# Vector Solutions for Linearly Coupled Choquard Type Equations with Lower Critical Exponents

Authors: Huiling Wu
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

The existence, nonexistence, and multiplicity of vector solutions of the linearly coupled Choquard type equations−Δu+V1xu=Iα∗uN+α/Nuα/N−1u+λv,x∈ℝN,−Δv+V2xv=Iα∗vN+α/Nvα/N−1v+λu,x∈ℝN,u,v∈H1ℝN, are proved, where α∈0,N, N≥3, V1xV2x∈L∞ℝN are positive functions, and Iα denotes the Riesz potential.


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## Body

## 1. Introduction

We deal with the linearly coupled Choquard type equations:(1)−Δu+V1xu=Iα∗uN+αNuαN−1u+λv,x∈ℝN,−Δu+V2xv=Iα∗vN+αNvαN−1v+λu,x∈ℝN,u,v∈H1ℝN,where N≥3, α∈0,N, and V1,V2∈L∞ℝN are positive functions, N+α/N is the lower critical exponent with respect to a Hardy-Littlewood-Sobolev inequality (see ([1], Theorem 3.1) or ([2], Theorem 4.3)), and Iα denotes the Riesz potential defined on ℝN\0 by
(2)Iαx=ΓN−α/22απN/2Γα/2xN−α.The single equation(3)−Δu+Vxu=Iα∗upup−2u,u∈H1ℝN,appears in various physical contexts (see [3–6]). Mathematically, equations of this type have received considerable attention due to the appearance of the nonlocal term Iα∗upup−2u, which makes the problem challenging and interesting. The readers can refer to [4, 7–18] and references therein for research on related problems.Recently, Chen and Liu [19] established the existence and asymptotic behavior of the vector ground state of the linearly coupled system:
(4)−Δu+u=Iα∗upup−2u+λv,x∈ℝN,−Δv+v=Iα∗vqvq−2v+λu,x∈ℝN,u,v∈H1ℝN,where 0<λ<1, N+α/N<p,q<N+α/N−2. Xu, Ma and Xing [20] extended the results in [19] to (4) in the case that Iα∗upup−2u and Iα∗uquq−2u are replaced with general subcritical nonlinearities Iα∗FuF′u and Iα∗GuG′u, respectively. Yang et al. [21] obtained the existence of the vector ground state of (4) in the following three cases:
(5)p=N+αN,N+αN<q<N+αN−2,p=N+αN−2,N+αN<q<N+αN−2,p=N+αN,q=N+αN−2.They also proved that (4) has no nontrivial solutions if p=q=N+α/N or p=q=N+α/N−2.As we know, whenα⟶0, the local system
(6)−Δu+u=up−2u+λv,x∈ℝN,−Δv+v=vq−2v+λu,x∈ℝN,u,v∈H1ℝN,which has application in a large number of physical problems such as in nonlinear optics, can be regarded as a limiting system of (4). Systems of this type have received great attention in recent years (see [22–28] for instance). However, linearly coupled systems with nonlocal nonlinearities have been less studied.In this paper, we are interested in the existence, nonexistence, and multiplicity of solutions of system (1) with positive nonconstant potentials. We assume that(H1)Vix≥C>0,Vix∈L∞ℝN and limx⟶∞Vix=1,i=1,2(H2)liminf∣x∣⟶∞1−Vixx2≥N2N−2/4N+1, i=1,2(H3)0<∣λ∣<infx∈ℝNV1xV2xFor simplicity, the integral∫ℝN·dx is denoted by ∫·. According to (H1), the norm in H≔H1ℝN×H1ℝN can be defined by
(7)u,v≔u12+v22,where
(8)u12≔∫∇u2+V1xu2,u22≔∫∇u2+V2xu2.Then, a solution of system (1) can be found as a critical point of the energy functional E:H↦ℝ defined by
(9)Eλu,v≔12u,v2−∫λuv−N2N+α∫Iα∗uN+α/NuN+α/N+Iα∗vN+α/NvN+α/N.Set(10)N≔u,v∈H\0,0∣Eλ′u,v,u,v=0,cλ≔infNEλu,v.We first show thatcλ is attained.Theorem 1.
Assume that (H1), (H2), and (H3) hold. Then, there exists a vector ground stateuλ,vλ of system (1). Additionally, if λn⊂0,infx∈ℝNV1xV2x is a sequence satisfying λn⟶0+ as n⟶+∞, then up to a subsequence, either uλn,vλn⟶u^,0 or uλn,vλn⟶0,v^ in H as n⟶∞, where u^ is a ground state of
(11)−Δu+V1xu=Iα∗uN+α/Nuα/N−1u,u∈H1ℝN,and v^ is a ground state of
(12)−Δu+V2xu=Iα∗uN+α/Nuα/N−1u,u∈H1ℝN.Remark 2.
We call a solutionu,v∈H of system (1) a nontrivial solution if u,v≠0,0 and a vector solution if u≠0 and v≠0. A nontrivial solution u,v satisfying Eλu,v≤Eλh,k for any nontrivial solutions h,k∈H of system (1) is called a ground state.Remark 3.
Under assumptions (H1) and (H2), the existence of ground states of equations (11) and (12) has been proved by Moroz and Van Schaftingen ([17], Theorem 3 and Theorem 6).To prove Theorem1, it is crucial to give an estimate of the upper bound of the least energy cλ due to the lack of compactness. In our case, the estimate is quite involved, since we are dealing with a coupled system, which is more complex than a single equation. The method we follow can be sketched as follows. We first study the minimizing problem
(13)S0=infu,v∈L2ℝN×L2ℝN\0,0∫u2+v2−2λuv∫Iα∗uN+α/NuN+α/N+∫Iα∗vN+α/NvN+α/NN/N+α,which can be considered an extension of the classical problem
(14)S1=infu∈L2ℝN\0∫u2∫Iα∗uN+α/NuN+α/NN/N+α.By the results thatS1 is attained if and only if
(15)ux=Ubx≔Abb2+x−a2N/2,where A>0 is a fixed constant, a∈ℝN, and b∈0,∞ (see ([1], Theorem 3.1) or ([2], Theorem 4.3)), and studying the minimum point of a function hτ defined on 0,+∞ by
(16)hτ=1+τ2−2∣λ∣τ1+τ2N+α/NN/N+α,we show that S0 is attained at Ub,τminUb if 0<λ<1 and at Ub,−τminUb if −1<λ<0 (see Theorem 7 in Section 2), which combined with the existence of ground states for equations (11) and (12) enables us to obtain the precise upper bound of cλ.Our second goal is to show the existence of a higher energy vector solution of (1).Theorem 4.
Assume that (H1) and (H2) hold. Then, for someλ∗∈0,infx∈ℝNV1xV2x, there exists a vector solution u¯λ,v¯λ of system (1) if 0<λ<λ∗. Additionally, if λn⊂0,λ∗ is a sequence satisfying λn⟶0+ as n⟶+∞, then up to a subsequence, u¯λn,v¯λn⟶u¯,v¯ in H, where u¯ is a ground state of (11) and v¯ is a ground state of (12).Remark 5.
Forλ>0 sufficiently small, it is trivial to see that the solutions obtained in Theorem 1 and Theorem 4 are different, which implies that there exists at least two vector solutions of system (1) if λ>0 is small enough.Finally, we prove the nonexistence of the nontrivial solution of system (1) by establishing the Pohozaev type identity.Theorem 6.
Assume that (H3) holds. IfVix∈Wloc1,1ℝN∩L∞ℝN,i=1,2 and
(17)supx∈ℝNx2∇Vix·x<N−222,i=1,2,then, system (1) has no nontrivial solutions in H.This paper is structured as follows. Some preliminary results are provided in Section2. The proofs of Theorems 1 and 4 are presented in Section 3 and Section 4, respectively. In Section 5, we show the nonexistence of nontrivial solutions.


## 2. Preliminary Results

In this section, we show the sharp constantS0 defined in (13) is attained and give an estimate of the upper bound of cλ.Theorem 7.
If0<∣λ∣<1, then S0 is attained. Moreover, Ub,τminUb (or Ub,−τminUb) is a solution of (13) for 0<λ<1 (or −1<λ<0), where τmin>0 is a minimum point of hτ defined on 0,+∞ by
(18)hτ=1+τ2−2∣λ∣τ1+τ2N+α/NN/N+α.Proof.
First, we show that there existsτmin>0 such that
(19)hτmin=minτ≥0hτ.
Calculating directly, we have(20)h′τ=τ−∣λ∣+∣λ∣τ2N+α/N−τN+2α/N21+τ2N+α/N2N+α/N+α.
Setfτ=τ−∣λ∣+∣λ∣τ2N+α/N−τN+2α/N. It can be easily seen that fτ⟶−∣λ∣ as τ⟶0, and fτ⟶+∞ as τ⟶+∞. Then, there is τmin>0 such that fτmin=0, and hτmin=minτ≥0hτ.
In the next step, we prove(21)S0=hτminS1,where S1 is defined in (14). We employ the idea in ([29], Theorem 5) to prove (21). For the case λ>0, taking u,v=Ub,τminUb gives
(22)S0≤1+τmin2−2λτmin∫Ub21+τmin2N+α/N∫Iα∗UbN+α/NUbN+α/NN/N+α=hτminS1.
Letun,vn∈L2ℝN×L2ℝN be a minimizing sequence for S0. Set zn=tnun, where tn=∫vn2/∫un21/2. Then,
(23)∫zn2=tn2∫un2=∫vn2,(24)∫znvn=tn∫unvn≤∫vn2=∫zn2.
Collecting (23) and (24) leads to
(25)S0+o1=∫un2+vn2−2λunvn∫Iα∗unN+α/NunN+α/N+∫Iα∗vnN+α/NvnN+α/NN/N+α≥tn−2∫zn2+∫vn2−2λtn−1∫vn2tn−2N+α/N∫Iα∗znN+α/NznN+α/N+∫Iα∗vnN+α/NvnN+α/NN/N+α≥htn−1S1≥hτminS1.
Then, (21) follows from (22) and (25). For the case λ<0, the conclusions follow by replacing Ub,τminUb with Ub,−τminUb and repeating the proof previously.Lemma 8.
Assume that (H1) and (H3) holds, then for anyu,v∈H\0,0, there exists t0>0 such that t0u,v∈N and
(26)Eλt0u,t0v=maxt≥0Eλtu,tv.Proof.
This result is standard and the proof can be found in ([30], Lemma 12). We omit it.For equations (11) and (12), we set
(27)Jiu=12ui2−N2N+α∫Iα∗uN+α/NuN+α/N,and Bi=infNiJiu, where
(28)Ni=u∈H1ℝN/0∣Ji′u,u=0,i=1,2.Then, according to ([17], Theorem 3 and Theorem 6), we have
(29)Bi<α2N+αS1N+α/α,i=1,2,and Bi is achieved, where S1 is defined in (14). By Theorem 7 and ([17], Theorem 3 and Theorem 6), we are able to get the following estimate.Lemma 9.
Assume that (H1), (H2), and (H3) hold. Then,(30)0<cλ<minB1,B2,α2N+αS0N+α/α.Proof.
We first show the positivity ofcλ. By (H3), we have
(31)Cu,v2≤u,v2−2∫λuv=∫Iα∗uN+α/NuN+α/N+Iα∗vN+α/NvN+α/N≤S1−1∫u2N+α/N+S1−1∫v2N+α/N≤S1−N+α/Nu,v2N+α/N,for some C>0, which suggests that there exists M1>0 such that ∥u,v∥>M1. Thus, we obtain
(32)cλ=infNEλu,v=infNα2N+αu,v2−2∫λuv≥α2N+αCM12>0.
Second, we show(33)cλ<α2α+NS0N+α/α.
From the assumptions (H1)–(H3), we see that0<∣λ∣<1, and so Theorem 7 holds. For the case λ>0, by Lemma 8, there is t>0 such that tUb,τminUb∈N; then, we have
(34)cλ≤EλtUb,tτminUb=t22∫1+τmin2∇Ub2+V1x+τmin2V2x−2τminλUb2−Nt2N+α/N2N+α1+τmin2N+α/N∫Iα∗UbN+α/NUbN+α/N=t221+τmin2−2τminλ∫Ub2−Nt2N+α/N2N+α1+τmin2N+α/N∫Iα∗UbN+α/NUbN+α/N+t22∫1+τmin2∇Ub2+V1x−1+τmin2V2x−1Ub2≤α2α+NS0N+α/α+t22∫1+τmin2∇Ub2+V1x−1+τmin2V2x−1Ub2.
The last inequality in (34) follows from Theorem 7 and direct calculation. Denote
(35)Liu=t22∫∇u∣2+Vix−1u2,i=1,2.
To prove (33), it is enough to show
(36)LiUb<0,i=1,2,for some b>0. Since
(37)∫x21+x2N+2=N−24N+1∫1x21+x2N,we have
(38)∫∇Ub2=N2N−24N+1∫Ub2x2.
Then, by a transformationx=a+bz, we get
(39)Liub=∫N2N−24N+1z2−b21−Via+bzC21+z2Ndz.
Taking the assumption (H2) into consideration, we see that that (36) holds. Then, (33) follows from (34).
Now, it remains to show(40)cλ<minB1,B2.
Denote ground states of (11) and (12) by U and V, respectively. Since U,0∈N and 0,V∈N, we have cλ≤minB1,B2. If cλ=minB1,B2, then we see that at least one of U,0 and 0,V is a solution of system (1), which is impossible since λ≠0, so (40) holds.


## 3. Proof of Theorem 11

Lemma 10.
Assume that (H1), (H2), and (H3) hold. Then, there exists a vector ground state of system (1).Proof.
According to Ekeland’s variational principle, there existsun,vn⊂N such that
(41)Eλun,vn⟶cλ,Eλ′un,vnN⟶0asn⟶∞.
For simplicity, we denoteIλu,v≔Eλ′u,v,u,v. First, we prove Eλ′un,vn⟶0. Indeed,
(42)o1=Eλ′un,vnN=Eλ′un,vn−σnIλ′un,vnfor some σn and sufficiently large n. Particularly,
(43)o1=Eλ′un,vn,un,vn−σnIλ′un,vn,un,vn=−σnIλ′un,vn,un,vn.
From the proof of Lemma9, we observe that there exists M1,M2>0 such that M1≤∥un,vn∥≤M2. Then, we have
(44)Iλ′un,vn,un,vn=2un,vn2−2∫λunvn−2N+αN∫Iα∗unN+α/NunN+α/N+Iα∗vnN+α/NvnN+α/N−2αNun,vn2−2∫λunvn≤−Cun,vn2≤−CM1<0.
Taking (43) into consideration, we obtain that σn⟶0 as n⟶∞. Then, from (42), we get Eλ′un,vn⟶0.
We may assume that(45)un,vn⇀u,vinH,un,vn→u,vinLlocrℝN×LlocrℝN2≤r<2∗,un,vn→u,va.einℝN.
Then,E′λu,v=0. To complete the proof, it is sufficient to prove that u,v≠0,0. Actually, if u,v≠0,0. By Fatou’s lemma, we get
(46)cλ≤Eλu,v=α2N+α∫Iα∗uN+α/NuN+α/N+Iα∗vN+α/NvN+α/N≤liminfn→∞α2N+α∫Iα∗unN+α/NunN+α/N+Iα∗vnN+α/NvnN+α/N=cλ.
Furthermore, sinceEλ′u,v=0 and λ≠0, we see from (1) that u≡0 and v≡0, that is, u,v is a vector ground state of (1).
Suppose the assertion is false, that is,u,v=0,0. On the one hand, we know from (H1) that
(47)∫V1xun2+V2xvn2=∫Br0V1xun2+V2xvn2+∫ℝN\Br0V1xun2+V2xvn2=∫Br0V1xun2+V2xvn2+∫ℝN\Br0un2+vn2+o1=∫un2+vn2+o1,as n⟶∞. Then, it follows
(48)∫∇un2+∇vn2+un2+vn2−2λunvn+o1=un,vn2−2∫λunvnas n⟶∞. Using Theorem 7, we obtain
(49)cλ=Eλun,vn−12Eλ′un,vn,un,vn+o1=α2N+α∫Iα∗unN+α/NunN+α/N+Iα∗vnN+α/NvnN+α/N+o1≤α2N+αS0−N+α/N∫un2+vn2−2λunvnN+α/N+o1≤α2N+αS0−N+α/N∫∇un2+∇vn2+un2+vn2−2λunvnN+α/N+o1=α2N+αS0−N+α/Nun,vn2−2∫λunvnN+α/N+o1.
On the other hand,(50)cλ=α2N+αun,vn2−2∫λunvn+o1.
Collecting (49) and (50) yields
(51)cλ≥α2N+αS0N+α/α,which contradicts Lemma 9. Thus, u,v≠0,0.Proof of Theorem 11.
By Lemma10, we need only show the asymptotic behavior of the vector ground state when λ⟶0+. First, we claim that cλ decreases strictly monotonically with respect to λ∈0,infx∈ℝNV1xV2x. Indeed, fix λ1,λ2∈0,infx∈ℝNV1xV2x with λ1<λ2. Denoting a vector ground state of system (1) when λ=λ1 by uλ1,vλ1 and letting t>0 be the constant such that tuλ1,tvλ1∈Nλ=λ2, we obtain
(52)uλ1,vλ12−2∫λ1uλ1vλ1=∫Iα∗uλ1N+α/Nuλ1N+α/N+Iα∗vλ1N+α/Nvλ1N+α/N,uλ1,vλ12−2∫λ2uλ1vλ1=t2α/N∫Iα∗uλ1N+α/Nuλ1N+α/N+Iα∗vλ1N+α/Nvλ1N+α/N.
Then, byλ1<λ2, we deduce that t<1, which gives
(53)cλ2≤Eλ2tuλ1,tvλ1=α2N+αt2N+α/N∫Iα∗uλ1N+α/Nuλ1N+α/N+Iα∗vλ1N+α/Nvλ1N+α/N<α2N+α∫Iα∗uλ1N+α/Nuλ1N+α/N+Iα∗vλ1N+α/Nvλ1N+α/N=cλ1.
The claim is proved.
Now, chooseλn⊂0,infx∈ℝNV1xV2x satisfying λn⟶0+ as n⟶∞ and denote a vector ground state of system (1) when λ=λn by uλn,vλn. Then ∀h1,h2∈H, we have
(54)limn⟶∞E0′uλn,vλn,h1,h2=limn⟶∞Eλn′uλn,vλn,h1,h2=0,limn⟶∞E0uλn,vλn=limn⟶∞Eλnuλn,vλn.
Bycλn<minB1,B2 and (29), we obtain
(55)limn⟶∞E0uλn,vλn=limn⟶∞cλn≤minB1,B2<α2N+αS1N+α/α.
Repeating an argument as in Lemma10, we deduce that uλn,vλn⟶u^,0 or uλn,vλn⟶0,v^ in H, where u^ and v^ are ground states of (11) and (12), respectively.


## 4. Proof of Theorem 18

In this section, we study the existence of a higher energy vector solution of system (1) for λ>0 sufficiently small. We suppose that B1≤B2 without loss of generality. Let U, V be ground states of (11) and (12), respectively. Then, we may assume that U and V are positive since ∣U∣ and ∣V∣ are also ground states of (11) and (12), respectively. Now, we set
(56)A=A1×A2,where
(57)Ai=u∈H1ℝN,Ji′u=0,Jiu=Bi.ThenU,V∈A. Moreover, by a similar argument as that in ([28], Lemma 12), we obtain the following.Lemma 12.
Assume that (H1) and (H2) hold. Then,A⊂H is compact, and there exist 0<a1<a2 such that
(58)a1≤u1,v2≤a2,∀u,v∈A.Proof.
The proof can be found in ([28], Lemma 12) and will be omitted here.By the definition ofU and V, we know that
(59)B1=J1U=maxt>0J1tU,B2=J2V=maxt>0J2tV,(60)J1tU≤B14,∀t∈0,t1∪t2,+∞,J2sV≤B24,∀s∈0,s1∪s2,+∞,for some t1, t2, s1, ands2 satisfying 0<t1<1<t2 and 0<s1<1<s2. Denote A≔0,t2×0,s2 and define γ^:A↦H by
(61)γ^t,s=γ^1t,γ^2s≔tU,sV.Then,maxt,s∈A∥γ^t,s∥≤a0 for some a0>0. Define
(62)c^λ≔infγ∈Γ^maxt,s∈AEλγt,s,mλ≔maxt,s∈AEλγ^t,s,where
(63)Γ^≔γ∈ℂA,H∣maxt,s∈Aγt,s≤2a2+a0,γt,s=γ^t,sfort,s∈A\t1,t2×s1,s2,and a2 is defined in Lemma 12. Obviously, γ^t,s∈Γ^. Moreover, we havethe following.Lemma 13.
Assume that (H1), (H2), and0<λ<infx∈ℝNV1xV2x hold. Then,
(64)limλ⟶0+c^λ=limλ⟶0+mλ=c^0=m0=B1+B2.Proof.
Byλ>0, we see that Eλγ^t,s≤E0γ^t,s and
(65)mλ≤m0=maxt,s∈AE0γ^t,s=maxt∈0,t2J1γ^1t+maxs∈0,s2J2γ^2s=J1U+J2V=B1+B2.
Observing thatc^λ≤mλ since γ^∈Γ^, we deduce
(66)limsupλ⟶0+c^λ≤liminfλ⟶0+mλ≤limsupλ⟶0+mλ≤m0,c^0≤m0.
Now, forγt,s=γ1t,γ2s∈Γ^, define a function fγ on t1,t2×s1,s2 by
(67)fγt,s≔ϕ1γ1t−ϕ2γ2s,ϕ1γ1t+ϕ2γ2s−2,where ϕ1,ϕ2:H↦ℝ are given by
(68)ϕ1u≔∫Iα∗uN+α/NuN+α/N∫∇u2+V1xu2,ifu≠0,0,ifu=0,ϕ2u≔∫Iα∗uN+α/NuN+α/N∫∇u2+V2xu2,ifu≠0,0,ifu=0.
Noting thatϕ1,ϕ2 are continuous and fγ^1,1=0, we deduce degfγ^,t1,t2×s1,s2,0,0=1. Moreover, we know from (60) that fγt,s=fγ^t,s≠0,0 for any t,s∈∂t1,t2×s1,s2, which implies degfγ,t1,t2×s1,s2,0,0 is well defined and
(69)degfγ,t1,t2×s1,s2,0,0=degfγ^,t1,t2×s1,s2,0,0=1.
Therefore, there existst∗,s∗∈t1,t2×s1,s2 satisfying fγt∗,s∗=0,0, that is, ϕ1γ1t∗=ϕ2γ2s∗=1. Recalling the definition of ϕ1,ϕ2, we have γ1t∗∈N1,γ2s∗∈N2. Then, it follows
(70)maxt,s∈AE0γt,s≥E0γt∗,s∗=J1γ1t∗+J2γ2s∗≥B1+B2=m0.
Thus,c^0≥m0. Taking account of (66), we obtain c^0=m0. Now, it remains to prove
(71)liminfλ⟶0+c^λ≥m0.
If (12) is not true, there exists a sequence λn⟶0+, γnt,s=γ1,nt,γ2,ns∈Γ^ and ε>0 satisfying
(72)maxt,s∈AEλnγnt,s≤m0−2ε.
For theε given above, the definition of Γ^ leads to
(73)maxt,s∈Aλn∫∣γ1,ntγ2,ns∣≤Cλn<ε,∀n≥N0for some N0>0 sufficiently large. Then, it follows
(74)maxt,s∈AE0γnt,s≤maxt,s∈AEλnγnt,s+ε≤m0−ε,∀n≥N0,which contradicts (70), implying that (71) holds.Set(75)Ad≔u,v∈H:distu,v,A≤d,Eλc≔u,v∈H:Eλu,v≤c.Then, we show the compactness of the PS sequence.Lemma 14.
Assume that (H1) and (H2) hold. Denoted0≔1/22N+α/αB11/2 and let d∈0,d0.Ifλn satisfies λn>0 and λn⟶0 as n⟶∞ and un,vn⊂Ad is a sequence with
(76)limn→∞Eλnun,vn≤c^0,limn→∞Eλn′un,vn=0,then, there exists u,v∈A such that un,vn⟶u,v in H.Proof.
Observing thatun,vn is bounded by the choice of d and Lemma 12, we assume un,vn⇀u,v in H. Then, by a similar argument as in ([28], Lemma 14), we obtain that
(77)u,v∈A2d.
Moreover, using the definition ofd again, we get u≠0,v≠0.
We now proveun,vn⟶u,v∈A. Actually, for h1,h2∈H, we have
(78)E0′u,v,h1,h2=limn⟶∞E0′un,vn,h1,h2=limn⟶∞Eλn′un,vn,h1,h2=0,(79)limn⟶∞E0un,vn=limn⟶∞Eλnun,vn≤c^0.
Then, it holds(80)E0u,v=α2N+α∫Iα∗uN+α/NuN+α/N+Iα∗vN+α/NvN+α/N≤liminfn→∞α2N+α∫Iα∗unN+α/NunN+α/N+Iα∗vnN+α/NvnN+α/N=liminfn→∞E0un,vn.
Note that from (78), u,v∈A. Then, combining Lemma 13 with (79) and (80), we have E0u,v=c^0 and un,vn⟶u,v∈A in H.Next, we will construct a PS sequence using a perturbation approach.Lemma 15.
Assume that (H1) and (H2) hold. Then, for ad∈0,d0/2, where d0 was defined in Lemma 14, there are λ¯∈0,infx∈ℝNV1xV2x and a∈0,1 such that
(81)∥Eλ′u,v∥≥a,∀u,v∈Eλmλ∩Ad\Ad2,λ∈0,λ¯.Proof.
We prove indirectly. Suppose that there existsλn satisfying limn⟶∞λn=0 and un,vn⊂Eλnmλn∩Ad\Ad/2 with ∥Eλn′un,vn∥⟶0 as n⟶∞. Then, we see immediately that Eλnun,vn≤c^0 from Lemma 13, and un,vn⟶u,v in H for some u,v∈A by Lemma 14. Thus, un,vn∈Ad/2 for n sufficiently large, which is in contradiction with un,vn⊂Eλnmλn∩Ad\Ad/2, so the conclusion holds.In the sequel, we assume thatd,a,λ¯ be fixed such that Lemma 15 holds.Lemma 16.
Assume that (H1) and (H2) hold. Then, there existλ^∈0,λ¯ and δ>0 such that ∀λ∈0,λ^(82)Eλγ^t,s≥c^λ−δimpliesγ^t,s∈Ad/2.Proof.
Arguing indirectly, we suppose that there existλn⟶0,δn⟶0 and tn,sn∈A satisfying
(83)Eλnγ^tn,sn≥c^λn−δn,γ^tn,sn∈Ad/2.
We may supposetn,sn⟶t^,s^∈A. Then, from Lemma 13 and (83), we deduce
(84)E0γ^t^,s^≥limn⟶∞c^λn−δn=B1+B2.
Recalling the definition ofγ^t,s, we have t^,s^=1,1, and so
(85)limn⟶∞γ^tn,sn−γ^1,1=limn⟶∞γ^tn,sn−U,V=0,which is in contradiction with γ^tn,sn∈Ad/2.Forδ and λ^ given in Lemma 16, we define
(86)δ0≔minδ2,B14,18da2.Then,(87)∣c^λ−mλ∣<δ0,∣c^λ−B1+B2∣<δ0,∀λ∈0,λ∗,for some λ∗∈0,λ^.Lemma 17.
Assume that (H1) and (H2) hold. For fixedλ∈0,λ∗, there is un,vn⊂Ad∩Eλmλ with
(88)Eλ′un,vn⟶0,asn⟶∞.Proof.
Forλ∈0,λ∗, suppose contradictorily that ∥Eλ′u,v∥≥lλ for all u,v∈Ad∩Eλmλ and some 0<lλ<1. Then, there is a pseudogradient vector field hλ for Eλ on neighborhood Sλ of Ad∩Eλmλ such that
(89)hλu,v≤2min1,Eλ′u,v,Eλ′u,v,hλu,v≥min1,Eλ′u,vEλ′u,v.
Define a functionηλ on H satisfying 0≤ηλ≤1, ηλ≡1 on Ad∩Eλmλ and ηλ≡0 on H\Sλ, and a function ζλ on ℝ with 0≤ζλt≤1, ζλt≡1 if ∣t−c^λ∣≤δ/2, and ζλt≡0 if ∣t−c^λ∣≥δ. Then both ηλ and ζλ are Lipschitz continuous. Set
(90)gλu,v=−ηλu,vζλEλu,vhλu,v,u,v∈Sλ,0,u,v∈H\Sλ.
Then, the initial problem(91)ddθψλu,v,θ=gλψλu,v,θ,ψλu,v,0=u,v,has a global solution ψλ on H×0,+∞ with the properties:
(i)
ψλu,v,θ=u,v if θ=0 or u,v∈H\Sλ or ∣Eλu,v−c^λ∣≥δ(ii)
d/dθψλu,v,θ≤2(iii)
d/dθEλψλu,v,θ=Eλ′ψλu,v,θ,gλψλu,v,θ≤0
Now, the proof can be divided into two steps.
Step 1. We show that there existsθ0≥0 such that
(92)ψλγ^t,s,θ0∈Eλc^λ−δ0for t,s∈A and δ0 defined in (86).
Arguing indirectly, we suppose(93)Eλψλγ^t,s,θ>c^λ−δ0,∀θ≥0,for some t,s∈A. Noting δ0<δ and applying Lemma 16, we get γ^t,s∈Ad/2. By property (iii) and the fact that
(94)Eλγ^t,s≤mλ<c^λ+δ0,we have
(95)c^λ−δ0<Eλψλγ^t,s,θ≤mλ<c^λ+δ0,∀θ≥0.
Then,ζλEλψλγ^t,s,θ≡1. If ψλγ^t,s,θ)∈Ad for all θ≥0, then
(96)ηλψλγ^t,s,θ≡1,∥Eλ′ψλγ^t,s,θ∥≥lλ,∀θ>0.
Thus,(97)Eλψλγ^t,s,δlλ2≤c^λ+δ2−∫0δ/lλ2l2λdθ≤c^λ−δ2,which contradicts (93). Hence, ψλγ^t,s,θ0∈Ad for some θ0>0. Observing that γ^t,s∈Ad/2, we deduce that for some θ1,θ2 with 0<θ1<θ2≤θ0, it holds ψλγ^t,s,θ1∈∂Ad/2, ψλγ^t,s,θ2∈∂Ad, and ψλγ^t,s,θ∈Ad\Ad/2 for all θ∈θ1,θ2. Then, according to Lemma 15, we get
(98)Eλ′ψλγ^t,s,θ≥a,∀θ∈θ1,θ2.
Moreover, we deduce from property (ii) that(99)d2≤ψλγ^t,s,θ1−ψλγ^t,s,θ2≤2θ1−θ2,which yields ∣θ1−θ2∣≥d/4. Then, we obtain
(100)Eλψλγ^t,s,θ2≤Eλψλγ^t,s,θ1+∫θ1θ2ddθEλψλu,v,θ≤c^λ+δ0−a2θ2−θ1≤c^λ+δ0−14da2,≤c^λ−δ0,which contradicts (93), and the proof of this step is complete.
By step 1, we define(101)Gt,s≔infθ≥0:Eλψλγ^t,s,θ≤c^λ−δ0and γt,s≔ψλγ^t,s,Gt,s. Obviously, Eλγt,s≤c^λ−δ0 for all t,s∈A.
Step 2. We prove(102)γt,s∈Γ^.
Noting that(103)Eλγ^t,s≤E0γ^t,s=J1γ^1t+J2γ^2s≤B14+B2≤B1+B2−3δ0<c^λ−δ0,for all t,s∈A\t1,t2×s1,s2, we have Gt,s=0 and γt,s=γ^t,s.
It remains to showγt,s≤2a2+a0, ∀t,s∈A, and Gt,s is a continuous function of t,s∈A. For any t,s∈A, if Eλγ^t,s≤c^λ−δ0, then Gt,s=0 and γt,s=γ^t,s, so γt,s=γ^t,s≤2a2+a0. If Eλγ^t,s>c^λ−δ0, then γ^t,s∈Ad/2 and
(104)c^λ−δ0<Eλψλγ^t,s,θ≤mλ<c^λ+δ0,∀θ∈0,Gt,s.
So we getζλEλψλγ^t,s,θ≡1 for θ∈0,Gt,s. Then, we can prove that
(105)γt,s=ψλγ^t,s,Gs,t∈Ad.
Indeed, if not, by similar arguments as in step 1, we know thatEλψλγ^t,s,θ2≤c^λ−δ0 for some θ1,θ2 satisfying 0<θ1<θ2<Gt,s, which is in contradiction with the definition of Gt,s. Thus, γs,t∈Ad and
(106)γs,t−u,v≤d≤a02,for some u,v∈A. Hence, from Lemma 12,
(107)γs,t≤u,v+a02≤2a2+a0.
Now, we prove thatGt,s is continuous. For fixed t¯,s¯∈A, if Eλγt¯,s¯<c^λ−δ0, then Gt¯,s¯=0 and Eλγ^t¯,s¯<c^λ−δ0. Since γ^ is continuous, we have
(108)Eλγ^t,s<c^λ−δ0,∀t,s∈t¯−τ,t¯+τ×s¯−τ,s¯+τ∩A,for some τ>0, which implies Gt,s=0,∀t,s∈t¯−τ,t¯+τ×s¯−τ,s¯+τ∩A. Thus, the continuity of G at t¯,s¯ is proved. If Eλγt¯,s¯=c^λ−δ0, then we see from the proof previously that γt¯,s¯=ψλγ^t¯,s¯,Gt¯,s¯∈Ad, so
(109)Eλ′ψλγ^t¯,s¯,Gt¯,s¯≥lλ>0,and Eλψλγ^t¯,s¯,Gt¯,s¯+ω<c^λ−δ0 for any ω>0. By the continuity of ψλ, we obtain Eλψλγ^t,s,Gt¯,s¯+ω<c^λ−δ0, ∀t,s∈t¯−τ,t¯+τ×s¯−τ,s¯+τ∩A for some τ>0. Therefore, Gt,s≤Gt¯,s¯+ω and
(110)0≤limsupt,s⟶t¯,s¯Gt,s≤Gt¯,s¯.
IfGt¯,s¯=0, then
(111)limt,s⟶t¯,s¯Gt,s=Gt¯,s¯.
IfGt¯,s¯>0, then Eλψλγ^t¯,s¯,Gt¯,s¯−ω>c^λ+δ0 for any ω with 0<ω<Gt¯,s¯. Then, since ψλ is continuous, we deduce
(112)liminft,s⟶t¯,s¯Gt,s≥Gt¯,s¯.
Combining with (110), we obtain the continuity of Gs,t at t¯,s¯. Consequently, (102) holds.
By Step 1 and Step 2, we have showed thatγt,s∈Γ^ and maxt,s∈AEλγt,s≤c^λ−δ0, which is in contradiction with the definition of c^λ. Thus, the conclusion holds.Proof of Theorem 18.
Denoted0≔1/22N+α/αB11/2. From Lemma 17, we obtain that there exists unλ,vnλ⊂Ad such that
(113)Eλunλ,vnλ≤mλ,Eλ′unλ,vnλ⟶0,for fixed λ∈0,λ∗, where λ∗∈0,infx∈ℝNV1xV2x. Then, by Lemma 14, unλ,vnλ⇀uλ,vλ in H for some uλ,vλ∈A2d and Eλ′uλ,vλ=0. Moreover, recalling the definition of d, we have uλ≠0,vλ≠0, that is, uλ,vλ is a vector solution of system (1.1).
Now, choosingλn⊂0,λ∗ such that λn⟶0 as n⟶∞, by a repeat of the proof in Lemma 14, we obtain uλn,vλn⟶u¯,v¯∈A in H, with u¯ and v¯ being ground states of (11) and (12), respectively, which completes the proof.


## 5. Proof of Theorem 20

Lemma 19.
LetN≥3, Vix∈Wloc1,1ℝN∩L∞ℝN,i=1,2, and u,v∈H be a solution of system (1). If
(114)supx∈ℝN∇Vix·x<∞,i=1,2,then u,v satisfies the Pohozaev identity
(115)N−22∫∇u2+∇v2+N2∫V1xu2+V2xv2−2λuv+12∫∇V1x·xu2+∇V2x·xv2=N2∫Iα∗uN+α/NuN+α/N+Iα∗vN+α/NvN+α/N.Proof.
The lemma can be proved by a similar argument as that in ([17], Proposition 11).Proof of Theorem 20.
Letu,v be a solution of system (1). By Lemma 19, we have
(116)∫∇u2+∇v2=12∫∇V1x·xu2+∇V2x·xv2.
Then, the conclusion follows from a classical Hardy inequality (see ([31], Theorem 6.4.10))
(117)N−224∫u2x2≤∫∇u2.



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*Source: 6623902-2020-12-21.xml*