋯,yn+y′n,t+t′+2(⟨x′,y⟩−⟨x,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=N∑i=1(X2i+Y2i), |
where
Xi=∂∂xi+2yi∂∂t,Yi=∂∂yi−2xi∂∂tfori=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={A⊂E∖{0}:Ais closed and A=−A}. |
For any A∈Σ, the Krasnoselskii's genus of A is defined by
γ(A)=inf{k:∃ϕ∈C(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 0∉E and γ(A)≥k, that is,
Σk={A:A⊂Eis closed symmetric,0∉Eandγ(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 A⊂B, 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¯(A∖B)≥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)={u∈E: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 J∈C1(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 limn→∞J(un)=c<ˉc and limn→∞‖J′(un)‖E′=0 has a convergent subsequence;
(2) For each k∈N, there exists Ak∈Σk such that supu∈AkJ(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, un≠0, limn→∞un=0; J′(vn)=0, J(vn)<0,limn→∞J(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 c∈R, H be a Banach space and Iλ1,λ2,β∈C1(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)→candI′λ1,λ2,β(un,vn)→0inH−1as 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)andI′λ1,λ2,β(un,vn)=on(1) in H−1asn→∞. |
By Young inequality and Hölder inequality, we have
∫Gf(z)ψγ|un|p1|vn|p2d(z)γdz≤p1p∫Gf(z)ψγ|un|pd(z)γdz+p2p∫Gf(z)ψγ|vn|pd(z)γdz≤p1p(∫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)≤‖f‖Lp∗(G,ψγd(z)γdz)(p1pS−p2γ,μ1‖un‖pμ1+p2pS−p2γ,μ2‖vn‖pμ2)≤‖f‖Lp∗(G,ψγd(z)γdz)(S−p2γ,μ1+S−p2γ,μ2)‖(un,vn)‖pH. |
Then,
on(1)+|c|+on(‖(un,vn)‖H)≥Iλ1,λ2,β(un)−12∗(α)⟨I′λ1,λ2,β(un,vn),(un,vn)⟩=(12−12∗(α))‖(un,vn)‖2H−β(1−p2∗(α))∫Gf(z)ψγ|un|p1|vn|p2d(z)γdz≥2−α2(Q−α)‖(un,vn)‖2H−β2∗(α)−p2∗(α)‖f‖Lp∗(G,ψγd(z)γdz)(S−p2γ,μ1+S−p2γ,μ2)‖un,vn‖pH, |
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, ˉρ∈R(G∪{∞}); at most countable set J and ˉJ; real numbers ˆμj, ˆνj(j∈J), ˉμk, ˉνk(k∈ˉJ), ˆμ0, ˆν0, ˆρ0, ˉμ0, ˉν0, ˉρ0 and different points zj∈G∖{0} (j∈J), ˉzk∈G∖{0} (k∈ˉJ) such that
|∇Gun|2dz⇀ˆμ≥|∇Gu0|2dz+∑j∈Jδzjˆμj+δ0ˆμ0, | (3.1) |
|∇Gvn|2dz⇀ˉμ≥|∇Gv0|2dz+∑k∈ˉJδˉzkˉμk+δ0ˉμ0, | (3.2) |
ψα|un|2∗(α)d(z)αdz⇀ˆν=ψα|u0|2∗(α)d(z)αdz+∑j∈Jδzjˆνj+δ0ˆν0, | (3.3) |
ψα|vn|2∗(α)d(z)αdz⇀ˉν=ψα|v0|2∗(α)d(z)αdz+∑k∈ˉJδˉzkˉνk+δ0ˉν0, | (3.4) |
ψ2|un|2d(z)2dz⇀ˆρ=ψ2|u0|2d(z)2dz+δ0ˆρ0, | (3.5) |
ψ2|vn|2d(z)2dz⇀ˉρ=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
ˆμj≥S(α,G)⋅ˆν22∗(α)j for all j∈J∪{0}, and ˆμ0≥μG⋅ˆρ0, | (3.7) |
ˉμk≥S(α,G)⋅ˉν22∗(α)k for all k∈ˉJ∪{0}, and ˉμ0≥μG⋅ˉρ0, | (3.8) |
where S(α,G) is the best Hardy-Sobolev constant, i.e.,
S(α,G)=infu∈S10(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∞:=limR→∞lim supn→∞∫G∩{d(z)>R}|∇Gun|2dz, | (3.9) |
ν∞:=limR→∞lim supn→∞∫G∩{d(z)>R}ψα|un|2∗(α)d(z)αdz, | (3.10) |
ρ∞:=limR→∞lim supn→∞∫G∩{d(z)>R}ψ2|un|2d(z)2dz, | (3.11) |
and
ˉμ∞:=limR→∞lim supn→∞∫G∩{d(z)>R}|∇Gvn|2dz,ˉν∞:=limR→∞lim supn→∞∫G∩{d(z)>R}ψα|vn|2∗(α)d(z)αdz,ˉρ∞:=limR→∞lim supn→∞∫G∩{d(z)>R}ψ2|vn|2d(z)2dz. |
For the sequence {un}, let ϕj(z)∈C∞0(G,[0,1]) be a cut-off function centered at zj∈G∖{0} with ϕj=1 on Bd(zj,1), ϕj=0 on G∖Bd(zj,2). Let ϕj,ε(z)=ϕj(δ1ε(z)). Then |∇Gϕj,ε|≤Cε and {ϕj,εun} is bounded in S10(G). Testing I′λ1,λ2,β(un,vn) with (ϕj,εun,0), we obtain limn→∞⟨I′λ1,λ2,β(un,vn),(ϕj,εun,0)⟩=0, that is,
limn→∞(∫G|∇Gun|2ϕj,εdz−μ1∫Gψ2|un|2ϕj,εd(z)2dz−λ1∫Gψα|un|2∗(α)ϕj,εd(z)αdz−βp1∫Gf(z)ψγ|un|p1ϕj,ε|vn|p2d(z)γdz)=limn→∞∫Gun∇Gun∇Gϕj,εdz. | (3.12) |
Now, we estimate each term in (3.12). From (3.1)–(3.6), we get
limn→∞∫G|∇Gun|2ϕj,εdz=∫Gϕj,εdˆμ≥∫G|∇Gu0|2ϕj,εdz+ˆμj, | (3.13) |
limn→∞∫Gψα|un|2∗(α)ϕj,εd(z)αdz=∫Gϕj,εdˆν=∫Gψα|u0|2∗(α)ϕj,εd(z)αdz+ˆνj, | (3.14) |
limε→0limn→∞|∫Gψ2|un|2ϕj,εd(z)2dz|≤limε→0limn→∞∫Bd(zj,2ε)ψ2|un|2d(z)2dz=0, | (3.15) |
and
limε→0limn→∞∫Gf(z)ψγ|un|p1ϕj,ε|vn|p2d(z)γdz≤limε→0limn→∞∫Bd(zj,2ε)f(z)ψγ|un|p1ϕj,ε|vn|p2d(z)γdz≤limε→0limn→∞‖f‖Lp∗(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
0≤limε→0limn→∞|∫Gun∇Gun∇Gϕj,εdz|≤limε→0limn→∞(∫G|∇Gun|2dz)12(∫G|∇Gϕj,ε|2|un|2dz)12≤Climε→0(∫G|∇Gϕj,ε|2|u0|2dz)12≤Climε→0(∫Bd(zj,2ε)|∇Gϕj,ε|Qdz)1Q(∫Bd(zj,2ε)|u0|2∗dz)12∗≤Climε→0(∫Bd(zj,2ε)|u0|2∗dz)12∗=0. | (3.17) |
Consequently, from the above arguments (3.13)–(3.17), we get
0=limε→0⟨I′λ1,λ2,β(un,vn),(ϕεun,0)⟩≥ˆμj−λ1ˆνj,∀j∈J. |
Combining with (3.7), we have
either(1)ˆνj=0,or(2)ˆνj≥(S(α,G)λ1)Q−α2−α, |
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)Q−α2−αfork∈ˉJ. |
On the other hand, choosing a suitable cutoff function centered at the origin, by the analogous argument we can prove that
ˆμ0−μ1ˆρ0≤λ1ˆν0 and ˉμ0−μ1ˉρ0≤λ1ˉν0. | (3.18) |
It follows from the definition of Sα,μ1 and Sα,μ2 that
ˆμ0−μ1ˆρ0≥Sα,μ1⋅ˆν22∗(α)0 | (3.19) |
ˉμ0−μ2ˉρ0≥Sα,μ2⋅ˉν22∗(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)Q−α2−α | (3.21) |
and
either(3)′ˉν0=0,or(4)′ˉν0≥(Sα,μ2λ2)Q−α2−α. | (3.22) |
Furthermore, the Hardy inequality (1.2) implies that
0≤μGˆρ0≤ˆμ0,0≤(1−μ1μG)ˆμ0≤ˆμ0−μ1ˆρ0, | (3.23) |
and
0≤μGˉρ0≤ˉμ0,0≤(1−μ2μG)ˉμ0≤ˉμ0−μ2ˉρ0. | (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 ϕ∈C∞1(G) such that 0≤ϕ≤1, ϕ(z)=0 on Bd(0,1), ϕ(z)=1 on G∖Bd(0,2) and |∇Gϕ|≤cR. Set ϕR(z)=ϕ(δ1R(z)), then {ϕRun}⊂S10(G) is bounded. Testing I′λ1,λ2,β(un,vn) with (ϕRun,0) we obtain limn→∞⟨I′λ1,λ2,β(un,vn),(ϕRun,0)⟩=0, i.e.,
−limn→∞∫G⟨∇Gun,∇GϕR⟩undz=limn→∞[∫G(|∇Gun|2ϕR−μ1ψ2|un|2d(z)2ϕR)dz−λ1∫Gψα|un|2∗(α)d(z)αϕRdz−βp1∫Gf(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)|2−μ1ψ2|unϕR|2d(z)2)dz, |
we conclude that
μ1∫Gψ2|unϕR|2d(z)2dz+Sα,μ1(∫Gψα|unϕR|2∗(α)d(z)αdz)22∗(α)≤∫G|∇G(unϕR)|2dz≤∫G|∇Gun|2|ϕR|2dz+∫G|∇GϕR|2|un|2dz+2∫G|∇GunϕRun∇GϕR|dz. | (3.26) |
By Hölder inequality, it is easy to get that
limR→∞lim supn→∞∫G|ϕR∇Gun||un∇GϕR|dz≤limR→∞lim supn→∞(∫Bd(0,2R)∖Bd(0,R)|∇Gun|2dz)12(∫Bd(0,2R)∖Bd(0,R)|un∇GϕR|2dz)12≤ClimR→∞(∫Bd(0,2R)∖Bd(0,R)|∇GϕR|2|u0|2dz)12≤ClimR→∞(∫Bd(0,2R)∖Bd(0,R)|∇Gϕε|Qdz)1Q(∫Bd(0,2)∖Bd(0,R)|u0|2∗dz)12∗≤ClimR→∞(∫Bd(0,2R)∖Bd(0,R)|u0|2∗dz)12∗=0. | (3.27) |
Similarly,
limR→∞lim supn→∞∫G|∇GϕR|2|un|2dz=0. | (3.28) |
Thus, we see from(3.27), (3.28) and (3.26), we have
μ∞−μ1ρ∞≥Sα,μ1⋅ν22∗(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|≤|∫G∖Bd(0,R)f(z)ψγ|un|pd(z)γϕRdz|+|∫G∖Bd(0,R)f(z)ψγ|vn|pd(z)γϕRdz|≤(∫G∖Bd(0,R)ψγ|f(z)|2∗(γ)2∗(γ)−pd(z)γdz)2∗(γ)−p2∗(γ)[(∫G∖Bd(0,R)ψγ|un|2∗(γ)d(z)γϕRdz)p2∗(γ)+(∫G∖Bd(0,R)ψγ|vn|2∗(γ)d(z)γϕRdz)p2∗(γ)]≤(∫G∖Bd(0,R)ψγd(z)γ|f(z)|2∗(γ)2∗(γ)−pdz)2∗(γ)−p2∗(γ)[S−p2γ,μ1‖un‖pμ2+S−p2γ,μ1‖un‖pμ2]. |
Since f∈Lp∗(G,ψγd(z)γdz), it follows that
limR→∞lim supn→∞|∫Gf(z)ψγ|un|p1|vn|p2d(z)γϕRdz|≤limR→∞C(∫G∖Bd(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
μ∞−μ1ρ∞≤λ1ν∞. | (3.30) |
Hence, it follows from (3.29) and (3.30) that
either(5)ν∞=0,or(6)ν∞≥(Sα,μ1λ1)Q−α2−α. |
In contrast, the Hardy inequality implies that
0≤μGρ∞≤μ∞,0≤(1−μ1μG)μ∞≤μ∞−μ1ρ∞. | (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,
ˉμ∞−μ2ˉρ∞≥Sα,μ2⋅ˉν22∗(α)∞, |
ˉμ∞−μ1ˉρ∞≤λ2ˉν∞, |
and
either(5)′ˉν∞=0,or(6)′ˉν∞≥(Sα,μ2λ2)Q−α2−α. |
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)⟨I′λ1,λ2,β(un,vn),(un,vn)⟩)=limn→∞((12−12∗(α))‖(un,vn)‖2H−β(1−p2∗(α))∫Gf(z)ψγ|un|p1|vn|p2d(z)γdz)≥2∗(α)−22⋅2∗(α)‖(u0,v0)‖2H−β(2∗(α)−p)2∗(α)‖f‖Lp∗(G,ψγd(z)γdz)(‖u0‖pL2∗(γ)(G,ψγd(z)γdz)+‖v0‖pL2∗(γ)(G,ψγd(z)γdz))≥2∗(α)−22⋅2∗(α)(Sγ,μ1‖u0‖2L2∗(γ)(G,ψγd(z)γdz)+Sγ,μ2‖v0‖2L2∗(γ)(G,ψγd(z)γdz))−β(2∗(α)−p)2∗(α)‖f‖Lp∗(G,ψγd(z)γdz)(‖u0‖pL2∗(γ)(G,ψγd(z)γdz)+‖v0‖pL2∗(γ)(G,ψγd(z)γdz)). | (3.32) |
Since
‖u0‖pL2∗(γ)(G,ψγd(z)γdz)+‖v0‖pL2∗(γ)(G,ψγd(z)γdz)≤2(‖u0‖L2∗(γ)(G,ψγd(z)γdz)+‖v0‖L2∗(γ)(G,ψγd(z)γdz))p, |
‖u0‖2L2∗(γ)(G,ψγd(z)γdz)+‖v0‖2L2∗(γ)(G,ψγd(z)γdz)≥12(‖u0‖L2∗(γ)(G,ψγd(z)γdz)+‖v0‖L2∗(γ)(G,ψγd(z)γdz))2, |
which and (3.32) yield that
2β(2∗(α)−p)2∗(α)‖f‖Lp∗(G,ψγd(z)γdz)(‖u0‖L2∗(γ)(G,ψγd(z)γdz)+‖v0‖L2∗(γ)(G,ψγd(z)γdz))p≥2∗(α)−24⋅2∗(α)min{Sγ,μ1,Sγ,μ2}(‖u0‖L2∗(γ)(G,ψγd(z)γdz)+‖v0‖L2∗(γ)(G,ψγd(z)γdz))2, |
namely,
‖u0‖L2∗(γ)(G,ψγd(z)γdz)+‖v0‖L2∗(γ)(G,ψγd(z)γdz)≤(8(2∗(α)−p)‖f‖Lp∗(G,ψγd(z)γdz)(2∗(α)−2)min{Sγ,μ2,Sγ,μ2})12−pβ12−p. | (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)⟨I′λ1,λ2,β(un,vn),(un,vn)⟩)≥2∗(α)−22⋅2∗(α)(μ∞−μ1ρ∞+ˉμ∞−μ2ˉρ∞)−22∗(α)(8(2∗(α)−2)min{Sγ,μ1,Sγ,μ2})p2−p((2∗(α)−p)‖f‖Lp∗(G,ψγd(z)γdz))22−p⋅β22−p≥2∗(α)−22⋅2∗(α)(Sα,μ1ν22∗(α)∞+Sα,μ2ˉν22∗(α)∞)−22∗(α)(8(2∗(α)−2)min{Sγ,μ1,Sγ,μ2})p2−p((2∗(α)−p)‖f‖Lp∗(G,ψγd(z)γdz))22−p⋅β22−p≥2∗(α)−22⋅2∗(α)(Sα,μ1[(Sα,μ1λ1)Q−α2−α]22∗(α)+Sα,μ2[(Sα,μ2λ2)Q−α2−α]22∗(α))−22∗(α)(8(2∗(α)−2)min{Sγ,μ1,Sγ,μ2})p2−p((2∗(α)−p)‖f‖Lp∗(G,ψγd(z)γdz))22−p⋅β22−p=2∗(α)−22⋅2∗(α)((Sα,μ1)Q−α2−αλ−Q−22−α1+(Sα,μ2)Q−α2−αλ−Q−22−α2)−22∗(α)(8(2∗(α)−2)min{Sγ,μ1,Sγ,μ2})p2−p((2∗(α)−p)‖f‖Lp∗(G,ψγd(z)γdz))22−p⋅β22−p, |
that is,
0>2∗(α)−22⋅2∗(α)((Sα,μ1)Q−α2−αλ−Q−22−α1+(Sα,μ2)Q−α2−αλ−Q−22−α2)−22∗(α)(8(2∗(α)−2)min{Sγ,μ1,Sγ,μ2})p2−p((2∗(α)−p)‖f‖Lp∗p∗(G,ψγd(z)γdz))22−p⋅β22−p. | (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
limn→∞∫Gψα|un|2∗(α)d(z)αdz=∫Gψα|u0|2∗(α)d(z)αdz |
and
limn→∞∫Gψα|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
limn→∞∫Gψα|un−u0|2∗(α)d(z)αdz=0,limn→∞∫Gψα|vn−v0|2∗(α)d(z)αdz=0. |
We are now going to prove that (un,vn)→(u0,v0) strongly in H. First, we have
‖(un−u0,vn−v0)‖2H=⟨(I′λ1,λ2,β(un,vn)−I′λ1,λ2,β(u0,v0)),(un−u0,vn−v0)⟩+λ1∫Gψα(|un|2∗(α)−2un−|u0|2∗(α)−2u0)(un−u0)d(z)αdz+λ2∫Gψα(|vn|2∗(α)−2vn−|v0|2∗(α)−2v0)(vn−v0)d(z)αdz+βp1∫Gf(z)ψγ[|un|p1−2un|vn|p2−|u0|p1−2u0|v0|p2](un−u0)d(z)γdz+βp2∫Gf(z)ψγ[|un|p1|vn|p2−2vn−|u0|p1|v0|p2−2v0](vn−v0)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)(un−u0)d(z)αdz|≤∫Gψα|un|2∗(α)−1|un−u0|d(z)αdz+∫Gψα|u0|2∗(α)−1|un−u0|d(z)αdz≤(∫Gψα|un|2∗(α)d(z)αdz)2∗(α)−12∗(α)(∫Gψα|un−u0|2∗(α)d(z)αdz)12∗(α)+(∫Gψα|u0|2∗(α)d(z)αdz)2∗(α)−12∗(α)(∫Gψα|un−u0|2∗(α)d(z)αdz)12∗(α)→0asn→∞. | (3.36) |
Similarly,
|∫Gψα(|vn|2∗(α)−2vn−|v0|2∗(α)−2v0)(vn−v0)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|p1−2un|vn|p2−|u0|p1−2u0|v0|p2](un−u0)d(z)γdz|≤∫Gψγ|f(z)||un|p−1|un−u0|d(z)γdz+∫G|f(z)|ψγ|u0|p−1|un−u0|d(z)γdz≤(∫Gψγ|f(z)|2∗(γ)2∗(γ)−pd(z)γdz)2∗(γ)−p2∗(γ)(∫Gψγ|un|2∗(γ)d(z)γdz)p−12∗(γ)(∫Gψγ|un−u0|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)p−12∗(γ)(∫Gψγ|un−u0|2∗(γ)d(z)γdz)12∗(γ)→0asn→∞, | (3.38) |
and
|∫Gf(z)ψγ[|un|p1|vn|p2−2vn−|u0|p1|v0|p2−2v0](vn−v0)d(z)γdz|→0asn→∞, | (3.39) |
Combining (3.36), (3.37), (3.38), (3.39), (3.35) with limn→∞⟨I′λ1,λ2,β(un,vn),(un−u0,vn−v0)⟩=0 and limn→∞⟨I′λ1,λ2,β(u0,v0),(un−u0,vn−v0)⟩=0, we deduce that
limn→∞‖(un−u0,vn−v0)‖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)αdz≤S−2∗(α)2α,μ1‖u‖2∗(α)μ1+S−2∗(α)2α,μ2‖v‖2∗(α)μ2≤(S−2∗(α)2α,μ1+S−2∗(α)2α,μ2)‖(u,v)‖2∗(α)H, |
and
∫Gf(z)ψγ|un|p1|vn|p2d(z)γdz≤p1p∫Gf(z)ψγ|un|pd(z)γdz+p2p∫Gf(z)ψγ|vn|pd(z)γdz≤p1p(∫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)≤‖f‖Lp∗(G,ψγd(z)γdz)(p1pS−p2γ,μ1‖un‖pμ1+p2pS−p2γ,μ2‖vn‖pμ2)≤‖f‖Lp∗(G,ψγd(z)γdz)(S−p2γ,μ1+S−p2γ,μ2)‖(un,vn)‖pH. | (3.40) |
Then,
Iλ1,λ2,β(u,v)=12‖(u,v)‖2H−λ12∗(α)∫Gψα|u|2∗(α)d(z)αdz−λ22∗(α)∫Gψα|v|2∗(α)d(z)αdz−β∫Gf(z)ψγ|u|p1|v|p2d(z)γdz≥12‖(u,v)‖2H−(λ1+λ2)(S−2∗(α)2α,μ1+S−2∗(α)2α,μ2)2∗(α)‖(u,v)‖2∗(α)H−β‖f‖Lp∗(G,ψγd(z)γdz)(S−p2γ,μ1+S−p2γ,μ2)‖(u,v)‖pH. |
Define the function
g(t)=12t2−C1(λ1+λ2)t2∗(α)−C2βtp,∀t>0, |
where
C1:=(S−2∗(α)2α,μ1+S−2∗(α)2α,μ2)2∗(α),C2:=‖f‖Lp∗(G,ψγd(z)γdz)(S−p2γ,μ1+S−p2γ,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 ϕ∈C∞0([0,∞),R) such that 0≤ϕ(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
{−ΔGu−μ1ψ2ud(z)2=λ1ϕ(‖(u,v)‖H)ψα|u|2∗(α)−2ud(z)α+βp1f(z)ψγ|u|p1−2u|v|p2d(z)γin G,−ΔGv−μ2ψ2vd(z)2=λ2ϕ(‖(u,v)‖H)ψα|v|2∗(α)−2vd(z)α+βp2f(z)ψγ|u|p1|v|p2−2vd(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:H→R defined as
Jλ1,λ2,β(u,v)=12‖(u,v)|2H−12∗(α)∫Gϕ(‖(u,v)‖H)(λ1ψα|u|2∗(α)d(z)α+λ2ψα|v|2∗(α)d(z)α)dz−β∫Gf(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),v≤0 and ‖(u,v)‖H≥t1. If ‖(u,v)‖H≥t2, then we have
Jλ1,λ2,β(u,v)≥12‖(u,v)‖2H−β‖f‖Lp∗(G,ψγd(z)γdz)(S−p2α,μ1+S−p2α,μ2)‖(u,v)‖pH>0. |
This contradicts Jλ1,λ2,β(u,v)<0.
If t1≤‖(u,v)‖H<t2, since 0≤ϕ(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 J′λ1,λ2,β(un,vn)→0 in H−1. 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
J−ελ1,λ2,β={(u,v)∈H:Jλ1,λ2,β(u,v)≤−ε}. |
Lemma 3.3. Given k∈N, there exists ε=ε(k)>0 such that γ(J−ελ1,λ2,β)≥k for any λ1, λ2, β>0.
Proof. Fix λ1, λ2>0, k∈N 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)γdz≥ck>−∞. |
Thus, for each (u,v)=rk(ω1,ω2) with rk∈(0,t1), we have
Jλ1,λ2,β(u,v)=Jλ,β(rk(ω1,ω2))=r2k2−r2∗(α)k2∗(α)ϕ(rk)∫G(λ1ψα|ω1|2∗(α)d(z)α+λ2ψα|ω2|2∗(α)d(z)α)dz−βrpk∫Gf(z)ψγ|ω1|p1|ω2|p2d(z)γdz≤12r2k−βckrpk. |
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 k−1 dimensional sphere Sk−1 and Sk∩Ek⊂J−ελ1,λ2,β. By Proposition 2.1 (2) and (4) it follows that
γ(J−ελ1,λ2,β)≥γ(Sk∩Ek)=k, |
concluding the proof.
Let us set the number
ck=infA∈Γksup(u,v)∈AJλ1,λ2,β(u,v), |
with
Γk={A⊂H:Ais closed,A=−Aandγ(A)≥k}. |
Clearly, ck≤ck+1 for each k∈N. Before proving our main result, we state the following technical results.
Lemma 3.4. ck<0 for all k∈N.
Proof. Fix k∈N. By Lemma 3.3, there exists ε>0 such that γ(J−ελ1,λ2,β)≥k. This and Jλ1,λ2,β is a continuous even functional imply that J−ελ1,λ2,β∈Γk. Then
(0,0)∉J−ελ1,λ2,βandsup(u,v)∈J−ελ1,λ2,βJλ1,λ2,β(u,v)≤−ε<0. |
Therefore, taking into account that Jλ1,λ2,β is bounded from below, we get
−∞<ck=infA∈Γksup(u,v)∈AJλ1,λ2,β(u,v)≤sup(u,v)∈J−ελ1,λ2,βJλ1,λ2,β(u,v)≤−ε<0. |
Let
Kc={(u,v)∈H:J′λ1,λ2,β(u,v)=0andJλ1,λ2,β(u,v)=c}. |
Lemma 3.5. For any λ1, λ2, β>0, the critical values {ck}k∈N of Jλ1,λ2,β satisfy ck→0 as k→∞.
Proof. Fix μ1, μ2∈[0,μG) and λ1, λ2, β>0. By Lemma 3.4 it follows that ck<0. Since ck≤ck+1 we can assume that limk→∞ck→c0≤0. 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
Kc0⊂E:={A⊂H∖{(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:H→H such that
η(Jc0+ελ1,λ2,β∖Nδ(Kc0))⊂Jc0−ελ1,λ2,β,for someε∈(0,−c0) | (3.43) |
Taking into account that ck+1≤ck and ck→c0 as k→∞, we can find k∈N such that ck>c0−ε and ck+k0≤c0, where k0 given in (3.42). Take A∈Γk+k0 such that sup(u,v)∈AJλ1,λ2,β(u,v)≤ck+k0<c0+ε, by using Properties 2.1 (4), we have
γ(¯A∖Nδ(Kc0))≥γ(A)−γ(Nδ(Kc0))≥kandγ(η(¯A∖Nδ(Kc0)))≥k, |
from which we have η(¯A∖Nδ(Kc0))∈Γk. Hence
sup(u,v)∈η(¯A∖Nδ(Kc0))Jλ1,λ2,β(u,v)≥ck>c0−ε. | (3.44) |
On the other hand, in view of (3.43) and A⊂Jc0+ελ1,λ2,β, we see that
η(A∖Nδ(Kc0))⊂η(Jc0+ελ1,λ2,β∖Nδ(Kc0))⊂Jc0−ελ1,λ2,β, |
which gives a contradiction in virtue of (3.44). Hence, c0=0 and limk→∞ck=0 hold.
Lemma 3.6. Let λ1, λ2, β be as in (ii) or (iii) of Lemma 3.2. If k,l∈N 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:H→H such that
η(Jc+ελ1,λ2,β∖Nδ(Kc))⊂Jc−ελ1,λ2,βfor someε∈(0,−c). | (3.45) |
From the definition of c=cn+l, we know there exists A∈Γn+l such that
sup(u,v)∈AJλ1,λ2,β(u,v)<c+ε, |
that is, A⊂Jc+ελ1,λ2,β, and so by (3.45) we get
η(A∖Nδ(Kc))⊂η(Jc+ελ1,λ2,β∖Nδ(Kc))⊂Jc−ελ1,λ2,β. |
This yields
supu∈η(¯A∖Nδ(Kc))Jλ1,λ2,β(u,v)≤c−ε, | (3.46) |
On the other hand, by parts (1), (3) of Proposition 2.1 we have
γ(η(¯A∖Nδ(Kc)))≥γ(¯A∖Nδ(Kc))≥γ(A)−γ(Nδ(Kc))≥n. |
Hence, we conclude that η(¯A∖Nδ(Kc))∈Γn and so
supu∈η(¯A∖Nδ(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 k∈N there exists l∈N such that ck=ck+1=⋯=ck+l=c, then γ(Kc)≥l+1≥2 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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