Ground state of a K-component critical system with linear and nonlinear couplings: an attractive case

Abstract

Consider the system:
\begin{equation*}
\left\{\aligned&-\Delta u_i+\mu_i u_i=\nu_iu_i^{2^*-1}+\beta\sum_{j=1,j\not=i}^ku_j^{\frac{2^*}{2}}u_i^{\frac{2^*}{2}-1}+\lambda \sum_{j=1,j\not=i}^ku_j\quad\text{in }\Omega,\\
&u_i>0\quad\text{in }\Omega,\\
&u_i=0\quad\text{on }\partial\Omega,\quad i=1,2,\cdots,k,\endaligned\right.
\end{equation*}
where k≥2k\geq2, \Omega\subset\bbr^N(N\geq3) is a bounded domain, 2∗=2NN−22^*=\frac{2N}{N-2}, \mu_i\in\bbr and νi>0\nu_i>0 are constants, and β,λ>0\beta,\lambda>0 are parameters. By showing a unique result of the limit system, we prove existence and nonexistence results of ground states to this system by variational methods, which generalize the results in \cite{CZ131,PSW17}. Concentration behaviors of ground states for β,λ\beta,\lambda are also established.

Full text

Adv. Nonlinear Stud. 2019; 19(3): 595–623
Research Article
Yuanze Wu*
Ground States of a K-Component Critical
System with Linear and Nonlinear Couplings:
The Attractive Case
https://doi.org/10.1515/ans-2019-2049
Received January 7, 2019; revised April 29, 2019; accepted May 30, 2019
Abstract: Consider the system










−∆ui+μiui=νiu2∗−1
i+β
k

j=1,j=i
u2∗
2
ju2∗
2−1
i+λ
k

j=1,j=i
ujin Ω,
ui>0in Ω,
ui=0on ∂Ω, i=1,2,...,k,
where k≥2,Ω⊂ℝN(N≥3) is a bounded domain, 2∗=2N
N−2,μi∈ℝand νi>0are constants, and β,λ>0
are parameters. By showing a unique result of the limit system, we prove existence and nonexistence results
of ground states to this system by variational methods, which generalize the results in [7, 18]. Concentration
behaviors of ground states for β,λare also established.
Keywords: Elliptic System, Critical Sobolev Exponent, Ground State Solution, Variational Method,
Asymptotic Property
MSC 2010: 35B09, 35B33, 35B40, 35J50
||
Communicated by: David Ruiz
1 Introduction
In this paper, we consider the system










−∆ui+μiui=νiu2∗−1
i+β
k

j=1,j=i
u2∗
2
ju2∗
2−1
i+λ
k

j=1,j=i
ujin Ω,
ui>0in Ω,
ui=0on ∂Ω, i=1,2,...,k,
(1.1)
where k≥2,Ω⊂ℝN(N≥3) is a bounded domain with a smooth boundary ∂Ω,2∗=2N
N−2is the critical
Sobolev exponent, μi∈ℝand νi>0for all i=1,2,...,kare constants, and β,λ>0are two parameters.
Let 𝔽=diag(−∆+μ1,−∆+μ2,...,−∆+μk)
*Corresponding author: Yuanze Wu, School of Mathematics, China University of Mining and Technology, Xuzhou 221116,
P. R. China, e-mail: wuyz850306@cumt.edu.cn. http://orcid.org/0000-0002-8263-7878

596 |Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings
and
ℤ=




ν1β β . . . β
β ν2β...β
β β ν3
...β
.
.
...........
.
.
β β β . . . νk





,𝕀=




0 1 1 . . . 1
1 0 1 ...1
1 1 0 ...1
.
.
...........
.
.
1 1 1 . . . 0





.
Then system (1.1) is equivalent to the following equation in H=(H1
0(Ω))k:
𝔽u=λ∇1
2uT𝕀u+∇1
2∗(u2∗
2)Tℤu2∗
2,(1.2)
where u=(u1,u2,...,uk)∈His a vector function, up=(up
1,up
2,...,up
k), and uTis the transposition of the
vector u. Thus system (1.1) is the generalization of the following well-known Brezis–Nirenberg equation:
−∆u=λu +|u|2∗−2uin Ω,
u=0on ∂Ω, (1.3)
from the viewpoint of linear algebra. Therefore, similar to the well-known Brezis–Nirenberg equation (1.3),
it appears from (1.2) that the parameter λplays an important role in studying the existence and nonexistence
results of system (1.1). Now our nonexistence results which can be stated as follows reveal such a property.
Theorem 1.1. Let α1>0be the first eigenvalue of −∆in H1
0(Ω). Then system (1.1) has no solution in one of the
following three cases:
(1) min{μi}≤−α1,
(2) min{μi}>−α1and λ≥λ1, where λ1is the unique solution of
k

j=1
λ
α1+μj+λ=1,(1.4)
(3) min{μi}>0,0<λ≤λ0, and Ωis star-shaped, where λ0is the unique solution of
k

j=1
λ
μj+λ=1.(1.5)
Remark 1.1. By (1.5), it is easy to see that λ0→0with λ0
min{μi}→+∞as min{μi}→0+. For the sake of sim-
plicity, we re-define
λ0=

the unique solution of (1.5) for min{μi}>0,
0for min{μi}≤0.(1.6)
Let
J(u)=k

i=11
2(‖∇ui‖2
2+μi‖ui‖2
2)−νi
2∗‖ui‖2∗
2∗−2β
2∗
Ω
L(u)−λ
Ω
Q(u),
where H=(H1
0(Ω))kis the Hilbert space with the inner product
⟨u,v⟩=k

i=1
Ω∇ui∇vi
and ui,viare respectively the i-th component of uand v, and ‖u‖p=∫Ω|u|p1
pis the usual norm in Lp(Ω)for
all p≥1,
L(u)=k

i,j=1,i<j|ui|2∗
2|uj|2∗
2,(1.7)
Q(u)=k

i,j=1,i<j
uiuj.(1.8)

Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings |597
Clearly, it is easy to see that J(u)is of class C1in H=(H1
0(Ω))k. Let
C=inf

u∈M
J(u)(1.9)
with
M=u∈H\{0}|J󸀠(u)u=0.(1.10)
Then Cis well defined, and M=0contains all nonzero critical points of J(u).
Definition 1.1. Let v∈Hbe a critical point of J(u), that is, J󸀠(v)=0in H−1, where J󸀠(u)is the Fréchet deriva-
tive of J(u)and H−1is the dual space of H. Then vis called nontrivial if vi=0for all i=1,2,...,k;vis called
nonzero if v= 0in H;vis called semi-trivial if vis nonzero but not nontrivial; vis called positive if vi>0for
all i=1,2,...,k;vis called a ground state if vis nontrivial and J(v)=C.
Clearly, the positive critical points of J(u)are the solutions of (1.1). Thus we could use the variational method
to study the existence of the solutions of system (1.1).
Definition 1.2. vis called a ground state solution of (1.1) if vis a positive ground state of J(u).
Since the nonlinearities of J(u)are of critical growth in the sense of Sobolev embedding, it is well known
that the major diculty in proving the existence of the solutions of system (1.1) by the variational method
is the lack of compactness. A typical idea in overcoming such diculty, which is contributed by Brezis and
Nirenberg in [2], is to control the energy level to be less than a special threshold which is always generated by
the energy level of ground states to the pure critical “limit” functional. In such an argument, the negativity of
the subcritical terms in the energy functional plays an important role in controlling the energy value to be less
than the threshold. Even though this idea has already been used in elliptic system (1.1) in [18, 24] and the
references therein for those only with linear couplings and in [5, 7, 23] and the references therein for those
only with nonlinear couplings, to apply this idea to study system (1.1) is still nontrivial, and some new ideas
are needed since it has both linear and nonlinear couplings. Indeed, we note that the methods for the critical
systems with only linear couplings in the recent work [18, 24] and the references therein are invalid for our
situation since the least energy of the single equation is not the threshold for system (1.1) with β>0. Thus
we cannot control the least energy level Cto be less than the threshold by testing it with a semi-trivial ground
state. On the other hand, the methods for the critical systems with only nonlinear couplings in [5, 7, 23]
and the references therein are also invalid for our situation since the subcritical terms of J(u)can only be
negative for a very special vector function u. Thus we also cannot control the least energy level Cto be less
than the threshold by testing it with the ground state of the pure critical “limit” functional. To overcome such
diculty, our idea is to drive a uniqueness result for the ground state of the limit functional (see Lemma 3.3
for more details). To the best of our knowledge, such a unique result has only been obtained for N=4and
k=2(cf. [5]), whose proof strongly depends on the precise algebraic expression of the least energy value
of the limit functional (see the proof of [5, Theorem 1.2]). However, even for the case N≥5and k=2, the
precise algebraic expression of the least energy value of the limit functional is not easy to obtain, which
causes the similar energy estimates to be much more complex by applying the same ideas (cf. [7]). In the
current paper, we develop a more simple and direct method to prove such a unique result for all N≥4with
β>0large enough by applying the variational argument to the minimizing problem (3.7) and the implicit
function theorem to the related system (3.11) (see Propositions 3.2 and 3.3 for more details).
As a by-product of our study of Propositions 3.2 and 3.3, we actually obtain a result for the elliptic system






−∆ui=νi|ui|2∗−2ui+β
k

j=1,j=i|uj|2∗
2|ui|2∗
2−2uiin ℝN,
ui∈D1,2(ℝN)i=1,2,...,k,
(1.11)
which can be stated as follows.
Theorem 1.2. Let N≥4. Then the ground state solution of (1.11) must be the “least energy” synchronized type
solution of the form U=(t1Uε,z,t2Uε,z,..., tkUε,z),

598 |Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings
where
Uε,z(x)=[N(N−2)ε2]N−2
4
(ε2+|x−z|2)N−2
2
is the Talanti function that satisfies −∆U=U2∗−1in ℝNand t=(t1,t2,..., tk)is a constant vector with ti>0
for all i=1,2,...,kin one of the following cases:
(1) N=4and β∈(0,min{νi})∪(max{νi},+∞),
(2) N≥5and β>0.
Moreover, there exists βk>0such that the ground state solution must be unique for β>βk.
Remark 1.2. Theorem 1.2 generalizes [4, Theorem 1.5] and [7, Theorem 1.6] to arbitrary k≥2. Moreover,
Theorem 1.2 also improves [7, Theorem 1.6] in the sense that it asserts that the ground state of (1.11) must
be the “least energy” synchronized type solution for all β>0and the ground state solution must be unique
for β>0large enough in the case of N≥5. We also believe that Theorem 1.2 can be used in other studies on
elliptic systems since (1.11) can be regarded as the limit system of many other elliptic systems.
Let us come back to our study on (1.1) now. Before we state our existence results, we assume without loss
of generality that μ1≤μ2≤⋅⋅⋅≤μk. Note that, in the symmetric case μ1=μ2=⋅⋅⋅=μk=μ, system (1.1) is
always expected to have the synchronized type solutions. Thus our existence results can be stated as follows.
Theorem 1.3. Let −α1<μ1≤μ2≤⋅⋅⋅≤μkand λ0<λ<λ1, where α1is the first eigenvalue of −∆in H1
0(Ω),
and λ0,λ1are respectively given by (1.6) and Theorem 1.1.
(a) If μ1=μ2=⋅⋅⋅=μk=μ, then (1.1) has the synchronized type solutions if and only if ν1=ν2=⋅⋅⋅=νk=ν.
(b) System (1.1) has a ground state solution in one of the following cases:
(1) N=3and μk<0with μk+α1>0small enough,
(2) N≥4and μk<0,
(3) N≥4and β>βk, where βkis given by Proposition 3.3.
Remark 1.3. (1) From Theorem 1.3, it can be seen that, dierent from the system that is coupled with only
nonlinear couplings (cf. [5, 7, 13, 17, 21]), further linear couplings make system (1.1) have the synchronized
type solutions in a more symmetric situation.
(2) By Theorems 1.1 and 1.3, λ1, the first eigenvalue of the equation
𝔽u=λ∇1
2uT𝕀u
in His the upper bound of λfor the existence of solutions to system (1.1), while λ0, the upper bound of λsuch
that the L2norm of J(u)is positive definite for min{μi}≥0(see Proposition A.2), is the lower bound of λfor
the existence of solutions to system (1.1) if Ωis star-shaped. Such properties coincide with the well-known
Brezis–Nirenberg equation (1.3).
We also study the concentration behavior of the ground state solution of (1.1) for the parameters βand λ
in this paper. For this purpose, we denote the ground state solution and its energy value, respectively, by
uλ,β=(uλ,β
1,uλ,β
2,...,uλ,β
k)and C(λ,β). In considering the case β→0, by Theorem 1.3, we need the further
conditions −α1<μ1≤μ2≤⋅⋅⋅≤μk<0. Thus, by a standard perturbation argument, it is easy to show that
uλ,β→uλ,0strongly in H=(H1
0(Ω))kas β→0up to a subsequence. Therefore, we shall mainly study the
cases β→+∞,λ→λ0and λ→λ1by Theorem 1.3 in what follows.
We first consider the case β→+∞. By a standard argument, it is not very dicult to show that uλ,β→0
strongly in H=(H1
0(Ω))kand C(λ,β)→0as β→+∞. To capture the precise decay rate of C(λ,β)as β→+∞,
we turn to consider the equivalent minimization problem
C(λ,β)=∑k
i=1(‖∇uλ,β
i‖2
2+μi‖uλ,β
i‖2
2)−2λ∫ΩQ(uλ,β)N
2
N∑k
i=1νi‖uλ,β
i‖2∗
2∗+2β∫ΩL(uλ,β)N−2
2
=inf

u∈H\{ 
0}∑k
i=1(‖∇ui‖2
2+μi‖ui‖2
2)−2λ∫ΩQ(u)N
2
N∑k
i=1νi‖ui‖2∗
2∗+2β∫ΩL(u)N−2
2
.(1.12)

Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings |599
Recall that uλ,β→0strongly in H=(H1
0(Ω))k. Thus ‖uλ,β
i‖2∗
2∗=o(‖∇uλ,β
i‖2
2+μi‖uλ,β
i‖2
2). This yields
C(λ,β)∼∑k
i=1(‖∇uλ,β
i‖2
2+μi‖uλ,β
i‖2
2)−2λ∫ΩQ(uλ,β)N
2
N2β∫ΩL(uλ,β)N−2
2∼Cβ−N−2
2(1.13)
as β→+∞. On the other hand, to capture the precise decay rate of uλ,β, it is natural to re-scale uλ,βin
a suitable way based on the precise energy estimate. Now our results on this aspect can be stated as follows.
Theorem 1.4. Let −α1<μ1≤μ2≤⋅⋅⋅≤μkand λ0<λ<λ1, where α1is the first eigenvalue of −∆in H1
0(Ω),
and λ0,λ1are respectively given by (1.6) and Theorem 1.1. Then C(λ,β)=C(λ)β−N−2
2+o(β−N−2
2)as β→+∞in
one of the following cases:
(1) N=3and μk<0with μk+α1>0small enough,
(2) N≥4.
Here
C(λ)=inf

u∈H\{ 
0}∑k
i=1(‖∇ui‖2
2+μi‖ui‖2
2)−2λ∫ΩQ(u)N
2
N2∫ΩL(u)N−2
2
,(1.14)
where L(u)and Q(u)are given by (1.7) and (1.8), respectively. If we have −α1<μ1≤μ2≤⋅⋅⋅≤μk≤0, then
vλ,β→vλ,∞strongly in Has β→+∞up to a subsequence, where vλ,β
i=βN−2
4uλ,β
ifor all i=1,2,...,kand
vλ,∞is a ground state solution of the system













−∆ui+μiui=k

j=1,j=i
u2∗
2
ju2∗
2−1
i+λ
k

j=1,j=i
ujin Ω,
ui>0in Ω,
ui=0on ∂Ω, i=1,2,...,k,
(1.15)
We next consider the case λ→λ1. Similar to the case β→+∞, by a similar argument to the one used for [24,
Theorem 1.10], we can show that uλ,β→0strongly in H=(H1
0(Ω))kand C(λ,β)→0as λ→λ1. However, the
decay rate of C(λ,β)as λ→λ1cannot be simply conjectured as in (1.13) for the case β→+∞. Now, by re-
scaling uλ,βtwice and combining minimizing problems (1.12) and (A.2), we can obtain the following results,
which surprisingly yield that, by a suitable re-scaling, uλ,βwill strongly converge to a nonzero eigenfunction
of the first eigenvalue λ1of the equation 𝔽u=λ∇(1
2uT𝕀u)in H.
Theorem 1.5. Let −α1<μ1≤μ2≤⋅⋅⋅≤μkand λ0<λ<λ1, where α1is the first eigenvalue of −∆in H1
0(Ω),
and λ0,λ1are respectively given by (1.6) and Theorem 1.1. Then we have
C(λ,β)=1
N[(λ1−λ)P(β)]N
2+o((λ1−λ)N
2)as λ→λ1
in one of the following cases:
(1) N=3and μk<0with μk+α1>0small enough,
(2) N≥4and μk<0,
(3) N≥4and β>βk, where βkis given by Proposition 3.3.
Here
P(β)≡2∫ΩQ(u)
∑k
i=1νi‖ui‖2∗
2∗+2β∫ΩL(u)N−2
N
is a constant that depends only on βfor all u∈N∗
1\{0}with N∗
1given by Proposition A.1, while L(u)and Q(u)
are respectively given by (1.7) and (1.8). Moreover, we also have 
wλ,β→
w0,βstrongly in Has λ→λ1, where
wλ,β
i=1
(λ1−λ)N
4
uλ,β
ifor all i=1,2,...,kand 
w0,β∈N∗
1\{0}.
Remark 1.4. To the best of our knowledge, the precise decay estimate of C(λ,β)and the strong convergence of
the re-scaled functions vλ,βand 
wλ,β, stated in Theorems 1.4 and 1.5, respectively, for β→+∞and λ→λ1,
are completely new in studies on the elliptic system. Moreover, we also observe in Theorems 1.4 and 1.5 that

600 |Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings
systems (1.15) and (A.1) are the limit systems of (1.1) under some suitable scalings as β→+∞and λ→λ1,
respectively, which is also novel to the best of our knowledge.
We finally consider the case λ→λ0. As we stated in Remark 1.3, λ0is the lower bound of λfor the existence
of solutions to system (1.1) in the case min{μi}≥0if Ωis star-sharped. Recall that the ground state solution
of the well-known Brezis–Nirenberg equation (1.3) is a spiked solution as λ→0, where 0is the lower bound
for the existence of solutions if Ωis star-shaped (cf. [14]). Thus it is natural to conjecture that uλ,βis also
a spiked solution as λ→λ0at least for min{μi}≥0. Our next result reveals such a property of uλ,β.
Theorem 1.6. Let −α1<μ1≤μ2≤⋅⋅⋅≤μkand λ0<λ<λ1, where α1is the first eigenvalue of −∆in H1
0(Ω),
and λ0,λ1are respectively given by (1.6) and Theorem 1.1.
(a) If −α1<μ1<0, then uλ,β→u0,βstrongly in Has λ→0such that J(u0,β)=C(β)in one of the following
cases:
(1) N=3and μk<0with μk+α1>0small enough,
(2) N≥4and μk<0,
(3) N≥4and β>βk, where βkis given by Proposition 3.3.
Here C(β)=inf 
MJ(u)with
J(u)=k

i=11
2(‖∇ui‖2
2+μi‖ui‖2
2)−νi
2∗‖ui‖2∗
2∗−2β
2∗
Ω
L(u)and 
M=u∈H\{0}| J󸀠(u)u=0,
where L(u)is given by (1.7). Moreover, if
(a) either N=4and β>max{νj},
(b) or N≥5,
then u0,βmust be nontrivial.
(b) If 0≤μ1≤μ2≤⋅⋅⋅≤μk,N≥4and β>βk, where βkis given by Proposition 3.3, then we have Aλ→+∞
and vλ,β
i→tiUε,zstrongly in D1,2(ℝN)for all i=1,2,...,kas λ→λ0, where Uε,zand t=(t1,t2,..., tk)
are respectively given by (3.8) and Proposition 3.2 and
vλ,β
i=1
(Aλ)N−2
2
uλ,β
ix
Aλ+yλfor all i=1,2,...,k
with Aλ=maxi=1,2,...,k{‖uλ,β
i‖2
N−2
∞,ℝN}.
Remark 1.5. (1) Theorem 1.6 implies that if min{μi}≥0, then the ground state solution of (1.1) is actually
a spiked solution and system (1.11) is the limit system of (1.1) as λ→λ0, where λ0, given by (1.6), is the
lower bound of λfor the existence of solutions to system (1.1) in this case if Ωis star-shaped. Such properties
coincide with the results obtained in [14] for the well-known Brezis–Nirenberg equation (1.3). However, if
{μi}<0, then, by Theorem 1.6, λ0=0, given by (1.6), will not be the lower bound of λfor the existence of
solutions to system (1.1), and it seems to be very interesting to find out the lower bound of λfor the existence of
solutions to system (1.1) and the limit system of (1.1) in such a case. On the other hand, it is also worthwhile
to point out that our method, based on Theorem 1.2, to prove Theorem 1.6 is dierent from that in [3], in
which a two-component critical system with only nonlinear couplings was considered.
(2) Compared with Theorems 1.4, 1.5 and Theorem 1.6, it can be seen that, even though we need to re-
scale uλ,βfor both the vanishing case and the blow-up case in capturing the precise decay or blow-up rate,
the re-scaling manners are quite dierent for the vanishing case and the blow-up case. The major dierence
is that we do not need to re-scale the domain Ωin the vanishing case.
We close this section by recalling some recent studies on critical system (1.1). The recent studies on critical
system (1.1) for λ=0appear to start from [5], where, by regarding such a system as equation (1.3) coupled
with nonlinear couplings and establishing several fundamental energy estimates, the Brezis–Nirenberg type
variational argument has been generalized to the case of elliptic systems to obtain a ground state solution
of system (1.1) for λ=0,k=2,ν1,ν2>0and −α1<μ1,μ2<0with βbeing in a wide range. Here α1is the
first eigenvalue of −∆in H1
0(Ω). The following related studies can be seen in [6, 7, 23, 25] and the references

Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings |601
therein. System (1.1) with λ=0,k=2,ν1,ν2>0and μ1=μ2=0in ℝN, that is,








−∆u1=ν1|u1|2∗−2u1+β|u2|2∗
2|u1|2∗
2−2u1in ℝN,
−∆u2=ν2|u2|2∗−2u1+β|u1|2∗
2|u2|2∗
2−2u2in ℝN,
u1,u2∈D1,2(ℝN),
(1.16)
has also been studied in recent years. In [5, 7], (1.16) was treated as the limit system of system (1.1) for λ=0,
k=2,ν1,ν2>0and −α1<μ1,μ2<0. In [8], by focusing on the conformal invariance, several interesting
results of system (1.16), including phase separation, were obtained, and radial and nonradial solutions of
system (1.16) were obtained by using the bifurcation method in [10, 11], while infinitely many positive non-
radial solutions were obtained by using the reduction method in [12]. The spiked solutions of system (1.1) for
λ=0and k=2were also studied in [3, 20], where it was proved that the ground state solution will blow-up
and concentrate at some x0∈∂Ωfor a wide range of β. We also remark that some other spiked solutions,
for example, the Bahri–Coron type, of a critical system similar to (1.1) or (1.16), which are only coupled
with nonlinear couplings, have been studied in [17, 19] and the references therein. On the other hand, the
recent studies on critical system (1.1) for β=0and k=2can be found in [4, 18] and the references therein,
where such systems were always considered to be the Brezis–Nirenberg equation (1.3) coupled with linear
couplings. By using the variational method, some existence and nonexistence results were established. In the
very recent work [24], by introducing a similar viewpoint of (1.2), some existence and nonexistence results
were obtained for system (1.1) with β=0and arbitrary k≥2also by using the variational method.
Organization of the Paper. For the convenience of the readers, we sketch the organization of this paper here.
In Section 2, we shall study the nonexistence of solutions of (1.1) by directly proving Theorem 1.1. In Sec-
tion 3, we will devote ourselves to the existence of solutions of (1.1). For the sake of clarity, we divide this
section into two parts, where, in the first part, we consider the synchronized type solutions in the symmetric
case, while, in the second part, we study the ground state solution in the general case. In Section 4, we prove
various kinds of the concentration behavior of the ground state solution of (1.1) stated in Theorems 1.4–1.6.
Notations. Throughout this paper, Cand C󸀠are indiscriminately used to denote various absolute positive
constants. We also list some notations used frequently below.
‖u‖p
p=
Ω|u|pdx,‖u‖p
p,ℝN=
ℝN|u|pdx,
u=(u1,u2,...,uk),t∘u=(t1u1,t2u2,...,tkuk),𝔹r(x)=y∈ℝN||y−x|<r,
tu=(t u1,tu2,...,tuk),un=(un
1,un
2,...,un
k),ℝ+=(0,+∞),
L(u)=k

i,j=1,i<j|ui|2∗
2|uj|2∗
2,Q(u)=k

i,j=1,i<j
uiuj,
H=(H1
0(Ω))k,(ℝN)+=x=(x1,x2,...,xN)∈ℝN|xN>0.
2 Nonexistence Results
In this section, we will establish the nonexistence results that are summarized in Theorem 1.1.
Proof of Theorem 1.1. (1) Without loss of generality, we assume that μ1≤−α1. Suppose now that system (1.1)
has a solution u=(u1,u2,...,uk), and let φ1be the corresponding eigenfunction of α1. Then, multiplying
system (1.1) with v=(φ1,0,0,...,0)and integrating by parts, we have
0≥(α1+μ1)
Ω
u1φ1=
Ω∇u1∇φ1+μ1u1φ1=
Ω
ν1u2∗−1
1φ1+β
k

j=2
u2∗
2
ju
2∗
2−1
1φ1+λ
k

j=2
ujφ1>0,
which is impossible.

602 |Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings
(2) By Proposition A.1,
k

j=1
λ
α1+μj+λ=1
has a unique solution λ1, which is also the first eigenvalue of the operator T=F∘Iwith
F=diag(−∆+μ1)−1,(−∆+μ2)−1,...,(−∆+μk)−1,
I=




0 1 1 . . . 1
1 0 1 ...1
1 1 0 ...1
.
.
...........
.
.
1 1 1 . . . 0





.
Now let us also suppose that system (1.1) has a solution u=(u1,u2,...,uk). Let v1=(e1φ1,e2φ1,...,ekφ1)
be the corresponding eigenfunction of λ1given by Proposition A.1. Then, by Proposition A.1 once more, we
can choose ei>0for all i=1,2,...,k. Now, multiplying system (1.1) with v1and integrating by parts, we
have from λ≥λ1that
2λ1
Ω
k

i,j=1,i<j
ejuiφ1=
Ω
k

i=1
ei(∇ui∇φ1+μiuiφ1)
=
Ω
k

i=1
eiνiu2∗−1
i+β
k

j=1,j=i
u2∗
2
ju2∗
2−1
iφ1+2λ
Ω
k

i,j=1,i<j
ejuiφ1
>2λ1
Ω
k

i,j=1,i<j
ejuiφ1,
which is impossible.
(3) By Proposition A.2,
k

j=1
λ
μj+λ=1
has a unique solution λ0∈(0,λ1)for min{μj}>0. We also suppose now that system (1.1) has a solution
u=(u1,u2,...,uk)for 0<λ≤λ0. Then, by the classical regularity theories, we know that the uiare all
of class C2. Now, without loss of generality, we assume that Ωis star-shaped for 0. Then, by the Pohozaev
identity (cf. [4]), it can be seen that
N−2
2N
k

j=1
Ω|∇uj|2+1
2N
k

j=1
∂Ω(x,n)|∇uj|2=−1
2
Ωk

j=1
μju2
j−2λ
k

i,l=1;i<l
uiul
+N−2
2N
Ωk

j=1
νju2∗
j+2β
k

i,j=1,i<j
u2∗
2
ju2∗
2
i,
where nis the unit outer normal vector of Ω. It follows from u∈Hbeing a solution to system (1.1) that
1
2N
k

j=1
∂Ω(x,n)|∇uj|2=−1
N
Ωk

j=1
μj|uj|2−2λ
k

i,l=1;i=l
uiul.
Since min{μj}>0and 0<λ≤λ0, by Proposition A.2, we known that the quadratic form

Ωk

j=1
μj|uj|2−2λ
k

i,l=1;i=l
uiul≥0,
which contradicts the fact that Ωis star-shaped for 0.

Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings |603
3 Existence Results
Recall that, without loss of generality, we assume that μ1≤μ2≤⋅⋅⋅≤μk. Thus, owing to the nonexistence
results given by Theorem 1.1, we always consider the case −α1<μ1≤μ2≤⋅⋅⋅≤μkand λ0<λ<λ1in this
section, where α1is the first eigenvalue of −∆in H1
0(Ω)and λ0,λ1are given by (1.6) and Theorem 1.1.
3.1 The Symmetric Case μ1=μ2=⋅⋅⋅ =μk=μ
By (1.4), (1.5) and (1.6), we have λ1=μ+α1
k−1and
λ0=

μ
k−1for 0<μ,
0for 0≥μ.
Since λ0<λ<λ1, for the sake of clarity, we re-denote λ=μ
k−1+α1
k−1λ, where
λ∈

(0,1)for μ≥0,
−μ
α1,1for μ<0.(3.1)
Proposition 3.1. Let μ1=μ2=⋅⋅⋅=μk=μ. Then system (1.1) has synchronized type solutions if and only if
ν1=ν2=⋅⋅⋅=νk=ν.
Proof. If ν1=ν2=⋅⋅⋅=νk=ν, then it is easy to see that system (1.1) has synchronized type solutions. Next
we shall show that ν1=ν2=⋅⋅⋅=νk=νis also the necessary condition for the existence of the synchronized
type solutions. Let v=(t1v,t2v,...,tkv)be a synchronized type solution of system (1.1), where vis a function
satisfying some equations and ti>0for all i=1,2,...,k. Then we must have
tj(−∆v+μv)=νjt2∗−1
j+β
k

i=1,i=j
t2∗
2
it2∗
2−1
jv2∗−1+μ
k−1+α1
k−1λk

i=1,i=j
tivin Ω(3.2)
for all j=1,2,...,k. It follows that
−∆v=∑k
j=1νjt2∗−1
j+β∑k
i=1,i=jt2∗
2
it2∗
2−1
j
∑k
j=1tj
v2∗−1+α1λv in Ω.
Thus v=sw, where s>0and wis a positive solution of
−∆w=w2∗−1+α1λw in Ω, w∈H1
0(Ω).(3.3)
Recall that λis given by (3.1), it is well known (cf. [2]) that (3.3) has a positive solution. Now, by (3.2) once
more, we can see that
tjw2∗−1+α1λtjw=tj(−∆w)
=s2∗−2νjt2∗−1
j+β
k

i=1,i=j
t2∗
2
it2∗
2−1
jw2∗−1−tjμw
+μ
k−1+α1
k−1λk

i=1,i=j
tiw
for all j=1,2,...,k. Thus we must have from μ
k−1+α1
k−1λ>0and s>0that
0=k

i=1,i=j(ti−tj)and νjt2∗
2
j+β
k

i=1,i=j
t2∗
2
i=t2−2∗
2
js2−2∗
for all j=1,2,...,k, which implies t1=t2=⋅⋅⋅=tkand ν1=ν2=⋅⋅⋅=νk.

604 |Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings
3.2 The General Case μ1≤μ2≤⋅ ⋅ ⋅ ≤μk
In this section, we will use the Nehari manifold Mgiven by (1.10) to prove the existence of a ground state
solution of (1.1). Since the functional
k

i=1
1
2(‖∇ui‖2
2+μi‖ui‖2
2)−λ
Ω
Q(u)
is positive definite for −α1<μ1≤μ2≤⋅⋅⋅≤μkand λ0<λ<λ1by Proposition A.1, it is standard (cf. [15,
Lemma 2.3]) to drive the following result, where Q(u)is given by (1.8), α1is the first eigenvalue of −∆in
H1
0(Ω)and λ0,λ1are respectively given by (1.6) and Theorem 1.1.
Lemma 3.1. Let −α1<μ1≤μ2≤⋅⋅⋅≤μkand λ0<λ<λ1, where α1is the first eigenvalue of −∆in H1
0(Ω)and
λ0,λ1are respectively given by (1.6) and Theorem 1.1. Then, for every u∈H\{0}, there exists a unique t>0
such that tu=(tu1,tu2,...,tuk)∈M. Moreover, we also have C>0, where C=inf 
u∈MJ(u).
To use Mgiven by (1.10), we must also attain the following lemma, which yields that system (1.1) is still
strongly coupled.
Lemma 3.2. Let −α1<μ1≤μ2≤⋅⋅⋅≤μkand λ0<λ<λ1, where α1is the first eigenvalue of −∆in H1
0(Ω)and
λ0,λ1are respectively given by (1.6) and Theorem 1.1. Suppose that v=(v1,v2,...,vk)is the minimizer of
J(u)on M. Then 
w=(|v1|,|v2|,...,|vk|)is a ground state solution of system (1.1).
Proof. Let v=(v1,v2,...,vk)be the minimizer of J(u)on M. Since |∇|vi||≤|∇vi|a.e. in ℝNand
k

i,j=1,i<j
vivj≤k

i,j=1,i<j|vi||vj|,
by Lemma 3.1, it is standard (cf. [24, Proposition 5.2]) to show that 
w=(w1,w2,...,wk)is also a minimizer
of J(u)on M, where wi=|vi|for all i=1,2,...,k. Clearly, Mis a C1manifold. Moreover, since 2∗>2, it is
also standard (cf. [24, Proposition 5.2]) to show that Mis a natural constraint. Thus 
wis a critical point of
J(u)by the method of the Lagrange multiplier. It follows that 
wsatisfies the system










−∆wi+μiwi=νiw2∗−1
i+β
k

j=1,j=i
w2∗
2
jw2∗
2−1
i+λ
k

j=1,j=i
wjin Ω,
wi≥0in Ω,
wi=0on ∂Ω, i=1,2,...,k.
(3.4)
It follows from the maximum principle that, for every i=1,2,...,k, we have either wi>0or wi≡0. Sup-
pose that 
wis not a solution of system (1.1). Then there exists at least one j∈{1,2,...,k}such that wj≡0.
Without loss of generality, we may assume that wj>0for j=1,2,...,i0and wj≡0for j=i0+1,...,kwith
i0∈{1,2,...,k−1}. Then (3.4) is equivalent to





















−∆wi+μiwi=νiw2∗−1
i+β
i0

j=1,j=i
w2∗
2
jw2∗
2−1
i+λ
i0

j=1,j=i
wjin Ω,
i0

i=1
wi=0in Ω,
wi>0in Ω,
wi=0on ∂Ω, i=1,2,...,i0,
which is impossible. Thus 
w=(|v1|,|v2|,...,|vk|)must be a ground state of system (1.1).
By Lemma 3.1, the Nehari manifold Mis a natural constraint. It follows from the Ekeland variational principle
that there exists a (PS) sequence {un}⊂Mat the least energy level C. Here Cis given by Lemma 3.1. Note that
the embedding map from H1
0(Ω)to L2∗(Ω)is not compact. Thus we shall use the Brezis–Nirenberg argument

Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings |605
(cf. [2]) to recover the compactness of {un}, which leads us to first study the minimizing problem
ck=inf
Nk
Ek(u).(3.5)
Here
Ek(u)=k

i=1
1
2‖∇ui‖2
2,ℝN−νi
2∗‖ui‖2∗
2∗,ℝN−2β
2∗
ℝN
L(u)
is a functional defined in D=(D1,2(ℝN))k, and Dis a Hilbert space equipped with the inner product
⟨u,v⟩ℝN=k

i=1
ℝN∇ui∇vi,
L(u)is given by (1.7), ‖u‖p,ℝN=∫ℝN|u|p1
pis the usual norm in Lp(ℝN)for all p≥2and
Nk=u∈D\{0}|E󸀠
k(u)u=0.(3.6)
Proposition 3.2. Let
dk=inf
Pk
Gk(t),(3.7)
where
Gk(t)=k

i=1t2
i
2−νi|ti|2∗
2∗−2β
2∗
k

i,j=1,i<j|tj|2∗
2|ti|2∗
2,
Pk=t∈ℝk\{0}k

i=1(t2
i−νi|ti|2∗)−2β
k

i,j=1,i<j|tj|2∗
2|ti|2∗
2=0.
Then ck=dkSN
2is attained by Uif and only if
U=(t1Uε,z,t2Uε,z,..., tkUε,z),
where Sis the best Sobolev embedding constant from D1,2(ℝN)to L2∗(ℝN),
Uε,z(x)=[N(N−2)ε2]N−2
4
(ε2+|x−z|2)N−2
2
(3.8)
is the Talanti function that satisfies −∆U=U2∗−1in ℝNand t=(t1,t2,..., tk)satisfies (3.11). Moreover, if
(1) either N=4and β>max{νj},
(2) or N≥5,
then ti>0for all i=1,2,...,k.
Proof. By a standard argument (cf. [17]), we can see that
ck=inf

u∈(D1,2(ℝN))k\{ 
0}∑k
i=1‖∇ui‖2
2,ℝNN
2
N∑k
i=1νi‖ui‖2∗
2∗,ℝN+2β∫ℝ4L(u)N−2
2
,(3.9)
which, together with the Hölder and Sobolev inequalities, implies
ck≥inf

u∈D\{ 
0}∑k
i=1‖ui‖2
2∗,ℝNN
2
N∑k
i=1νi‖ui‖2∗
2∗,ℝN+2β∑k
i<j‖ui‖2∗
2
2∗,ℝN‖uj‖2∗
2
2∗,ℝNN−2
2
SN
2.(3.10)
Here L(u)is given by (1.7). Clearly, we also have
dk=inf

t∈ℝk\{ 
0}∑k
i=1t2
iN
2
N∑k
i=1νi|ti|2∗+2β∑k
i,j=1,i<j|tj|2∗
2|ti|2∗
2N−2
2
,

606 |Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings
which can be attained by some twith ti≥0for all i=1,2,...,kand ti>0for some i. By the method of
Lagrange’s multiplier, talso satisfies the system













ti=νit2∗−1
i+β
k

j=1,j=it2∗
2
jt2∗
2−1
ifor all i=1,2,...,k,
ti≥0and
k

i=1ti>0for all i=1,2,...,k.
(3.11)
Thus ck=dkSN
2can be attained by U=(t1Uε,z,t2Uε,z,..., tkUε,z),
where Uε,zis given by (3.8). Suppose now that ckis attained by some nonzero v. Then, by the Hölder and
Sobolev inequalities, we must have from (3.10) and ck=dkSN
2that ‖∇vi‖2
2,ℝN=S‖vi‖2
2∗,ℝNfor all i=1,2, . . ., k,
which implies either vi=Uε,zfor some ε>0and z∈ℝNor vi=0. Moreover, we also have that
s=(‖v1‖2∗,ℝN,‖v2‖2∗,ℝN,...,‖vk‖2∗,ℝN)
attains dk. Thus ck=dkSN
2is attained by Uif and only if
U=(t1Uε,z,t2Uε,z,..., tkUε,z),
where Uε,zis given by (3.8) and t=(t1,t2,..., tk)satisfies (3.11). In what follows, we shall borrow some
ideas from [1] to show that ti>0for all i=1,2,...,kin one of the following cases:
(1) N=4and β>max{νj},
(2) N≥5.
We set m=1,2,...,k−1and lm={l1,l2,...,lm}⊂{1,2,...,k}with l1<l2<⋅⋅⋅<lm. We also define
clm,m=inf
Nlm,m
Elm,m(u).
Here
Elm,m(u)=m

i=1
1
2‖∇uli‖2
2,ℝN−νli
2∗‖uli‖2∗
2∗,ℝN−2β
2∗
ℝN
Llm,m(u),
Llm,m(u)=m

i,j=1,i<j|uli|2∗
2|ulj|2∗
2,
and
Nlm,m=u∈D\{0}|E󸀠
lm,m(u)u=0.
If ck<clm,mfor all m=2,3,...,k−1and lm={l1,l2,...,lm}⊂{1,2,...,k}with l1<l2<⋅⋅⋅<lm, then
we can see that ti>0for all i=1,2,...,k. Without loss of generality, we assume ck−1=min{clm,m}, which
is attained by 
w=(w1,w2,...,wk−1). Let 
w=(w1,w2,...,wk−1,0). Now, similar to the proof of [1, Theo-
rem 2.2], by considering 
w+s
ϕ, we can show that there exists a unique
t(s)=1−(1+o(1))2βs 2∗
2∫ℝN∑k−1
i=1|wi|2∗
2|ϕk|2∗
2−s2‖∇ϕ‖2
2,ℝN
(2∗−2)∑k−1
i=1‖∇wi‖2
2,ℝN
(3.12)
for s>0small enough such that
t(s)( 
w+s
ϕ)=t(s)w1,...,t(s)wk−1,t(s)sϕ∈Nk,
where 
ϕ=(0,0,...,0,ϕ). Using the fact that 2∗<4for N≥5, we have from (3.12) that
Jkt(s)( 
w+s
ϕ)=[t(s)]2
Nk−1

i=1‖∇wi‖2
2,ℝN+s2‖∇ϕ‖2
2,ℝN
=1
N
k−1

i=1‖∇wi‖2
2,ℝN−4βs 2∗
2∫ℝN∑k−1
i=1|wi|2∗
2|ϕk|2∗
2
(2∗−2)∑k−1
i=1‖∇wi‖2
2,ℝN+O(s2)
<1
N
k−1

i=1‖∇wi‖2
2,ℝN=Jk−1(
w)

Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings |607
for s>0small enough. In the case N=4, we see from the fact that 
w=(w1,w2,...,wk−1)is a critical point
of Ek−1(u)that ‖∇wi‖2
2,ℝ4<β
ℝ4
k−1

j=1|wj|2|wi|2for all i=1,2,...,k−1
if β>max{νj}. It follows that β>β∗
k, where β∗
kis given by
β∗
k=inf‖∇ϕ‖2
2,ℝ4ϕ∈H1(ℝ4),
ℝ4
k−1

i=1|wi|2|ϕ|2=1.
Since wi∈D1,2(ℝ4), the eigenvalue β∗
kcan be attained by some ϕ∗
k. Thus we have
t(s)=1−(1+o(1))(2β−β∗
k)s2∫ℝ4∑k−1
i=1|wi|2|ϕ∗
k|2dx
2∑k−1
i=1‖∇wi‖2
2,ℝN
as s→0,
by taking ϕ=ϕ∗
kin (3.12). It follows that
Jkt(s)( 
w+s
ϕ)=t(s)2
4k−1

i=1‖∇wi‖2
2,ℝ4+s2‖∇ϕ∗
k‖2
2,ℝ4
=1
4
k−1

i=1‖∇wi‖2
2,ℝ4−2(β−β∗
k)
ℝ4
k−1

i=1|wi|2|ϕ∗
k|2s2+o(s2)
<Jk−1(
w)
for β>max{νj}and s>0small enough. This yields ck<min{clm,m}if
(1) either N=4and β>max{νj},
(2) or N≥5,
which completes the proof.
We re-denote t=(t1,t2,..., tk), which is given by Proposition 3.2, by tβ=(tβ
1,tβ
2,..., tβ
k).
Proposition 3.3. Let N≥4. Then there exists βk>0such that
tβ=(tβ
1,tβ
2,..., tβ
k)
is the unique solution of (3.11) for β>βk. Moreover, βk=max{νj}for N=4.
Proof. We first consider the case N=4. In this case, letting si=t2
i, system (3.11) is equivalent to the linear
system 

1=νisi+β
k

j=1,j=i
sjfor all i=1,2,...,k,
si>0for all i=1,2,...,k.(3.13)
By the Cramer rule, linear system (3.13) has a unique solution s=(s1,s2,...,sk)with
si=1
(νi−β)1+∑k
j=1
β
νi−βfor all i=1,2,...,k,
for β>max{νj}. In what follows, let us consider the case N≥5. Since tβ=(tβ
1,tβ
2,..., tβ
k)is a solution of
system (3.11), we have (tβ
i)2=νi(tβ
i)2∗+β
k

j=1,j=i(tβ
j)2∗
2(tβ
i)2∗
2for all i=1,2,...,k.(3.14)
This yields
k

i=1(tβ
i)4−2∗
2≥min{νj}+(k−1)βk

i=1(tβ
i)2∗
2,

608 |Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings
which, together with the fact that 2<2∗<4and the Young inequality, implies
k

i=1(tβ
i)2∗
2≤C1
min{νj}+(k−1)βN
4.
It follows that
k

i=1(tβ
i)2=O(β−N−2
2)(3.15)
for β>0large enough. Let sβ
i=βN−2
4tβ
ifor all i=1,2,...,k. Then, by (3.14) and (3.15), {sβ}is bounded for
βlarge enough in ℝk, and they satisfy
sβ
i=νi
β(sβ
i)2∗−1+k

j=1,j=i(sβ
j)2∗
2(sβ
i)2∗
2−1for all i=1,2,...,k.(3.16)
Without loss of generality, we assume that sβ→s0in ℝkas β→+∞up to a subsequence. Note that tβ
i>0
for all i=1,2,...,kand β>0by Proposition 3.2. Thus, by (3.14), we can see that s0is a solution of the
system 





(s0
i)2=k

j=1,j=i(s0
j)2∗
2(s0
i)2∗
2,
s0
i≥0for all i=1,2,...,k,
(3.17)
which is equivalent to 





(s0
i)4−2∗
2=k

j=1,j=i(s0
j)2∗
2,
s0
i≥0for all i=1,2,...,k.
(3.18)
System (3.18) yields (s0
i)4−2∗
2−(s0
l)4−2∗
2=(s0
l)2∗
2−(s0
i)2∗
2for all i,l=1,2,...,kwith i=l. Since 2<2∗<4
for N≥5, we must have s0
i=s0
lfor all i,l=1,2,...,kwith i=l, which, together with (3.18), implies
s0
i=(k−1)−1
2∗−2for all i,l=1,2,...,k. Let
s0=(k−1)−1
2∗−2,(k−1)−1
2∗−2,...,(k−1)−1
2∗−2.
Then we also have that s0is the unique solution of (3.17). Since sβ→s0in ℝkas β→+∞for every subse-
quence, we have sβ→s0in ℝkas β→+∞. Let
Γ(s,σ)=Γ1(s,σ), . . . , Γk(s,σ)
with
Γi(s,σ)=si−σν is2∗−1
i−k

j=1,j=i
s2∗
2
js2∗
2−1
ifor all i=1,2,...,k.
Since s0is the unique solution of (3.17), we have Γ(s0,0)=0. Moreover,
∂Γj
sj(s0,0)=4−2∗
2and ∂Γj
si(s0,0)=−2∗
2for all i,j=1,2,...,kwith i=j.
It follows from a direct calculation that
det∂Γj
si(s0,0)i,j=1,2,...,k=2k−2(4−k2∗),
which, together with k≥2and 2<2∗, implies det∂Γj
si(s0,0)i,j=1,2,...,k=0. By the implicit function theo-
rem, we can see from (3.16) that (sβ,1
β)is the unique curve bifurcated from (s0,0). Let tβbe any solution
of (3.11). Then, repeating the above argument as used for tβ, we can show that sβ
i=βN−2
4tβ
i→(k−1)−1
2∗−2as
β→+∞for all i=1,2,...,k. Thus there exists βk>0such that tβ=(tβ
1,tβ
2,..., tβ
k)is the unique solution
of (3.11) for β>βk.

Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings |609
Now we can give the proof of Theorem 1.2.
Proof of Theorem 1.2. Since the Cramer rule also works for (3.13) in the case 0<β<min{νi}, the conclusions
follow from Propositions 4.1 and 4.2.
With Propositions 3.2 and 3.3, we can also estimate C, which is given by Lemma 3.1 as follows.
Lemma 3.3. Let −α1<μ1≤μ2≤⋅⋅⋅≤μkand λ0<λ<λ1, where α1is the first eigenvalue of −∆in H1
0(Ω)and
λ0,λ1are respectively given by (1.6) and Theorem 1.1. Then C<ckin one of the following cases:
(1) N=3and μk<0with μk+α1>0small enough,
(2) N≥4and μk<0,
(3) N≥4and β>βk, where βkis given by Proposition 3.3.
Here ckis given by (3.5).
Proof. Without loss of generality, we assume 0∈Ω. Choose ρ>0such that 𝔹2ρ(0)⊂Ω, and let ψ∈C2
0(𝔹2ρ(0))
be a radial symmetric cut-o function satisfying 0≤ψ(x)≤1and ψ(x)≡1in 𝔹ρ(0). Furthermore, we define
Vε(x)=ψ(x)Uε,0(x)with Uε,0given by (3.8). Then it is well known (cf. [2]) that
‖∇Vε‖2
2=SN
2+O(εN−2),‖Vε‖2∗
2∗=SN
2+O(εN),(3.19)
and ‖Vε‖2
2≥

Cε2+O(εN−2),N≥5,
Cε2|ln ε|+O(ε2),N=4.(3.20)
For the sake of clarity, we consider the following two cases.
The Case N≥3and μk<0.Recall that −α1<μ1≤μ2≤⋅⋅⋅≤μk. Thus we have that the quadratic form
k

i=1
μia2
i−2λ
k

i,j=1,i<j
aiaj
is always negative definite for μk<0and λ>0. Now we choose the test function of Cby
Vε=(t1Vε,t2Vε,..., tkVε),
where tiare given by Proposition 3.2. Then, by a standard argument (cf. [2, 22]), we have from (3.19), (3.20)
that
C≤ck+

−Cε2+O(εN−2),N≥5,
−Cε2|ln ε|+O(ε2),N=4.
For the case N=3, we choose ρsuch that 𝔹ρ(x0)⊂Ω⊂𝔹2ρ(x0)for some x0∈Ω, and we take the cut-o
function ψ(|x−x0|)=cos(π
2|x−x0|)for |x−x0|≤ρ. Moreover, we also require dist(∂𝔹2ρ(x0),∂Ω)>0. With-
out loss of generality, we assume that x0=0. Then, by a similar calculation to the one used for [2, (1.27)
and (1.29)], we can see that
‖∇Vε‖2
2=S3
2+εω
ρ

0|ψ󸀠(r)|2+O(ε2)=S3
2+εωπ2
4ρ2
ρ

0|ψ(r)|2+O(ε2),
and ‖Vε‖2
2=εω
ρ

0|ψ(r)|2+O(ε2),
where ωis the area of the unit ball in ℝ3. Note that, by 𝔹ρ(x0)⊂Ω⊂𝔹2ρ(x0)and dist(∂𝔹2ρ(x0),∂Ω)>0,
we have α1>π2
4ρ2. Thus, letting μk+α1>0small enough and testing Cby Vε=(t1Vε,t2Vε,..., tkVε)once
more, we can use a similar calculation to the one used in [2, Lemma 1.3] to show that C≤ck−Cε +O(ε2)for
N=3. Hence, taking ε>0small enough, we always have C<ckfor μk<0.

610 |Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings
The Case N≥4and μk≥0.Since λ0<λ<λ1, by Proposition A.2, we can see that the quadratic form
k

i=1
μia2
i−2λ
k

i,j=1,i<j
aiaj
is non-positive definite. Moreover, a direct calculation (cf. [24]) yields that it has a unique negative eigenvalue
−γsatisfying
1=k

j=1
λ
μj+λ−γ.
It follows that there exists a constant vector a=(a1,a2,...,ak)such that
k

i=1
μia2
i−2λ
k

i,j=1,i<j
aiaj=−γ.
Now let us choose Vε=(a1Vε,a2Vε,...,akVε)as the test function. Then, by (3.19) and (3.20), it is easy to
see that there exists
s=∑k
i=1a2
i+o(ε)
∑k
i=1νia2∗
i+2β∑k
i,j=1,i=ja2∗
2
ia2∗
2
j1
2∗−2
such that sVε=(sa1Vε,sa2Vε,...,sakVε)∈M. It follows from (3.19) and (3.20) that
C≤J(sVε)=k

i=1
(sai)2
2(SN
2+O(εN−2))−γs2
2‖Vε‖2
2
−k

i=1
νi(sai)2∗
2∗+2β
2∗
k

i,j=1,i=j(sa i)2∗
2(saj)2∗
2(SN
2+O(εN)).(3.21)
Since 2∗>2, it is standard (cf. [23, Lemma 2.3]) to show that Gk(t), given by (3.7), has a global maximum
point t=(t1,t2,..., tk)in (ℝ+)k. Clearly, tis also a solution of system (3.11). By Proposition 3.3, t=tfor
β>βk, where tis given by Proposition 3.2. This, together with Proposition 3.2 once more, yields
k

i=1
(sai)2
2−k

i=1
νi(sai)2∗
2∗+2β
2∗
k

i,j=1,i=j(sa i)2∗
2(saj)2∗
2SN
2≤ck,
which, together with (3.21), implies C≤ck−C‖Vε‖2
2+O(εN−2)for β>βk. Thanks to (3.20), we have
C≤ck+

−Cε2+O(εN−2),N≥5,
−Cε2|ln ε|+O(ε2),N=4,
for β>βk. Thus we can obtain C<ckfor β>βkby taking ε>0small enough.
We close this section by the proof of Theorem 1.3.
Proof of Theorem 1.3. Conclusion (a) immediately follows from Proposition 3.1. In what follows, let us prove
conclusion (b). Recall that {un}⊂Mis a (PS) sequence of J(u)at the least energy level C. Since
k

i=1
1
2(‖∇ui‖2
2+μi‖ui‖2
2)−λ
Ω
Q(u)
is positive definite for −α1<μ1≤μ2≤⋅⋅⋅≤μkand λ0<λ<λ1given by Proposition A.1, it is standard to
show that {un}is bounded in H, where Q(u)is given by (1.8). Without loss of generality, we assume un⇀u0
weakly in Has n→∞. In what follows, we claim that u0= 0in one of the cases (1)–(3). Suppose to the
contrary that u0=0. Then, by the Sobolev embedding theorem, we may assume that un→0strongly in
(L2(Ω))kas n→∞, which implies
k

i=1‖∇un
i‖2
2=k

i=1
νi‖un
i‖4
4+2β
Ω
L(un)+on(1),(3.22)

Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings |611
where L(u)is given by (1.7). Since {un}⊂Mand
k

i=1
1
2(‖∇ui‖2
2+μi‖ui‖2
2)−λ
Ω
Q(u)
is positive definite by Proposition A.1, it is standard to show that ∑k
i=1‖∇un
i‖2
2≥C+on(1)by the Hölder and
Sobolev inequalities. Thus, by (3.22), there exists tn→1as n→∞such that
tnun=(tnun
1,tnun
2,...,tnun
k)∈Nk.
Here Nkis given by (3.6), and we regard H1
0(Ω)⊂D1,2(ℝN)by letting u≡0outside Ω. It follows that
C+on(1)=J(un)=Ek(tnun)+on(1)≥ck+on(1),
which contradicts Lemma 3.3 in one of the cases (1)–(3). Thus we must have u0= 0in these three cases. Now,
by a standard argument (cf. [24, Proposition 5.2]), we can see that u0is the minimizer of J(u)on M. Then,
by Lemma 3.2, system (1.1) has a ground state solution in these cases.
4 The Asymptotic Properties
This section is devoted to the concentration behavior of the ground state solution of system (1.1). For the sake
of clarity, we denote the ground state solution obtained by Theorem 1.3 by uλ,β. The corresponding energy
value Cwill be re-denoted by C(λ,β).
4.1 The Case β→ +∞
Without loss of generality, we always assume that β>βk, where βkis given by Proposition 3.3. Moreover, by
the definition of C(λ,β)given by (1.9), it is easy to see that
C(λ,β)=∑k
i=1(‖∇uλ,β
i‖2
2+μi‖uλ,β
i‖2
2)−2λ∫ΩQ(uλ,β)N
2
N∑k
i=1νi‖uλ,β
i‖2∗
2∗+2β∫ΩL(uλ,β)N−2
2
(4.1)
=inf

u∈H\{ 
0}∑k
i=1(‖∇ui‖2
2+μi‖ui‖2
2)−2λ∫ΩQ(u)N
2
N∑k
i=1νi‖ui‖2∗
2∗+2β∫ΩL(u)N−2
2
(4.2)
where L(u)and Q(u)are respectively given by (1.7) and (1.8). For the sake of clarity, we also re-denote ckand
dkas ck(β)and dk(β), respectively, where ckand dkare given by (3.5) and Proposition 3.2, respectively. By
Proposition 3.2, dk(β)is attained by some tβsatisfying (3.11). Moreover, we also have dk(β)=1
N∑k
i=1(tβ
i)2. By
(3.11), we have tβ→0as β→+∞. Thus we also have dk(β)→0as β→+∞. Note that, by Proposition 3.2
once more, we also have ck(β)=dk(β)SN
2. This, together with Lemma 3.3, yields C(λ,β)→0as β→+∞.
Clearly, we also have uλ,β→0strongly in Has β→+∞.
Lemma 4.1. Let −α1<μ1≤μ2≤⋅⋅⋅≤μkand λ0<λ<λ1, where α1is the first eigenvalue of −∆in H1
0(Ω)and
λ0,λ1are respectively given by (1.6) and Theorem 1.1. Then we have
k

i=1
νi‖uλ,β
i‖2∗
2∗=oβ
Ω
L(uλ,β)as β→+∞
in one of the following cases:
(1) N=3and μk<0with μk+α1>0small enough,
(2) N≥4.

612 |Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings
Proof. Since uλ,β→0strongly in Has β→+∞, we have from the Sobolev inequality that
k

i=1
νi‖uλ,β
i‖2∗
2∗≤Ck

i=1‖∇uλ,β
i‖2
22∗
2=ok

i=1‖∇uλ,β
i‖2
2as β→+∞.(4.3)
On the other hand, since −α1<μ1≤μ2≤⋅⋅⋅≤μkand λ0<λ<λ1, by Proposition A.1, we have
k

i=1(‖∇uλ,β
i‖2
2+μi‖uλ,β
i‖2
2)−2λ
Ω
Q(uλ,β)≥1−λ
λ1k

i=1(‖∇uλ,β
i‖2
2+μi‖uλ,β
i‖2
2)
≥1−λ
λ11−|μ1|
α1k

i=1‖∇uλ,β
i‖2
2.(4.4)
This, together with (4.3) and the fact that uλ,βis the ground state solution of (1.1), yields
1−λ
λ11−|μ1|
α1+o(1)k

i=1‖∇uλ,β
i‖2
2≤2β
Ω
L(uλ,β),
which implies ∑k
i=1νi‖uλ,β
i‖2∗
2∗=o(β∫ΩL(uλ,β))as β→+∞.
With Lemma 4.1 in hands, we can obtain the following precise estimates of C(λ,β).
Proposition 4.1. Let −α1<μ1≤μ2≤⋅⋅⋅≤μkand λ0<λ<λ1, where α1is the first eigenvalue of −∆in H1
0(Ω)
and λ0,λ1are respectively given by (1.6) and Theorem 1.1. Then C(λ,β)=C(λ)β−N−2
2+o(β−N−2
2)as β→+∞in
one of the following cases:
(1) N=3and μk<0with μk+α1>0small enough,
(2) N≥4.
Here C(λ)is given by (1.14).
Proof. For every 0<ε<1, we choose vε∈H\{0}such that
N2
Ω
L(vε)N−2
2=1and k

i=1(‖∇vε
i‖2
2+μi‖vε
i‖2
2)−2λ
Ω
Q(vε)N
2<C(λ)+ε.
By using a similar estimate to (4.4), we can show that
k

i=1‖∇vε
i‖2
2≤(C(λ)+1)2
N
1−λ
λ11−|μ1|
α1.
Now, by (4.2) and the Sobolev inequality, we can see that
C(λ,β)≤∑k
i=1(‖∇vε
i‖2
2+μi‖vε
i‖2
2)−2λ∫ΩQ(vε)N
2
N∑k
i=1νi‖vε
i‖4
4+2β∫ΩL(vε)N−2
2=(C(λ)+ε)β−N−2
2+o(β−N−2
2).
Letting ε→0, we have
C(λ,β)≤C(λ)β−N−2
2+o(β−N−2
2).(4.5)
It follows from the fact that uλ,βis the ground state solution of (1.1) and (4.1) that
k

i=1(‖∇uλ,β
i‖2
2+μi‖uλ,β
i‖2
2)−2λ
Ω
Q(uλ,β)≤NC(λ)β−N−2
2+o(β−N−2
2).
On the other hand, denote
γβ=∑k
i=1νi‖uλ,β
i‖2∗
2∗
β∫ΩL(uλ,β).

Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings |613
Then, by Lemma 4.1, we have γβ→0as β→+∞. Recall that uλ,βis the ground state solution of (1.1). Thus
we also have from Lemma 4.1 that
k

i=1(‖∇uλ,β
i‖2
2+μi‖uλ,β
i‖2
2)−2λ
Ω
Q(uλ,β)=2(1+o(1))β
Ω
L(uλ,β).(4.6)
Now, by (1.14), we have
C(λ)≤∑k
i=1(‖∇uλ,β
i‖2
2+μi‖uλ,β
i‖2
2)−2λ∫ΩQ(uλ,β)N
2
N2∫ΩL(uλ,β)N−2
2
=∑k
i=1(‖∇uλ,β
i‖2
2+μi‖uλ,β
i‖2
2)−2λ∫ΩQ(uλ,β)N
2
N2∫ΩL(uλ,β)N−2
2(1+γβ)N−2
2(1+γβ)N−2
2
=βN−2
2(C(λ,β)+o(1)),
which, together with (4.5), completes the proof.
By (4.3) and Proposition 4.1, we can see from (4.6) that
2(1+o(1))β
Ω
L(uλ,β)=k

i=1(‖∇uλ,β
i‖2
2+μi‖uλ,β
i‖2
2)−2λ
Ω
Q(uλ,β)=NC(λ)β−N−2
2+o(β−N−2
2).(4.7)
and
k

i=1
νi‖uλ,β
i‖2∗
2∗=O(β−N
2).(4.8)
Let vλ,β=(vλ,β
1,vλ,β
2,...,vλ,β
k)with vλ,β
i=βN−2
4uλ,β
ifor all i=1,2,...,k.
Proposition 4.2. Let −α1<μ1≤μ2≤⋅⋅⋅≤μk≤0and λ0<λ<λ1, where α1is the first eigenvalue of −∆in
H1
0(Ω)and λ0,λ1are respectively given by (1.6) and Theorem 1.1. Then vλ,β→vλ,∞strongly in Has β→+∞
up to a subsequence in one of the following cases:
(1) N=3and μk<0with μk+α1>0small enough,
(2) N≥4.
Here vλ,∞is a ground state solution of (1.15).
Proof. By (4.7) and a similar calculation to the one used for (4.4), we can see that {vλ,β}is bounded in H
for β. Thus, without loss of generality, we assume that vλ,β⇀vλ,∞weakly in Has β→+∞. We first claim
that vλ,∞= 0. Suppose the contrary; then vλ,β→0strongly in (L2(Ω))kas β→+∞owing to the Sobolev
embedding theorem. It follows from (4.7) that
2
Ω
L(vλ,β)=k

i=1‖∇vλ,β
i‖2
2=NC(λ)+o(1).
Thus
ck=inf

u∈(D1,2(ℝN))k\{ 
0}∑k
i=1‖∇ui‖2
2,ℝNN
2
N2∫ℝ4L(u)N−2
2≤∑k
i=1‖∇vλ,β
i‖2
2,ℝNN
2
N2∫ℝ4L(vλ,β)N−2
2=C(λ)+o(1).(4.9)
On the other hand, by the Sobolev and Hölder inequalities, we have
ck≥∑k
i=1‖ui‖2
2∗,ℝNN
2
N∑k
i,j=1,i<j‖ui‖2∗
2
2∗,ℝN‖uj‖2∗
2
2∗,ℝNN−2
2
SN
2,
where Sis the best Sobolev embedding constant from D1,2(ℝN)to L2∗(ℝN). By similar arguments to the ones
used for Proposition 3.2, we can show that ckis attained by U=(t1Uε,z,t2Uε,z,...,tkUε,z), where Uε,zis

614 |Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings
given by (3.8) and t=(t1,t2,...,tk)is a nonzero constant vector satisfying ti≥0for all i=1,2,...,k.
On the other hand, by the method of Lagrange’s multiplier, we also know that tis a solution of the system






si=dk
k

j=1,j=i
s2∗
2
js2∗
2−1
i,
si≥0for all i=1,2,...,k,
where
dk=inf

t∈ℝk\{ 
0}∑k
i=1t2
iN
2
N∑k
i,j=1,i<j|tj|2∗
2|ti|2∗
2N−2
2
.
By a similar argument to the one used in the proof of Proposition 3.3, we can show that ti=((k−1)dk)−N−2
4
for all i=1,2,...,k. Therefore, we recall that −α1<μ1≤μ2≤⋅⋅⋅≤μk≤0and λ0<λ<λ1, and we can use
a similar argument to the one used in the proof of Lemma 3.3 to show that C(λ)<ck, which contradicts (4.9).
Hence we must have vλ,∞= 0. Since uλ,βis the ground state solution of (1.1), we have from vλ,β
i=βN−2
4uλ,β
i
that 









−∆vλ,β
i+μivλ,β
i=νi(vλ,β
i)2∗−1
β+k

j=1,j=i(vλ,β
j)2∗
2(vλ,β
i)2∗
2−1+λ
k

j=1,j=i
vλ,β
jin Ω,
vλ,β
i>0in Ω,
vλ,β
i=0on ∂Ω, i=1,2,...,k.
Thanks to (4.8), we must have that vλ,∞is a nonzero solution of (1.15). Recall that, by a standard argument,
we have
C(λ)=inf

u∈Mλ
Jλ(u)
with
Jλ(u)=k

i=1
1
2(‖∇ui‖2
2+μi‖ui‖2
2)−2
2∗
Ω
L(u)−λ
Ω
Q(u),
and
Mλ=u∈H\{0}|J󸀠
λ(u)u=0.
Then, thanks to (4.7), there exists tβ→1as β→+∞such that

wλ,β=(wλ,β
1,wλ,β
2,...,wλ,β
k)∈Mλ
with wλ,β
i=tβvλ,β
ifor all i=1,2,...,k. Clearly, by the Sobolev embedding theorem, we also have that
wλ,β
i→vλ,∞
istrongly in L2(Ω)as β→+∞. Now, by a standard argument, we have from (4.7) once more that
C(λ)+o(1)=Jλ(
wλ,β)=1
Nk

i=1(‖∇wλ,β
i‖2
2+μi‖wλ,β
i‖2
2)−2λ
Ω
Q(
wλ,β)
≥1
Nk

i=1(‖∇vλ,∞
i‖2
2+μi‖vλ,∞
i‖2
2)−2λ
Ω
Q(vλ,∞)
=Jλ(vλ,∞)≥C(λ).
Thus we must have vλ,β→vλ,∞strongly in Has β→+∞. Finally, applying a similar argument to the one used
for Lemma 3.2, we can see from vλ,∞= 0that vλ,∞is nontrivial. Therefore, vλ,∞is a ground state solution
of (1.15).
We close this section by the proof of Theorem 1.4.
Proof of Theorem 1.4. It follows from Propositions 4.1 and 4.2.

Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings |615
4.2 The Case λ→λ1
Since uλ,βis positive, it is easy to show that C(λ,β)is decreasing for λ∈(λ0,λ1). Now, by a similar argument
to the one used for [24, Theorem 1.10], we can show that uλ,β→0strongly in Has λ→λ1. Let vλ,β
i=uλ,β
i
pλ,β,
where
pλ,β=max{‖∇uλ,β
1‖2,‖∇uλ,β
2‖2,...,‖∇uλ,β
k‖2}.
Clearly, pλ,β→0as λ→λ1.
Lemma 4.2. Let −α1<μ1≤μ2≤⋅⋅⋅≤μkand λ0<λ<λ1, where α1is the first eigenvalue of −∆in H1
0(Ω)and
λ0,λ1are respectively given by (1.6) and Theorem 1.1. Then we have vλ,β
i→v0,β
istrongly in H1
0(Ω)for all
i=1,2,...,kas λ→λ1up to a subsequence in one of the following cases:
(1) N=3and μk<0with μk+α1>0small enough,
(2) N≥4and μk<0,
(3) N≥4and β>βk, where βkis given by Proposition 3.3.
Here v0,β=(v0,β
1,v0,β
2,...,v0,β
k)∈N∗
1\{0}
with N∗
1given by Proposition A.1.
Proof. By the definition of vλ,β
i, it is easy to see that
1≤k

i=1‖∇vλ,β
i‖2
2≤k.(4.10)
Thus, without loss of generality, we assume that vλ,β
i⇀v0,β
iweakly in H1
0(Ω)as λ→λ1for all i=1,2,...,k.
Recall that uλ,βis the ground state solution of (1.1). Thus we have from vλ,β
i=uλ,β
i
pλ,βthat










−∆vλ,β
i+μivλ,β
i=νi(vλ,β
i)2∗−1+β
k

j=1,j=i(vλ,β
j)2∗
2(vλ,β
i)2∗
2p2∗−2
λ,β+λ
k

j=1,j=i
vλ,β
jin Ω,
vλ,β
i>0in Ω,
vλ,β
i=0on ∂Ω, i=1,2,...,k.
(4.11)
It follows that v0,β=(v0,β
1,v0,β
2,...,v0,β
k)is a nonnegative solution of the system




−∆v0,β
i+μiv0,β
i=λ1
k

j=1,j=i
v0,β
jin Ω,
v0,β
i=0on ∂Ω, i=1,2,...,k.
(4.12)
Thus, by Proposition A.1, we have v0,β∈N∗
1. It remains to show that v0,β= 0. Suppose to the contrary that
v0,β=0. Then, by the Sobolev embedding theorem, we have vλ,β
i→v0,β
istrongly in L2(Ω)as λ→λ1for all
i=1,2,...,k. Multiplying (4.11) with vλ,β
ifor every i=1,2,...,kand integrating by parts, we have from
pλ,β→0as λ→λ1that
k

i=1‖∇vλ,β
i‖2
2=o(1),
which contradicts (4.10).
By Proposition A.1, for every u∈N∗
1\{0}, we have
2∫ΩQ(u)
∑k
i=1νi‖ui‖2∗
2∗+2β∫ΩL(u)N−2
N≡constant,
where L(u)and Q(u)are respectively given by (1.7) and (1.8). We denote this constant by P(β).

616 |Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings
Proposition 4.3. Let −α1<μ1≤μ2≤⋅⋅⋅≤μkand λ0<λ<λ1, where α1is the first eigenvalue of −∆in H1
0(Ω)
and λ0,λ1are respectively given by (1.6) and Theorem 1.1. Then we have
(NC(λ,β))2
N=(λ1−λ)(P(β)+o(1)) as λ→λ1
in one of the following cases:
(1) N=3and μk<0with μk+α1>0small enough,
(2) N≥4and μk<0,
(3) N≥4and β>βk, where βkis given by Proposition 3.3.
Proof. Since uλ,β→0strongly in Has λ→λ1, by a similar argument to the one used for Lemma 4.1, we can
see that ‖uλ,β
i‖2∗
2∗=o(‖∇uλ,β
i‖2
2)and β
Ω
L(uλ,β)=ok

i=1‖∇uλ,β
i‖2
2as λ→λ1.
Now, by Proposition A.1 and (4.1), we have from Lemma 4.2 that
λ1≤∑k
i=1(‖∇vλ,β
i‖2
2+μi‖vλ,β
i‖2
2)
2∫ΩQ(vλ,β)=∑k
i=1(‖∇uλ,β
i‖2
2+μi‖uλ,β
i‖2
2)
2∫ΩQ(uλ,β)
=λ+∑k
i=1νi‖uλ,β
i‖2∗
2∗+2β∫ΩL(uλ,β)N−2
N
2∫ΩQ(uλ,β)(NC(λ,β))2
N
=λ+∑k
i=1νi‖vλ,β
i‖2∗
2∗+2β∫ΩL(vλ,β)N−2
N
2∫ΩQ(vλ,β)(NC(λ,β))2
N
=λ+1
P(β)+o(1)(NC(λ,β))2
N.(4.13)
On the other hand, by (4.2) and (4.12), we have
C(λ,β)≤∑k
i=1(‖∇v0,β
i‖2
2+μi‖v0,β
i‖2
2)−2λ∫ΩQ(v0,β)N
2
N∑k
i=1νi‖v0,β
i‖4
4+2β∫ΩL(v0,β)N−2
2
=2(λ1−λ)∫ΩQ(v0,β)N
2
N∑k
i=1νi‖v0,β
i‖4
4+2β∫ΩL(v0,β)N−2
2
=((λ1−λ)P(β))N
2.(4.14)
The conclusion follows from (4.13) and (4.14).
Let
wλ,β
i=1
(λ1−λ)N
4
uλ,β
ifor all i=1,2,...,k.
Proposition 4.4. Let −α1<μ1≤μ2≤⋅⋅⋅≤μkand λ0<λ<λ1, where α1is the first eigenvalue of −∆in H1
0(Ω)
and λ0,λ1are respectively given by (1.6) and Theorem 1.1. Then we have 
wλ,β→
w0,βstrongly in Has λ→λ1
in one of the following cases:
(1) N=3and μk<0with μk+α1>0small enough,
(2) N≥4and μk<0,
(3) N≥4and β>βk, where βkis given by Proposition 3.3.
Here 
w0,β∈N∗
1\{0}with N∗
1given by Proposition A.1.
Proof. Recall that uλ,βis the ground state solution of (1.1). Thus, by (4.1), we have
(NC(λ,β))2
N=k

i=1(‖∇uλ,β
i‖2
2+μi‖uλ,β
i‖2
2)−λ
Ω
Q(uλ,β)2
N
,

Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings |617
where Q(u)is given by (1.8). It follows from Proposition 4.3 that
k

i=1(‖∇wλ,β
i‖2
2+μi‖wλ,β
i‖2
2)−λ
Ω
Q(
wλ,β)2
N=P(β)+o(1),
Now we could follow the argument that is used for Lemma 4.2 step by step to obtain the conclusion.
We close this section by the proof of Theorem 1.5
Proof of Theorem 1.5. It follows from Propositions 4.3 and 4.4.
4.3 The Case λ→λ0
As we stated in the above section, we know that C(λ,β)is decreasing for λ∈(λ0,λ1). Thus, by Lemma 3.3,
we have
lim
λ→λ0k

i=1(‖∇uλ,β
i‖2
2+μi‖uλ,β
i‖2
2)−2λ
Ω
Q(uλ,β)=Nlim
λ→λ0
C(λ,β)>0,(4.15)
where Q(u)is given by (1.8). Now, by a similar argument to the one used for (4.4), we can see that {uλ,β}is
bounded in Hfor λ. Without loss of generality, we assume uλ,β⇀u0,βweakly in Has λ→λ0. Thanks to the
Sobolev embedding theorem, we also have uλ,β→u0,βstrongly in (L2(Ω))kas λ→λ0.
In what follows, let us first consider the case −α1<μ1<0. Without loss of generality, we assume
−α1<μ1≤μ2≤⋅⋅⋅≤μl<0for some l∈{1,2,...,k}. Let
C(β)=inf

MJ(u)(4.16)
with J(u)=k

i=11
2(‖∇ui‖2
2+μi‖ui‖2
2)−νi
2∗‖ui‖2∗
2∗−2β
2∗
Ω
L(u),
and 
M=u∈H\{0}| J󸀠(u)u=0,
where L(u)is given by (1.7). Recall that, by (1.6), λ0=0for −α1<μ1<0.
Proposition 4.5. Suppose −α1<μ1<0. Then uλ,β→u0,βstrongly in Has λ→0with J(u0,β)=C(β)in one of
the following cases:
(1) N=3and μk<0with μk+α1>0small enough,
(2) N≥4and μk<0,
(3) N≥4and β>βk, where βkis given by Proposition 3.3.
Moreover, if
(a) either N=4and β>max{νj},
(b) or N≥5,
then u0,βmust be nontrivial.
Proof. We first claim that u0,β= 0. Suppose to the contrary that u0,β=0. Then we have
γλ=k

i=1
μi‖uλ,β
i‖2
2−2λ
Ω
Q(uλ,β)=o(1)
by the Sobolev embedding theorem. It follows from (4.15) and the fact that uλ,βis the ground state solution
of (1.1) that
lim
λ→λ0k

i=1(νi‖uλ,β
i‖4
4,ℝ4)+2β
ℝ4
L(uλ,β)=Nlim
λ→λ0
C(λ,β)>0.

618 |Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings
Thus, by (3.9), we can see that
ck≤∑k
i=1‖∇uλ,β
i‖2
2,ℝ4N
2
N∑k
i=1(νi‖uλ,β
i‖4
4,ℝ4)+2β∫ℝ4L(uλ,β)N−2
2=C(λ,β)+o(1).
On the other hand, since λ>0, it is standard to show that C(β)≥C(λ,β), where C(β)is given by (4.16). Since
−α1<μ1<0, there exists u∈Hsuch that ∑k
i=1μi‖ui‖2
2<0. Let ti=‖ui‖2for all i=1,2,...,k. Then, using
U=(t1Uε,z,t2Uε,z,..., tkUε,z)
as the test function for C(β), we can see from a similar argument to the one used for Lemma 3.3 that C(β)<ck
in one of the cases (1)–(3), which is impossible. Here, Uε,zis given by (3.8). Thus we must have u0,β= 0. Now,
since uλ,β⇀u0,βweakly in Has λ→0, it is also standard to show that u0,βis a solution of the system










−∆ui+μiui=νiu2∗−1
i+β
k

j=1,j=i
u2∗
2
ju2∗
2−1
iin Ω,
ui≥0in Ω,
ui=0on ∂Ω, i=1,2,...,k.
Hence we obtain C(β)≥1
Nlim
λ→λ0(μ1)k

i=1(‖∇uλ,β
i‖2
2+μi‖uλ,β
i‖2
2)−2λ
Ω
Q(uλ,β)
≥1
Nk

i=1(‖∇ u0,β
i‖2
2+μi‖u0,β
i‖2
2)−2λ
Ω
Q(u0,β)
≥C(β).
It follows that uλ,β→u0,βstrongly in Has λ→0with J(u0,β)=C(β). By a similar argument to the one used
for Proposition 3.2, we can show that C(β)can be attained only by nontrivial circumstances in one of the
cases (a) and (b). Therefore, u0,βmust be nontrivial in one of the cases (a) and (b).
The case 0≤μ1≤μ2≤⋅⋅⋅≤μkstill needs to be considered, which, together with Theorem 1.3, implies that
we shall impose N≥4and β>βkin what follows. Here βkis given by Proposition 3.3. In the following, we
also always set u≡0outside Ωand regard every u∈H1
0(Ω)as in D1,2(ℝN). Recall that
γλ=k

i=1
μi‖uλ,β
i‖2
2−2λ
Ω
Q(uλ,β).
Then, thanks to the Sobolev embedding theorem, we have
γλ=k

i=1
μi‖u0,β
i‖2
2−2λ0
Ω
Q(u0,β)+o(1)=γ0+o(1).
By Proposition A.2, we also have γ0≥0.
Lemma 4.3. Suppose 0≤μ1≤μ2≤⋅⋅⋅≤μk,N≥4and β>βk, where βkis given by Proposition 3.3. Then we
have ck=C(λ,β)+o(1)and uλ,β⇀0weakly in Has λ→λ0, where ckis given by (3.9).
Proof. As in the proof of Proposition 4.5, by (3.9), we can see from γ0≥0that
ck≤∑k
i=1‖∇uλ,β
i‖2
2,ℝNN
2
N∑k
i=1(νi‖uλ,β
i‖2∗
2∗,ℝN)+2β∫ℝNL(uλ,β)N−2
2≤C(λ,β)+o(1).
It follows from Lemma 3.3 that ck=C(λ,β)+o(1)as λ→λ0, which also implies
γ0=k

i=1
μi‖u0,β
i‖2
2−λ0
Ω
Q(u0,β)=0.

Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings |619
We complete the proof by showing that u0,β=0. Indeed, it is easy to see that u0,βis a solution of the system










−∆ui+μiui=νiu2∗−1
i+β
k

j=1,j=i
u2∗
2
ju2∗
2−1
i+λ0
k

j=1,j=i
ujin Ω,
ui≥0in Ω,
ui=0on ∂Ω, i=1,2,...,k.
Suppose to the contrary that u0,β= 0. Then by (3.9), (4.1) and γ0=0, we can see from Lemma 3.3 and the
fact that uλ,βis the ground state solution of (1.1) that
Nck≤k

i=1‖∇ u0,β
i‖2
2,ℝN≤k

i=1‖∇uλ,β
i‖2
2,ℝN+o(1)≤NC(λ,β)+o(1)≤Nck+o(1).
Therefore, we must have ∑k
i=1‖∇ u0,β
i‖2
2,ℝN=Nckand uλ,β→u0,βstrongly in Has λ→λ0. Applying the
Hölder inequality similar to (3.10), we can see from Proposition 3.2 that ‖∇ u0,β
i‖2
2,ℝN=S‖u0,β
i‖2
2∗,ℝNfor
all i=1,2,...,k. Then we must have that u0,β
iattains the best Sobolev embedding constant for some
i∈{1,2,...,k}, which is impossible since u0,β
i∈H1
0(Ω)for all i=1,2,...,k.
Let
Aλ=max
i=1,2,...,k{‖uλ,β
i‖2
N−2
∞,ℝN},
where ‖⋅‖∞,ℝNis the usual norm in L∞(ℝN). Then, by Lemma 4.3, we must have Aλ→+∞. Without loss of
generality, we may assume that (Aλ)N−2
2=‖uλ,β
1‖∞,ℝN=uλ,β
1(yλ), where yλ∈Ω. We define
vλ,β
i=1
(Aλ)N−2
2
uλ,β
ix
Aλ+yλfor all i=1,2,...,k.
Then vλ,β
i∈H1
0(Ωλ), where Ωλ={x∈ℝN|x
Aλ+yλ∈Ω}. Moreover, we also have
‖∇uλ,β
i‖2
2,ℝN=‖∇vλ,β
i‖2
2,ℝN,‖uλ,β
i‖2∗
2∗,ℝN=‖vλ,β
i‖2∗
2∗,ℝN,(4.17)

ℝN
L(uλ,β)=
ℝN
L(vλ,β),(4.18)
‖uλ,β
i‖2
2,ℝN=1
A2
λ‖vλ,β
i‖2
2,ℝN.(4.19)
Lemma 4.4. Suppose 0≤μ1≤μ2≤⋅⋅⋅≤μk,N≥4and β>βk, where βkis given by Proposition 3.3. Then we
have Ωλ→ℝNas λ→λ0.
Proof. Since ∂Ωis smooth and Aλ→+∞, it is well known (cf. [16]) that either Ωλ→ℝNor Ωλ→(ℝN)+
as λ→λ0up to translations and rotations. Here (ℝN)+={x=(x1,x2,...,xN)∈ℝN|xN>0}. Suppose the
contrary. Then we must have Ωλ→(ℝN)+as λ→λ0. Recall that
lim
λ→λ0k

i=1(νi‖uλ,β
i‖2∗
2∗,ℝN)+2β
ℝN
L(uλ,β)=lim
λ→λ0k

i=1(‖∇uλ,β
i‖2
2,ℝN+μi‖uλ,β
i‖2
2,ℝN)−2λ
ℝN
Q(uλ,β)
=Nlim
λ→λ0
C(λ,β)>0,
where L(u)and Q(u)are respectively given by (1.7) and (1.8). By Lemma 4.3 and (4.17), (4.18), we can see
that
k

i=1(νi‖vλ,β
i‖2∗
2∗,ℝN)+2β
ℝN
L(vλ,β)=k

i=1‖∇vλ,β
i‖2
2,ℝN+o(1)=Nck+o(1).
Thus, by a similar argument to the one used in the proof of Lemma 4.3, we can show that
‖∇vλ,β
i‖2
2,ℝN=S‖vλ,β
i‖2∗
2∗,ℝN+o(1)for all i=1,2,...,k.

620 |Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings
It follows from Proposition 3.2 that
s=(‖vλ,β
1‖2∗,ℝN,‖vλ,β
2‖2∗,ℝN,...,‖vλ,β
k‖2∗,ℝN)
is a minimizing sequence of dk. By β>βk, we have from Proposition 3.2 once more that ‖vλ,β
1‖2∗
2∗,ℝN=C+o(1)
as λ→λ0up to a subsequence. Let
vλ,β
1=vλ,β
1
‖vλ,β
1‖2∗,ℝN
.
Then ‖∇vλ,β
1‖2
2,ℝN=S+o(1)and ‖vλ,β
1‖2∗,ℝN=1. By [22, Theorem 4.9], there exists Rλand yλ∈ℝNsuch that
wλ,β
1=1
(Rλ)N−2
2
vλ,β
1x−yλ
Rλ→Uε,z(4.20)
strongly in D1,2(ℝN)as λ→λ0, where Uε,zis given by (3.8). Recall that (Aλ)N−2
2=maxi=1,2,...,k{‖uλ,β
i‖∞,ℝN}.
Thus, by ‖vλ,β
1‖2∗
2∗,ℝN=C+o(1), we know that ‖vλ,β
1‖∞,ℝN≤C󸀠. It follows from (4.20) that {Rλ}is bounded
from above. Without loss of generality, we assume that Rλ→R0as λ→λ0. Let
Ωλ=x∈ℝNx−yλ
Rλ∈Ωλand wλ,β
i=1
(Rλ)N−2
2
vλ,β
ix−yλ
Rλfor all i∈{2,...,k}.
Then, by (Aλ)N−2
2=maxi=1,2,...,k{‖uλ,β
i‖∞,ℝN}and Rλ→R0as λ→λ0, we can see that ‖wλ,β
i‖∞,ℝN≤Cfor all
i∈{2,...,k}. By the fact that ‖vλ,β
1‖2∗
2∗,ℝN=C+o(1)as λ→λ0,wλ,β
1is a solution of the equation










−∆u≤Cν1u2∗−1+β
k

l=2(wλ,β
l)2∗
2u2∗
2−1+λ
R2
λA2
λ
k

l=2
wλ,β
lin Ωλ,
u>0in Ωλ,
u=0on ∂Ωλ,i=1,2,...,k.
Note that, by the fact that Ωλ→(ℝN)+as λ→λ0and (4.20), we can obtain Ωλ→ℝNand |yλ|→+∞as
λ→λ0. Therefore, by the elliptic estimates in [9] (see also [26, Lemma 2.4]), we have from (4.20) that
sup
𝔹r(y)wλ,β
1≤C
𝔹2r(y)|Uε,z|2∗dx1
2∗+o(1)for all y∈ℝN,
where 𝔹r(x)={y∈ℝN||y−x|<r}. This yields that wλ,β
1(x)<1
C(1
2R0)N−2
2for |x|large enough uniformly for
λ−λ0>0small enough. Now we have from |yλ|→+∞as λ→λ0that
o(1)+1
C1
R0N−2
2=wλ,β
1(yλ)<1
C1
2R0N−2
2
for λ−λ0>0small enough, which is impossible.
Proposition 4.6. Suppose 0≤μ1≤μ2≤⋅⋅⋅≤μk,N≥4and β>βk, where βkis given by Proposition 3.3. Then
we have vλ,β
i→tiUε,zstrongly in D1,2(ℝN)for all i=1,2,...,kas λ→λ0, where Uε,zand t=(t1,t2,..., tk)
are respectively given by (3.8) and Proposition 3.2.
Proof. Recall that
k

i=1(νi‖vλ,β
i‖2∗
2∗,ℝN)+2β
ℝ4
L(vλ,β)=k

i=1‖∇vλ,β
i‖2
2,ℝN+o(1)
=Nck+o(1).(4.21)
Thus, without loss of generality, we may assume that vλ,β⇀v0,βweakly in Das λ→λ0. On the other hand,
it is easy to see that vλ,βis a solution of the system










−∆ui+μi
A2
λ
ui=νiu2∗−1
i+β
k

j=1,j=i
u2∗
2
ju2∗
2−1
i+λ
A2
λ
k

j=1,j=i
ujin Ωλ,
ui>0in Ωλ,
ui=0on ∂Ωλ,i=1,2,...,k.
(4.22)

Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings |621
Recall that we also have ‖vλ,β
i‖∞,ℝN≤1for all i=1,2,...,k. Then, by Lemma 4.4 and the elliptic estimates
in [9], we can see that {vλ,β}is bounded in (C1,α
loc (ℝN))k. Hence, by the Arzelá–Ascoli theorem, vλ,β
1→v0,β
1
strongly in C(𝔹1(0))as λ→λ0, which, together with vλ,β
1(0)=1, implies v0,β= 0. Note that, by Lemmas 4.3
and 4.4 and (4.19), we can see from (4.22) that v0,βis a solution of the system










−∆ui=νiu2∗−1
i+β
k

j=1,j=i
u2∗
2
ju2∗
2−1
iin ℝN,
ui≥0in ℝN,
ui∈D1,2(ℝN),i=1,2,...,k.
Hence, by (3.9), (4.17) and (4.18), we have from (4.21) that
Nck≤k

i=1‖∇v0,β
i‖2
2,ℝN≤k

i=1‖∇vλ,β
i‖2
2,ℝN+o(1)=Nck+o(1).
It follows that Nck=∑k
i=1‖∇v0,β
i‖2
2,ℝNand vλ,β→v0,βstrongly in (D1,2(ℝN))kas λ→λ0. Applying the
Hölder inequality similarly to (3.10), we must have from Propositions 3.2 and 3.3 that v0,β
i=tiUε,zfor
all i=1,2,...,k.
We close this section by the proof of Theorem 1.6.
Proof of Theorem 1.6. It follows from Propositions 4.5 and 4.6.
A Appendix
In this section, we list some results that appear in the very recent work [24], which are used frequently in this
paper.
Let {αm}m∈ℕbe the eigenvalues of −∆in H1
0(Ω)which are increasing for m, and let Pmbe the correspond-
ing eigenspace of αm.
Proposition A.1 ([24, Theorem 1.4]).Let N≥1,μi>−α1for all i=1,2,...,kand λ>0. Then there exists
a sequence {λm}⊂ℝ+with λm↗+∞as m→∞such that the system






−∆ui+μiui=λ
k

j=1,j=i
ujin Ω,
ui=0on ∂Ω, i=1,2,...,k,
(A.1)
has a nonzero solution if and only if λ=λm. Moreover, we also have the following:
(a) For every m∈ℕ,λmis the unique solution to
k

j=1
λ
αm+μj+λ=1.
(b) Here u=(u1,u2,...,uk)is a solution to system (A.1) corresponding to λmif and only if
u∈N∗
m=φum|φ∈Pm,
where emis the unique basic of the algebra equation D∗
mX=0with
D∗
m=




αm+μ1−λm−λm. . . −λm
−λmαm+μ2−λm
...−λm
−λm−λmαm+μ3
...−λm
.
.
...........
.
.
−λm−λm−λm. . . αm+μk





.

622 |Y. Wu, Ground States of a K-Component Critical System with Linear and Nonlinear Couplings
(c) We have
λm=inf

u∈Mm−1
k

i=1
1
2(‖∇ui‖2
2+μi‖ui‖2
2),(A.2)
where Mm−1={u∈(
N∗
m−1)⊥|G(u)=1}with G(u)=∑k
i,j=1,i<j∫Ωujuidx and (
N∗
m−1)⊥=∞
l=mN∗
l. In par-
ticular, (
N∗
0)⊥=H.
Proposition A.2 ([24, Lemma 4.1]).Let N≥1and μi>0for all i=1,2,...,k. Then the quadratic form
k

i=1
μia2
i−2λ
k

i,j=1,i<j
aiaj,
is nonnegative if and only if 0<λ≤λ0, where λ0is the unique solution of
1=k

j=1
λ
μj+λ.
In particular, 
Ωk

j=1
μj|uj|2−2λ
k

i,l=1;i<l
uiuldx ≥0
for all u∈Hif and only if 0<λ≤λ0.
Acknowledgment: This paper was partly completed when Y. Wu was visiting the University of British
Columbia. He is grateful to the members of the Department of Mathematics at the University of British
Columbia for their invitation and hospitality.
Funding: Y. Wu is supported by NSFC (11701554, 11771319), the Fundamental Research Funds for the
Central Universities (2017XKQY091) and Jiangsu overseas visiting scholar program for university prominent
young & middle-aged teachers and presidents.
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