In this paper we study the existence of multiple nontrivial solutions of the coupled Schrödinger system with external sources terms as perturbations. This type of the system arises from Bose-Einstein condensate. As these external sources terms are nonlinear functions and small in some sense, we use fibre map to divide the Nehari manifold into threes parts, and then prove the existence of a nontrivial ground state solution and a bound state solution.
Citation: Xiaoyong Qian, Jun Wang, Maochun Zhu. Existence of solutions for a coupled Schrödinger equations with critical exponent[J]. Electronic Research Archive, 2022, 30(7): 2730-2747. doi: 10.3934/era.2022140
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In this paper we study the existence of multiple nontrivial solutions of the coupled Schrödinger system with external sources terms as perturbations. This type of the system arises from Bose-Einstein condensate. As these external sources terms are nonlinear functions and small in some sense, we use fibre map to divide the Nehari manifold into threes parts, and then prove the existence of a nontrivial ground state solution and a bound state solution.
In this paper we consider solitary wave solutions of the time-dependent coupled nonlinear Schrödinger system with perturbation
{−i∂Φ1∂t−ΔΦ1=μ1|Φ1|2Φ1+β|Φ2|2Φ1+f1(x),x∈Ω, t>0,−i∂Φ2∂t−ΔΦ2=μ2|Φ2|2Φ2+β|Φ1|2Φ2+f2(x),x∈Ω, t>0,Φ1(t,x)=Φ2(t,x)=0,x∈∂Ω, t>0, j=1,2, | (1.1) |
where Ω⊂RN is a smooth bounded domain, i is the imaginary unit, μ1,μ2>0 and β≠0 is a coupling constant. When N≤3, the system (1.1) appears in many physical problems, especially in nonlinear optics. Physically, the solution j denotes the j-th component of the beam in Kerr-like photorefractive media (see [1]). The positive constant μj is for self-focusing in the j th component of the beam. The coupling constant β is the interaction between the two components of the beam. The problem (1.1) also arises in the Hartree-Fock theory for a double condensate, that is, a binary mixture of Bose-Einstein condensates in two different hyperfine states, for more information, see [2].
If we looking for the stationary solution of the system (1.1), i.e., the solution is independent of time t. Then the system (1.1) is reduced to the following elliptic system with perturbation
{−Δu+λ1u=μ1|u|2u+βuv2+f1(x),x∈Ω,−Δv+λ2v=μ2|v|2v+βu2v+f2(x),x∈Ω,u=v=0,x∈∂Ω. | (1.2) |
In the case where N≤3 and f1=f2=0, then the nonlinearity and the coupling terms in (1.2) are subcritical, and the existence of solutions has recently received great interest, for instance, see [3,4,5,6,7,8,9,10,11] for the existence of a (least energy) solution, and [12,13,14,15,16] for semiclassical states or singularly perturbed settings, and [17,18,19,20,21,22] for the existence of multiple solutions.
In the present paper we consider the case when N=4 and p=2∗=4 is the Sobolev critical exponent. If f1=f2=0, the paper [23] proved the existence of positive least energy solution for negative β, positive small β and positive large β.Furthermore, for the case λ1=λ2, they obtained the uniqueness of positive least energy solutions and they studied the limit behavior of the least energy solutions in the repulsive case β→−∞, and phase separation is obtained. Later, the paper [24] studied the high dimensional case N≥5. The paper [25] proved the existence of sign-changing solutions of (1.2). Recently, the paper [26] considered the system (1.2) with perturbation in dimension N≤3. By using Nehari manifold methods, the authors proved the existence of a positive ground state solution and a positive bound state solution. To the best of our knowledge, the existence of multiple nontrivial solution to the system (1.2) with critical growth(N=4) is still unknown. In the present paper we shall fill this gap.
Another motivation to study the existence of multiple nontrivial solution of (1.2) is coming from studying of the scalar critical equation. In fact, the second-order semilinear and quasilinear problems have been object of intensive research in the last years. In the pioneering work [27], Brezis and Nirenberg have studied the existence of positive solutions of the scalar equation
{−Δu=up+f,x∈Ω,u>0,x∈Ω,u=0,x∈∂Ω, | (1.3) |
where Ω is a bounded smooth domain in RN, N≥3, p=N+2N−2, f(x,u) is a lower order perturbation of up. Particularly, when f=λu, where λ∈R is a constant, they have discovered the following remarkable phenomenon: the qualitative behavior of the set of solutions of (1.7) is highly sensitive to N, the dimension of the space. Precisely, the paper [27] has shown that, in dimension N≥4, there exists a positive solution of (1.3), if and only if λ∈(0,λ1); while, in dimension N=3 and when Ω=B1 is the unit ball, there exists a positive solution of (1.7), if and only if λ∈(λ1/4,λ1), where λ1>0 is the first eigenvalue of −Δ in Ω. The paper [28] proved the existence of both radial and nonradial solutions to the problem
{−Δu=b(r)up+f(r,u),x∈Ω,r=|x|,u>0,x∈Ω,u=0,x∈∂Ω | (1.4) |
under some assumptions on b(r) and f(r,u), p=N+2N−2, where Ω=B(0,1) is the unit ball in RN. In the paper [29], G. Tarantello considered the critical case for (1.3). He proved that (1.3) has at least two solutions under some conditions of f: f≠0, f∈H−1 and
‖f‖H−1<cNSN4,cN=4N−2(N−2N+2)N+24, | (1.5) |
and
S=infu∈H10(Ω)∖{0}|∇u|22|u|24 | (1.6) |
is the best Sobolev constant of the imbedding from H10(Ω) to Lp(Ω). For more results on this direction we refer the readers to [30,31,32,33,34,35] and the references therein.
Motivated by the above works, in the present paper, we are interested in the critical coupled Schrödinger equations in (1.2) with λ1=λ2=0
{−Δu=μ1u3+βuv2+f1,x∈Ω,−Δv=μ2v3+βu2v+f2,x∈Ω,u=v=0,x∈∂Ω. | (1.7) |
where Ω is a smooth bounded domain in R4, Δ is the Laplace operator and p=2∗=4 is the Sobolev critical exponent, and μ1>0,μ2>0,0<β≤min{μ1,μ2}.
Obviously, the energy functional is denoted by
I(u,v)=12∫Ω(|∇u|2+|∇v|2)dx−14∫Ω(μ1|u|4+μ2|v|4+2βu2v2)dx−∫Ω(f1u+f2v)dx | (1.8) |
for (u,v)∈H=H10(Ω)×H10(Ω). So, the critical point of I(u,v) is the solution of the system (1.7). We shall fill the gap and generalize the results of [26] to the critical case. Our main tool here is the Nehari manifold method which is similar to the fibering method of Pohozaev's, which was first used by Tarantello [29].
We define thee Nehari manifold
N={(u,v)∈H|I′(u,v)(u,v)=0}. | (1.9) |
It is clear that all critical points of I lie in the Nehari manifold, and it is usually effective to consider the existence of critical points in this smaller subset of the Sobolev space. For fixed (u,v)∈H∖{(0,0)}, we set
g(t)=I(tu,tv)=A2t2−B4t4−Dt,t>0. |
where
A=∫Ω(|∇u|2+|∇v|2)dx,B=∫Ω(μ1|u|4+μ2|v|4+2βu2v2)dx,D=∫Ω(f1u+f2v)dx. | (1.10) |
The mapping g(t) is called fibering map. Such maps are often used to investigate Nehari manifold for various semilinear problem. By using the relationship of I and g(t), we can divide N into three parts as follow:
N+={(u,v)∈N|A−3B>0},N0={(u,v)∈N|A−3B=0},N−={(u,v)∈N|A−3B<0}. |
In order to get our results, we assume that fi satisfies
fi≠0,fi∈L43(Ω),|fi|43<S323√3K12,i=1,2, | (1.11) |
where K=max{μ1,μ2}, S is defined in (1.6). Then we have the following main results.
Theorem 1.1. Assume that 0<β≤min{μ1,μ2}, and f1,f2 satisfies (1.11). Then
infNI=infN+I=c0 | (1.12) |
is achieved at a point (u0,v0)∈N. Furthermore, (u0,v0) is a critical point of I.
Next we consider then following minimization problem
infN−I=c1. | (1.13) |
Then we have the following result.
Theorem 1.2. Assume that 0<β≤min{μ1,μ2}, and f1,f2 satisfies (1.11). Then c1>c0 and the infimum in (1.13) is achieved at a point (u1,v1)∈N−, which is the second critical point of I.
Remark 1.3. We point out that to the best of our knowledge, the existence of multiple nontrivial solution to the system (1.2) with critical growth(N=4) is still unknown. In the present paper we shall fill this gap and generalized the results of [26] to the critical case.
Throughout the paper, we shall use the following notation.
● Let (⋅,⋅) be the inner product of the usual Sobolev space H10(Ω) defined by (u,v)=∫Ω∇u∇vdx, and the corresponding norm is ‖u‖=(u,u)12.
● Let S=infu∈H10(Ω)∖{0}(|∇u|22/|u|24) be the best Sobolev constant of the imbedding from H10(Ω) to L4(Ω).
● |u|p is the norm of Lp(Ω) defined by |u|p=(∫Ω|u|pdx)1p, for 0<p<∞.
● Let ‖(u,v)‖2=∫Ω(|∇u|2+|∇v|2)dx be the norm in the space of H=H10(Ω)×H10(Ω).
● Let C or Ci(i=1,2,...) denote the different positive constants.
We shall use the variational methods to prove the main results. In this section we shall prove some basic results for the system (1.2). The next lemma states the purpose of the assuptions (1.11).
Lemma 2.1. Assume that the conditions of Theorem 1.1 hold. Thenfor every (u,v)∈H∖{(0,0)}, there exists a uniquet1>0 such that (t1u,t1v)∈N−. In particular, wehave
t1>t0:=[A3B]12 | (2.1) |
and g(t1)=maxt≥t0g(t), where A and B are given in (1.10). Moreover, if D>0, then there exists a unique t2>0, such that (t2u,t2v)∈N+, where D>0 is given in (1.10). In particular, one has
t2<t0andI(t2u,t2v)≤I(tu,tv),∀t∈[0,t1]. | (2.2) |
Proof. We first define the fibering map by
g(t)=A2t2−B4t4−Dt,t>0. |
Then we have
g′(t)=At−Bt3−D=Φ(t)−D. |
We deduce from Φ′(t)=0 that
t=t0=[A3B]12. |
If 0<t<t0, we have g″(t)=Φ′(t)>0, and if t>t0, one sees g″(t)=Φ′(t)<0. A direct computation shows that Φ(t) achieves its maximum at t0 and Φ(t0)=23√3A32B12.
From the assumption (1.11), Sobolev's and Hölder's inequalities, we infer that
D=∫Ω(f1u+f2v)dx≤|f1|43|u|4+|f2|43|v|4≤√(|f1|243+|f2|243)(|u|24+|v|24)≤√2max{|f1|43,|f2|43}(|u|24+|v|24)12<√2S323√3K12(|u|24+|v|24)12. | (2.3) |
On the other hand, since 0<β≤min{μ1,μ2}, it follows that
Φ(t0)=23√3A32B12=23√3(∫Ω(|∇u|2+|∇v|2)dx)32(∫Ω(μ1|u|4+μ2|v|4+2βu2v2)dx)12≥23√3S32(|u|24+|v|24)32K12(∫Ω(|u|4+|v|4+2βu2v2)dx)12≥2S323√3K12(|u|24+|v|24)32√2(|u|44+|v|44)12≥√2S323√3K12(|u|24+|v|24)32|u|24+|v|24=√2S323√3K12(|u|24+|v|24)12, |
where K=max{μ1,μ2}. Hence we get
g′(t0)=Φ(t0)−D>0andg′(t)→−∞,ast→+∞. | (2.4) |
Thus, there exists an unique t1>t0 such that g′(t1)=0. We infer from the monotonicity of Φ(t) that for t1>t0
g″(t1)=Φ′(t1)<0,t21Φ′(t1)=t21(A−3Bt21)<0. |
This implies that (t1u,t1v)∈N−. If D>0, then we have g′(0)=Φ(0)−D=−D<0. Furthermore, there exists an unique t2∈[0,t0] such that g′(t2)=0 and Φ(t2)=D. A direct computation shows that (t2u,t2v)∈N+ and I(t2u,t2v)≤I(tu,tv),∀t∈[0,t1].
Next we study the structure of N0.
Lemma 2.2. Let fi≠0(i=1,2) satisfy (1.11). Then for every(u,v)∈N∖{(0,0)}, we have
∫Ω(|∇u|2+|∇v|2)dx−3∫Ω(μ1|u|4+μ2|v|4+2βu2v2)dx≠0. | (2.5) |
Hence we can get the conclusion that N0={(0,0)}.
Proof. In order to prove that N0={(0,0)}, we only need to show that for (u,v)∈H∖{(0,0)}, g(t) has no critical point that is a turning point. We use contradiction argument. Assume that there exists ∃(u,v)≠(0,0) such that (t0u,t0v)∈N0 and t0>0. Thus, we get
g′(t0)=At0−Bt30−D=0andg″(t0)=A−3Bt20=0. |
Then we have t0=[A3B]12. This contradicts (2.4). This finishes the proof.
In the next lemma, we shall prove the properties of Nehari manifolds N.
Lemma 2.3. Let fi≠0(i=1,2) satisfy (1.11). For (u,v)∈N∖{(0,0)}, then there exist ε>0 anda differentiable function t=t(w,z)>0,(w,z)∈H,‖(w,z)‖<ε, and satisfying the following conditions
t(0,0)=1,t(w,z)((u,v)−(w,z))∈N,∀‖(w,z)‖<ε, |
and
<t′(0,0),(w,z)>=2∫Ω(∇u∇w+∇v∇z)dx−4∫Ω[μ1|u|2uw+μ2|v|2vz+β(uv2w+u2vz)]dx−∫Ω(f1w+f2z)dx∫Ω(|∇u|2+|∇v|2)dx−3∫Ω(μ1|u|4+μ2|v|4+2βu2v2)dx. |
Proof. We define F:R×H→R by
F(t,(w,z))=t‖∇(u−w)‖22+t‖∇(v−z)‖22−t3∫Ω(μ1|u−w|4+μ2|v−z|4+2β(u−w)2(v−z)2)dx−∫Ω(f1(u−w)+f2(v−z))dx. |
We deduce from Lemma 2.2 and (u,v)∈N that F(1,(0,0))=0. Moreover, one has
Ft(1,(0,0))=∫Ω(|∇u|2+|∇v|2)dx−3∫Ω(μ1|u|4+μ2|v|4+2βu2v2)dx≠0. |
By applying the implicit function theorem at point (1, (0, 0)), we can obtain the results.
In this section we are devoted to proving Theorem 1.1. We begin the following lemma for the property of infI.
Lemma 3.1. Let
c0=infNI=infN+I. |
Hence I is bounded from below in N and c0<0.
Proof. For (u,v)∈N, we have ⟨I′(u,v),(u,v)⟩=0. We infer from (1.10) that A−B−D=0. Thus, one deduces from (2.3) and Hölder inequality that
D<C(|u|24+|v|24)12≤C1(|∇u|22+|∇v|22)12=C1A12. |
Hence, one deduces that
I(u,v)=A2−B4−D=A4−3D4>A4−C2A12. |
Thus, the infimum c0 in N+ is bounded from below. Next we prove the upper bound for c0. Let wi∈H10(Ω)(i=1,2) be the solution for −Δw=fi,(i=1,2). So, for fi≠0 one sees that
∫Ω(f1w1+f2w2)dx=|∇w1|22+|∇w2|22>0. |
We let t2=t2(u,v)>0 as defined by Lemma 2.1. Thus, we infer that (t2w1,t2w2)∈N+ and
t22∫Ω(|∇w1|2+|∇w2|2)dx−t42∫Ω(μ1|w1|4+μ2|w2|4+2βw21w22)dx−t2∫Ω(f1w1+f2w2)dx=0. |
Furthermore, it follows from (2.2) that
c0=inf(u,v)∈N+I(u,v)≤I(t2w1,t2w2))<I(0,0)=0. |
This completes the proof.
The next lemma studies the properties of the infimum c0.
Lemma 3.2.
(1) The level c0 can be attained. That is, there exists(u0,v0)∈N+ such that I(u0,v0)=c0.
(2) (u0,v0) is a local minimum for I in H.
Proof. From Lemma 3.1, we can apply Ekeland's variational principle to the minimization problem, which gives a minimizing sequence {(un,vn)}⊂N such that
(i) I(un,vn)<c0+1n,
(ii) I(w,z)≥I(un,vn)−1n(|∇(w−un)|2+|∇(z−vn)|2),∀(w,z)∈N.
For n large enough, by Lemma 3.1 and (i)-(ii) of the above, we can get
∃C1>0,C2>0,0<C1≤|∇un|22+|∇vn|22≤C2. |
In the following we shall prove that ‖I′(un,vn)‖→0 as n→∞. In fact, we can apply Lemma 2.3 with (u,v)=(un,vn) and (w,z)=δ(Iu(un,vn),Iv(un,vn))‖I′(un,vn)‖(δ>0). Then can find tn(δ) such that
(wδ,zδ)=tn(δ)[(un,vn)−δ(Iu(un,vn),Iv(un,vn))‖I′(un,vn)‖]∈N. |
Thus, we infer from the condition (ii) that
I(un,vn)−I(wδ,zδ)≤1n(|∇(wδ−un)|2+|∇(zδ−vn)|2). | (3.1) |
On the other hand, by using Taylor expansion we have that
I(un,vn)−I(wδ,zδ)=(1−tn(δ))(I′(wδ,zδ),(un,vn))+δtn(δ)(I′(wδ,zδ),I′(un,vn)‖I′(un,vn)‖)+o(δ). |
Dividing by δ>0 and letting δ→0, we get
1n(2+t′n(0)(|∇un|2+|∇vn|2))≥−t′n(0)(I′(un,vn),(un,vn))+‖I′(un,vn)‖=‖I′(un,vn)‖. | (3.2) |
Combining (3.1) and (3.2) we conclude that
‖I′(un,vn)‖≤Cn(2+t′n(0)). |
We infer from Lemma 2.3 and (un,vn)⊂N that t′n(0) is bounded. That is,
|t′n(0)|≤C. |
Hence we obtain that
‖I′(un,vn)‖→0asn→∞. | (3.3) |
Therefore, by choosing a subsequence if necessary, we have that
(un,vn)⇀(u0,v0)inHandI′(u0,v0)=0, |
and
c0≤I(u0,v0)=14(|∇u0|22+|∇v0|22)−∫Ω(f1u0+f2v0)dx≤limn→∞I(un,vn)=c0. |
Consequently, we infer that
(un,vn)→(u0,v0)inH,I(u0,v0)=c0=infNI. |
From Lemma 2.1 and (3.3), we deduce that (u0,v0)∈N+.
(2) In order to get the conclusion, it suffices to prove that ∀(w,z)∈H,∃ε>0, if ‖(w,z)‖<ε, then I(u0−w,v0−z)≥I(u0,v0). In fact, notice that for every (w,z)∈H with ∫Ω(f1u+f2v)dx>0, we infer from Lemma 2.1 that
I(su,sv)≥I(t1u,t1v),∀s∈[0,t0]. |
In particular, for (u0,v0)∈N+, we have
t2=1<t0=[∫Ω(|∇u0|2+|∇v0|2)dx3∫Ω(μ1|u0|4+μ2|v0|4+2βu20v20)dx]12. |
Let ε>0 sufficiently small. Then we infer that for ‖(w,z)‖<ε
1<[∫Ω(|∇(u0−w)|2+|∇(v0−z)|2)dx3∫Ω(μ1|u0−w|4+μ2|v0−z|4+2β(u0−w)2(v0−z)2)dx]12=˜t0. | (3.4) |
From Lemma 2.3, let t(w,z)>0 satisfy t(w,z)(u0−w,v0−z)∈N for every ‖(w,z)‖<ε. Since t(w,z)→1 as ‖(w,z)‖→0, we can assume that
t(w,z)<˜t0,∀‖(w,z)‖<ε. |
Hence we obtain that t(w,z)(u0−w,v0−z)∈N+ and
I(s(u0−w),s(v0−z))≥I(t(w,z)(u0−w),t(w,z)(v0−z))≥I(u0,v0),∀0<s<˜t0. |
From (3.4) we can take s=1 and conclude
I(u0−w,v0−z)≥I(u0,v0),∀(w,z)∈H,‖(w,z)‖<ε. |
This finishes the proof.
Proof of Theorem 1.1. From Lemma 3.2, we know that (u0,v0) is the critical point of I.
In this section we focus on the proof of Theorem 1.2. The main difficulty here is the lack of compactness(due to the embedding H↪L4(Ω)×L4(Ω) is noncompact). Motivated by previous works of [27,29,37], we shall seek the local compactness. Then by using the Mountain pass principle to find the second nontrivial solution of equation (1.7). The pioneering paper [29] has used this methods to find the second solution of the scalar Schrödinger equation. To this purpose, we first begin with the following lemma to find the threshold to recover the compactness.
Lemma 4.1. For every sequence (un,vn)∈H satisfying
(i) I(un,vn)→cwithc<c0+14min{S2μ1,S2μ2}, where c0 is defined in (1.12), S is the best Sobolevconstant of the imbedding from H10(Ω) to L4(Ω),
(ii)‖I′(un,vn)‖→0asn→∞.
Then {(un,vn)} has a convergent subsequence. This means thatthe (PS)c condition holds for all levelc<c0+14min{S2μ1,S2μ2}.
Proof. From condition (i) and (ii), it is easy to verify that ‖(un,vn)‖ is bounded. So, for the subsequence {(un,vn)}(which we still call {(un,vn)}, we can find a (w0,z0)∈H such that (un,vn)⇀(w0,z0) in H. Then from the condition (ii), we obtain that
(I′(w0,z0),(w,z))=0,∀(w,z)∈H. |
That is, (w0,z0) is a solution in H. Moreover, (w0,z0)∈N and I(w0,z0)≥c0. Let
(un,vn)=(w0+wn,z0+zn). |
Then (wn,zn)⇀(0,0) in H. Then it suffices to prove that
(wn,zn)→(0,0)inH. | (4.1) |
We use the indirect argument. Assume that (4.1) does not hold. Then we divide the following three cases to find the contradiction.
Case 1: wn→0 and zn↛0 in H. Since ‖(un,vn)‖ is bounded, it follows that
∫Ωw2nz2ndx=o(1). |
by (1.7), we can get
c0+14min{S2μ1,S2μ2}>I(un,vn)=I(w0+wn,z0+zn)=I(w0,z0)+12∫Ω|∇zn|2dx−μ24∫Ω|zn|4dx+o(1)≥c0+12|∇zn|22−μ24|zn|44+o(1), |
and then
12|∇zn|22−μ24|zn|44<S24μ2. | (4.2) |
We infer from the condition (ii) that
o(1)=(I′(un,vn),(un,vn))=(I′(w0,z0),(w0,z0))+|∇zn|22−μ2|zn|44+o(1). |
That is, we get
|∇zn|22=μ2|zn|44+o(1). |
By using the embedding from H10(Ω) to L4(Ω), we get
μ2|zn|44=|∇zn|22≥S|zn|24+o(1). |
Since zn↛0, we infer that |zn|24≥S/μ2++o(1). That is,
|zn|44≥S2μ22+o(1). |
Hence we get
12|∇zn|22−μ24|zn|44=μ24|zn|44+o(1)≥14S2μ2. | (4.3) |
This contradicts with the fact (4.2).
Case 2: wn↛0 and zn→0 in H. This can be accomplished by using same argument as in the proof of the Case 1.
Case 3: wn↛0 and zn↛0 in H. Similar to the Case 1, we infer from condition (ii) that
o(1)=(I′(un,vn),(un,0))=|∇un|2−μ1|un|44−β∫Ωu2nv2ndx−∫Ωf1undx=(I′(w0,z0),(w0,0))+|∇wn|22−μ1|wn|44−β∫Ωw2nz2ndx+o(1). |
Then we have
|∇wn|22=μ1|wn|44+β∫Ωw2nz2ndx+o(1). | (4.4) |
One infers from Hölder and Sobolev inequality that
S|wn|24≤|∇wn|22=μ1|wn|44+β∫Ωw2nz2ndx+o(1)≤μ1|wn|44+β|wn|24|zn|24+o(1). | (4.5) |
Since wn↛0, we have
S≤μ1|wn|24+β|zn|24+o(1)≤μ1(|wn|24+|zn|24)+o(1). | (4.6) |
Similarly, we obtain that
S≤μ2|zn|24+β|wn|24+o(1)≤μ2(|wn|24+|zn|24)+o(1). | (4.7) |
Thus, we conclude that
|wn|24+|zn|24≥max{Sμ1,Sμ2}+o(1). | (4.8) |
On the other hand, we infer from the condition (ii) that
o(1)=(I′(un,vn),(un,vn))=|∇un|2+|∇vn|2−μ1|un|44−μ2|vn|44−2β∫Ωu2nv2ndx−∫Ωf1undx−∫Ωf2vndx=(I′(w0,z0),(w0,z0))+|∇wn|22+|∇zn|22−μ1|wn|44−μ2|zn|44−2β∫Ωw2nz2ndx+o(1). | (4.9) |
From (4.6)-(4.9), we deduce that
c0+14min{S2μ1,S2μ2}>I(un,vn)=I(w0+wn,z0+zn)=I(w0,z0)+12|∇wn|22+12|∇zn|22−14(μ1|wn|44+μ2|zn|44+2β∫Ωw2nz2ndx)+o(1)≥c0+14(μ1|wn|44+μ2|zn|44+2β∫Ωw2nz2ndx)+o(1)≥c0+S4(|wn|24+|zn|24)+o(1)≥c0+14max{S2μ1,S2μ2}+o(1). |
This is a contradiction.
In order to applying Lemma 4.1 to get the compactness, we need to prove the following inequality
c1=infN−I<c0+14min{S2μ1,S2μ2}. |
Let
uε(x)=εε2+|x|2ε>0, x∈R4 |
be an extremal function for the Sobolev inequality in R4. Let uε,a=u(x−a) for x∈Ω and the cut-off function ξa∈C∞0(Ω) with ξa≥0 and ξa=1 near a. We set
Uε,a(x)=ξa(x)uε,a(x),x∈R4. |
Following [37], we let Ω1⊂Ω be a positive measure set such that u0>0,v0>0, where c0=I(u0,v0) is given in Theorem 1.1. Then we have the following conclusion.
Lemma 4.2. For every R>0, and a.e. a∈Ω1, there existsε0=ε0(R,a)>0, such that
min{I(u0+RUε,a,v0),I(u0,v0+RUε,a)}<c0+14min{S2μ1,S2μ2} | (4.10) |
for every 0<ε<ε0.
Proof. As in [37], a direct computation shows that
I(u0+RUε,a,v0)=12|∇u0|22+R∫Ω∇u0∇Uε,adx+R22|∇Uε,a|22+12|∇v0|22−μ14(|u0|44+R4|Uε,a|44+4R∫Ωu30Uε,adx+4R3∫ΩU3ε,au0dx)−μ24|v0|44−β2∫Ω(u20v20+2Ru0v20Uε,a+R2U2ε,av20)dx−∫Ω(f1u0+f2v0)dx−R∫ΩfUε,adx+o(ε). | (4.11) |
We infer from [27] that
|∇Uε,a|22=F+O(ε2)and|Uε,a|44=G+O(ε4), | (4.12) |
where
F=∫R4|∇u1(x)|2dx,G=∫R4dx(1+|x|2)4,S=FG12 |
If we let u0=0 outside Ω, then
∫ΩU3ε,au0dx=∫R4u0ξa(x)ε3(ε2+|x−a|2)3dx=ε∫R4u0ξa(x)1ε4φ(xε)dx |
where
φ(x)=1(1+|x|2)3∈L1(R4). |
Set
E=∫R41(1+|x|2)3dx. |
Then we can derive
∫R4u0ξa(x)1ε4φ(xε)dx→u0(a)E. |
Since (u0,v0) is the critical point of I, it follows that
∫Ω(|∇u0|2+|∇v0|2)dx−∫Ω(μ1u40+μ2v40+2βu20v20)dx−∫Ω(f1u0+f2v0)dx=0. | (4.13) |
We infer from (4.11)-(4.13) that
I(u0+RUε,a,v0)=I(u0,v0)+R22F−R44μ1G−μ1R3∫ΩU3ε,au0dx−βR22∫ΩU2ε,av20dx+o(ε)≤c0+R22F−R44μ1G−μ1εR3Eu0(a)+o(ε). | (4.14) |
In order to get the upper bound of (4.14), we define
q1(s)=F2s2−μ1G4s4−kεs3,k=μ1Eu0(a)>0, |
and
q2(s)=F2s2−μ1G4s4. |
It is easy to get the maximum of q2(s) is achieved at s0=(Fμ1G)12. Let the maximum of q1(s) is achieved at sε, so we can let sε=(1−δε)s0, and get δε→0(ε→0). Substituting sε=(1−δε)s0 into q′1(s)=0, we can get
F−F(1−δε)2=3s0(1−δε)kε. |
As in [29], we infer that
δε∼ε,ε→0. |
Then we can get the upper bound estimation of I(u0+RUε,a,v0):
I(u0+RUε,a,v0)≤c0+R22F−R44μ1G−kεR3+o(ε)≤c0+[(1−δε)s0]22F−[(1−δε)s0]44μ1G−kε[(1−δε)s0]3+o(ε)=c0+(s202F−s404μ1G)+(s40μ1G−s20F)δε−kεs30+o(ε)<c0+S24μ1+o(ε). |
Thus, for ε0>0 small, we get
I(u0+RUε,a,v0)<c0+S24μ1. |
Similarly, we obtain that
I(u0,v0+RUε,a)<c0+S24μ2. |
So, we prove
min{I(u0+RUε,a,v0),I(u0,v0+RUε,a)}<c0+14min{S2μ1,S2μ2},∀0<ε<ε0. |
This finishes the proof.
Without loss of generality, from above Lemma 4.2 we can assume
I(u0+RUε,a,v0)<c0+S24μ1,R>0,∀0<ε<ε0. |
Now we are ready to prove Theorem 1.2.
Proof of Theorem 1.2. It is clear that there exists an uniqueness of t1>0 such that
(t1u,t1v)∈N−andI(t1u,t1v)=maxt≥t0I(t1u,t1v), ∀(u,v)∈H, ‖(u,v)‖=1. |
Moreover, t1(u,v) is a continuous function of (u,v), and N− divides H into two components H1 and H2, which are disconnect with each other. Let
H1={(u,v)=(0,0)or(u,v):‖(u,v)‖<t1((u,v)‖(u,v)‖)}andH2={(u,v):‖(u,v)‖>t1((u,v)‖(u,v)‖)}. |
Obviously, we have H∖N−=H1∪H2. Furthermore, we obtain that N+⊂H1 for (u0,v0)∈H1. We choose a constant C0 such that
0<t1(u,v)≤C0,∀‖(u,v)‖=1. |
In the following we deduce that
(w,z)=(u0+R0Uε,a,v0)∈H2, | (4.15) |
where R0=(1F|C20−‖(u0,v0)‖2)12+1. Since
‖(w,z)‖2=‖(u0,v0)‖2+R20|∇Uε,a|2+2R0∫Ω|∇u0||∇Uε,a|dx=‖(u0,v0)‖2+R20F+o(1)>C20≥[t1((w,z)‖(w,z)‖)]2 |
for ε>0 small enough. We fix ε>0 small to make both (4.10) and (4.15) hold by the choice of R0 and a∈Ω1. Set
Γ1={γ∈C([0,1],H):γ(0)=(u0,v0),γ(1)=(u0+R0Uε,a,v0)}. |
We take h(t)=(u0+tR0Uε,a,v0). Then h(t)∈Γ1. From Lemma 4.1, we conclude that
c′=infh∈Γ1maxt∈[0,1]I(h(t))<c0+S24μ1. |
Since the range of every h∈Γ1 intersect N−, we have
c1=infN−I≤c′<c0+S24μ1. | (4.16) |
Set
Γ2={γ∈C([0,1],H):γ(0)=(u0,v0),γ(1)=(u0,v0+R0Uε,a)}. |
By using the same argument, we can get similar results
c″=infh∈Γ2maxt∈[0,1]I(h(t))<c0+S24μ2. |
Moreover, since the range of every h∈Γ2 intersect N−, we have
c1=infN−I≤c″<c0+S24μ2. | (4.17) |
Combining (4.16) and (4.17), we obtain that
c1<c0+14min{S2μ1,S2μ2}. |
Next by using Mountain-Pass lemma(see [36]) to obtain that there exist {(un,vn)}⊂N− such that
I(un)→c1,‖I′((un,vn))‖→0. |
From Lemma 4.1, we can obtain a subsequence (still denote {(un,vn)}) of {(un,vn)}, and (u1,v1)∈H such that
(un,vn)→(u1,v1)inH. |
Hence, we get (u1,v1) is a critical point for I,(u1,v1)∈N− and I(u1,v1)=c1.
The authors thank the referees's nice suggestions to improve the paper. This work was supported by NNSF of China (Grants 11971202, 12071185), Outstanding Young foundation of Jiangsu Province No. BK20200042.
The authors declare there is no conflicts of interest.
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