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BY 4.0 license Open Access Published by De Gruyter June 13, 2019

Ground States of a 𝐾-Component Critical System with Linear and Nonlinear Couplings: The Attractive Case

Yuanze Wu ORCID logo

Abstract

Consider the system

{ - Δ u i + μ i u i = ν i u i 2 * - 1 + β j = 1 , j i k u j 2 * 2 u i 2 * 2 - 1 + λ j = 1 , j i k u j in Ω , u i > 0 in Ω , u i = 0 on Ω , i = 1 , 2 , , k ,

where k2, ΩN (N3) is a bounded domain, 2*=2NN-2, μi and νi>0 are 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,λ are also established.

1 Introduction

In this paper, we consider the system

(1.1) { - Δ u i + μ i u i = ν i u i 2 * - 1 + β j = 1 , j i k u j 2 * 2 u i 2 * 2 - 1 + λ j = 1 , j i k u j in Ω , u i > 0 in Ω , u i = 0 on Ω , i = 1 , 2 , , k ,

where k2, ΩN (N3) is a bounded domain with a smooth boundary Ω, 2*=2NN-2 is the critical Sobolev exponent, μi and νi>0 for all i=1,2,,k are constants, and β,λ>0 are two parameters.

Let

𝔽 = diag ( - Δ + μ 1 , - Δ + μ 2 , , - Δ + μ k )

and

= ( ν 1 β β β β ν 2 β β β β ν 3 β β β β ν k ) , 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 =(H01(Ω))k:

(1.2) 𝔽 𝐮 = λ ( 1 2 𝐮 T 𝕀 𝐮 ) + ( 1 2 * ( 𝐮 2 * 2 ) T 𝐮 2 * 2 ) ,

where 𝐮=(u1,u2,,uk) is a vector function, 𝐮p=(u1p,u2p,,ukp), and 𝐮T is the transposition of the vector 𝐮. Thus system (1.1) is the generalization of the following well-known Brezis–Nirenberg equation:

(1.3) { - Δ u = λ u + | u | 2 * - 2 u in Ω , u = 0 on Ω ,

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 λ plays 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>0 be the first eigenvalue of -Δ in H01(Ω). Then system (1.1) has no solution in one of the following three cases:

  1. min { μ i } - α 1 ,

  2. min { μ i } > - α 1 and λ λ 1 , where λ 1 is the unique solution of

    (1.4) j = 1 k λ α 1 + μ j + λ = 1 ,

  3. min { μ i } > 0 , 0<λλ0, and Ω is star-shaped, where λ0 is the unique solution of

    (1.5) j = 1 k λ μ j + λ = 1 .

Remark 1.1.

By (1.5), it is easy to see that λ00 with λ0min{μi}+ as min{μi}0+. For the sake of simplicity, we re-define

(1.6) λ 0 = { the unique solution of (1.5) for min { μ i } > 0 , 0 for min { μ i } 0 .

Let

𝒥 ( 𝐮 ) = i = 1 k ( 1 2 ( u i 2 2 + μ i u i 2 2 ) - ν i 2 * u i 2 * 2 * ) - 2 β 2 * Ω ( 𝐮 ) - λ Ω 𝒬 ( 𝐮 ) ,

where =(H01(Ω))k is the Hilbert space with the inner product

𝐮 , 𝐯 = i = 1 k Ω u i v i

and ui,vi are respectively the i-th component of 𝐮 and 𝐯, and up=(Ω|u|p)1p is the usual norm in Lp(Ω) for all p1,

(1.7)   ( ) = i , j = 1 , i < j k | u i | 2 * 2 | u j | 2 * 2 ,
(1.8) 𝒬 ( 𝐮 ) = i , j = 1 , i < j k u i u j .

Clearly, it is easy to see that 𝒥(𝐮) is of class C1 in =(H01(Ω))k. Let

(1.9) = inf 𝐮 𝒥 ( 𝐮 )

with

(1.10) = { 𝐮 \ { 𝟎 } 𝒥 ( 𝐮 ) 𝐮 = 0 } .

Then is well defined, and contains all nonzero critical points of 𝒥(𝐮).

Definition 1.1.

Let 𝐯 be a critical point of 𝒥(𝐮), that is, 𝒥(𝐯)=𝟎 in -1, where 𝒥(𝐮) is the Fréchet derivative of 𝒥(𝐮) and -1 is the dual space of . Then 𝐯 is called nontrivial if vi0 for all i=1,2,,k; 𝐯 is called nonzero if 𝐯𝟎 in ; 𝐯 is called semi-trivial if 𝐯 is nonzero but not nontrivial; 𝐯 is called positive if vi>0 for all i=1,2,,k; 𝐯 is called a ground state if 𝐯 is nontrivial and 𝒥(𝐯)=.

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.

𝐯 is called a ground state solution of (1.1) if 𝐯 is 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 difficulty 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 difficulty, 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 to 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 to be less than the threshold by testing it with the ground state of the pure critical “limit” functional. To overcome such difficulty, 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=4 and 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 N5 and 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 N4 with β>0 large 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

(1.11) { - Δ u i = ν i | u i | 2 * - 2 u i + β j = 1 , j i k | u j | 2 * 2 | u i | 2 * 2 - 2 u i in N , u i D 1 , 2 ( N ) i = 1 , 2 , , k ,

which can be stated as follows.

Theorem 1.2.

Let N4. Then the ground state solution of (1.11) must be the “least energy” synchronized type solution of the form

𝐔 = ( t ~ 1 U ε , z , t ~ 2 U ε , z , , t ~ k U ε , z ) ,

where

U ε , z ( x ) = [ N ( N - 2 ) ε 2 ] N - 2 4 ( ε 2 + | x - z | 2 ) N - 2 2

is the Talanti function that satisfies -ΔU=U2*-1 in RN and t=(t~1,t~2,,t~k) is a constant vector with t~i>0 for all i=1,2,,k in one of the following cases:

  1. N = 4 and β ( 0 , min { ν i } ) ( max { ν i } , + ) ,

  2. N 5 and β > 0 .

Moreover, there exists βk>0 such 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 k2. 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>0 and the ground state solution must be unique for β>0 large enough in the case of N5. 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 μ1μ2μk. 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<μ1μ2μk and λ0<λ<λ1, where α1 is the first eigenvalue of -Δ in H01(Ω), and λ0,λ1 are respectively given by (1.6) and Theorem 1.1.

  1. If μ 1 = μ 2 = = μ k = μ , then ( 1.1 ) has the synchronized type solutions if and only if ν 1 = ν 2 = = ν k = ν .

  2. System ( 1.1 ) has a ground state solution in one of the following cases:

    1. N = 3 and μ k < 0 with μ k + α 1 > 0 small enough,

    2. N 4 and μ k < 0 ,

    3. N 4 and β > β k , where β k is given by Proposition 3.3.

Remark 1.3.

(1) From Theorem 1.3, it can be seen that, different from the system that is coupled with only nonlinear couplings (cf. [5, 7, 13, 21, 17]), 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

𝔽 𝐮 = λ ( 1 2 𝐮 T 𝕀 𝐮 )

in is the upper bound of λ for the existence of solutions to system (1.1), while λ0, the upper bound of λ such that the L2 norm of 𝒥(𝐮) is positive definite for min{μi}0 (see Proposition A.2), is the lower bound of λ for the existence of solutions to system (1.1) if Ω is 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 β and λ in this paper. For this purpose, we denote the ground state solution and its energy value, respectively, by 𝐮λ,β=(u1λ,β,u2λ,β,,ukλ,β) and (λ,β). In considering the case β0, by Theorem 1.3, we need the further conditions -α1<μ1μ2μk<0. Thus, by a standard perturbation argument, it is easy to show that 𝐮λ,β𝐮λ,0 strongly in =(H01(Ω))k as β0 up to a subsequence. Therefore, we shall mainly study the cases β+, λλ0 and λλ1 by Theorem 1.3 in what follows.

We first consider the case β+. By a standard argument, it is not very difficult to show that 𝐮λ,β𝟎 strongly in =(H01(Ω))k and (λ,β)0 as β+. To capture the precise decay rate of (λ,β) as β+, we turn to consider the equivalent minimization problem

(1.12) ( λ , β ) = ( i = 1 k ( u i λ , β 2 2 + μ i u i λ , β 2 2 ) - 2 λ Ω 𝒬 ( 𝐮 λ , β ) ) N 2 N ( i = 1 k ν i u i λ , β 2 * 2 * + 2 β Ω ( 𝐮 λ , β ) ) N - 2 2 = inf 𝐮 \ { 𝟎 } ( i = 1 k ( u i 2 2 + μ i u i 2 2 ) - 2 λ Ω 𝒬 ( 𝐮 ) ) N 2 N ( i = 1 k ν i u i 2 * 2 * + 2 β Ω ( 𝐮 ) ) N - 2 2 .

Recall that 𝐮λ,β𝟎 strongly in =(H01(Ω))k. Thus uiλ,β2*2*=o(uiλ,β22+μiuiλ,β22). This yields

(1.13) ( λ , β ) ( i = 1 k ( u i λ , β 2 2 + μ i u i λ , β 2 2 ) - 2 λ Ω 𝒬 ( 𝐮 λ , β ) ) N 2 N ( 2 β Ω ( 𝐮 λ , β ) ) N - 2 2 C β - N - 2 2

as β+. On the other hand, to capture the precise decay rate of 𝐮λ,β, it is natural to re-scale 𝐮λ,β in 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<μ1μ2μk and λ0<λ<λ1, where α1 is the first eigenvalue of -Δ in H01(Ω), and λ0,λ1 are respectively given by (1.6) and Theorem 1.1. Then C(λ,β)=C¯(λ)β-N-22+o(β-N-22) as β+ in one of the following cases:

  1. N = 3 and μ k < 0 with μ k + α 1 > 0 small enough,

  2. N 4 .

Here

(1.14) ¯ ( λ ) = inf 𝐮 \ { 𝟎 } ( i = 1 k ( u i 2 2 + μ i u i 2 2 ) - 2 λ Ω 𝒬 ( 𝐮 ) ) N 2 N ( 2 Ω ( 𝐮 ) ) N - 2 2 ,

where L(u) and Q(u) are given by (1.7) and (1.8), respectively. If we have -α1<μ1μ2μk0, then vλ,βvλ, strongly in H as β+ up to a subsequence, where viλ,β=βN-24uiλ,β for all i=1,2,,k and vλ, is a ground state solution of the system

(1.15) { - Δ u i + μ i u i = j = 1 , j i k u j 2 * 2 u i 2 * 2 - 1 + λ j = 1 , j i k u j 𝑖𝑛 Ω , u i > 0 𝑖𝑛 Ω , u i = 0 𝑜𝑛 Ω , i = 1 , 2 , , k ,

We next consider the case λλ1. Similar to the case β+, by a similar argument to the one used for [24, Theorem 1.10], we can show that 𝐮λ,β𝟎 strongly in =(H01(Ω))k and (λ,β)0 as λλ1. However, the decay rate of (λ,β) as λλ1 cannot be simply conjectured as in (1.13) for the case β+. Now, by re-scaling 𝐮λ,β twice 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,β will strongly converge to a nonzero eigenfunction of the first eigenvalue λ1 of the equation 𝔽𝐮=λ(12𝐮T𝕀𝐮) in .

Theorem 1.5.

Let -α1<μ1μ2μk and λ0<λ<λ1, where α1 is the first eigenvalue of -Δ in H01(Ω), and λ0,λ1 are respectively given by (1.6) and Theorem 1.1. Then we have

( λ , β ) = 1 N [ ( λ 1 - λ ) 𝔓 ( β ) ] N 2 + o ( ( λ 1 - λ ) N 2 ) 𝑎𝑠 λ λ 1

in one of the following cases:

  1. N = 3 and μ k < 0 with μ k + α 1 > 0 small enough,

  2. N 4 and μ k < 0 ,

  3. N 4 and β > β k , where β k is given by Proposition 3.3.

Here

𝔓 ( β ) 2 Ω 𝒬 ( 𝐮 ) ( i = 1 k ν i u i 2 * 2 * + 2 β Ω ( 𝐮 ) ) N - 2 N

is a constant that depends only on β for all uN1*\{0} with N1* given by Proposition A.1, while L(u) and Q(u) are respectively given by (1.7) and (1.8). Moreover, we also have wλ,βw0,β strongly in H as λλ1, where

w i λ , β = 1 ( λ 1 - λ ) N 4 u i λ , β for all i = 1 , 2 , , k 𝑎𝑛𝑑 𝐰 0 , β 𝒩 1 * \ { 𝟎 } .

Remark 1.4.

To the best of our knowledge, the precise decay estimate of (λ,β) and the strong convergence of the re-scaled functions 𝐯λ,β and 𝐰λ,β, stated in Theorems 1.4 and 1.5, respectively, for β+ and λλ1, are completely new in studies on the elliptic system. Moreover, we also observe in Theorems 1.4 and 1.5 that systems (1.15) and (A.1) are the limit systems of (1.1) under some suitable scalings as β+ and λλ1, respectively, which is also novel to the best of our knowledge.

We finally consider the case λλ0. As we stated in Remark 1.3, λ0 is the lower bound of λ for the existence of solutions to system (1.1) in the case min{μi}0 if Ω is star-sharped. Recall that the ground state solution of the well-known Brezis–Nirenberg equation (1.3) is a spiked solution as λ0, where 0 is the lower bound for the existence of solutions if Ω is star-shaped (cf. [14]). Thus it is natural to conjecture that 𝐮λ,β is also a spiked solution as λλ0 at least for min{μi}0. Our next result reveals such a property of 𝐮λ,β.

Theorem 1.6.

Let -α1<μ1μ2μk and λ0<λ<λ1, where α1 is the first eigenvalue of -Δ in H01(Ω), and λ0,λ1 are respectively given by (1.6) and Theorem 1.1.

  1. If - α 1 < μ 1 < 0 , then 𝐮 λ , β 𝐮 ^ 0 , β strongly in as λ 0 such that 𝒥 ˇ ( 𝐮 ^ 0 , β ) = ˇ ( β ) in one of the following cases:

    1. N = 3 and μ k < 0 with μ k + α 1 > 0 small enough,

    2. N 4 and μ k < 0 ,

    3. N 4 and β > β k , where β k is given by Proposition 3.3.

    Here ˇ ( β ) = inf ˇ 𝒥 ˇ ( 𝐮 ) with

    𝒥 ˇ ( 𝐮 ) = i = 1 k ( 1 2 ( u i 2 2 + μ i u i 2 2 ) - ν i 2 * u i 2 * 2 * ) - 2 β 2 * Ω ( 𝐮 ) 𝑎𝑛𝑑 ˇ = { 𝐮 \ { 𝟎 } 𝒥 ˇ ( 𝐮 ) 𝐮 = 0 } ,

    where ( 𝐮 ) is given by ( 1.7 ). Moreover, if

    1. either N = 4 and β > max { ν j } ,

    2. or N 5 ,

    then 𝐮 ^ 0 , β must be nontrivial.

  2. If 0 μ 1 μ 2 μ k , N4 and β>βk, where βk is given by Proposition 3.3, then we have 𝒜λ+ and viλ,βt~iUε,z strongly in D1,2(N) for all i=1,2,,k as λλ0, where Uε,z and 𝐭~=(t~1,t~2,,t~k) are respectively given by (3.8) and Proposition 3.2 and

    v i λ , β = 1 ( 𝒜 λ ) N - 2 2 u i λ , β ( x 𝒜 λ + y λ ) for all i = 1 , 2 , , k

    with 𝒜 λ = max i = 1 , 2 , , k { u i λ , β , N 2 N - 2 } .

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 λλ0, where λ0, given by (1.6), is the lower bound of λ for the existence of solutions to system (1.1) in this case if Ω is 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 λ for the existence of solutions to system (1.1), and it seems to be very interesting to find out the lower bound of λ for 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 different 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,β for both the vanishing case and the blow-up case in capturing the precise decay or blow-up rate, the re-scaling manners are quite different for the vanishing case and the blow-up case. The major difference is that we do not need to re-scale the domain Ω in 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=0 appear 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>0 and -α1<μ1,μ2<0 with β being in a wide range. Here α1 is the first eigenvalue of -Δ in H01(Ω). The following related studies can be seen in [7, 6, 23, 25] and the references therein. System (1.1) with λ=0, k=2, ν1,ν2>0 and μ1=μ2=0 in N, that is,

(1.16) { - Δ u 1 = ν 1 | u 1 | 2 * - 2 u 1 + β | u 2 | 2 * 2 | u 1 | 2 * 2 - 2 u 1 in N , - Δ u 2 = ν 2 | u 2 | 2 * - 2 u 1 + β | u 1 | 2 * 2 | u 2 | 2 * 2 - 2 u 2 in N , u 1 , u 2 D 1 , 2 ( N ) ,

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>0 and -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 nonradial solutions were obtained by using the reduction method in [12]. The spiked solutions of system (1.1) for λ=0 and k=2 were also studied in [3, 20], where it was proved that the ground state solution will blow-up and concentrate at some x0Ω for 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 [19, 17] and the references therein. On the other hand, the recent studies on critical system (1.1) for β=0 and k=2 can 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=0 and arbitrary k2 also 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 Section 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 1.41.6.

Notations

Throughout this paper, C and C are indiscriminately used to denote various absolute positive constants. We also list some notations used frequently below.

u p p = Ω | u | p d x , u p , N p = N | u | p d x ,
𝐮 = ( u 1 , u 2 , , u k ) , 𝐭 𝐮 = ( t 1 u 1 , t 2 u 2 , , t k u k ) , 𝔹 r ( x ) = { y N | y - x | < r } ,
t 𝐮 = ( t u 1 , t u 2 , , t u k ) , 𝐮 n = ( u 1 n , u 2 n , , u k n ) , + = ( 0 , + ) ,
( 𝐮 ) = i , j = 1 , i < j k | u i | 2 * 2 | u j | 2 * 2 , 𝒬 ( 𝐮 ) = i , j = 1 , i < j k u i u j ,
= ( H 0 1 ( Ω ) ) k , ( N ) + = { x = ( x 1 , x 2 , , x N ) N x N > 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 φ1 be the corresponding eigenfunction of α1. Then, multiplying system (1.1) with 𝐯=(φ1,0,0,,0) and integrating by parts, we have

0 ( α 1 + μ 1 ) Ω u 1 φ 1 = Ω u 1 φ 1 + μ 1 u 1 φ 1 = Ω ν 1 u 1 2 * - 1 φ 1 + β j = 2 k u j 2 * 2 u 1 2 * 2 - 1 φ 1 + λ j = 2 k u j φ 1 > 0 ,

which is impossible.

(2) By Proposition A.1,

j = 1 k λ α 1 + μ j + λ = 1

has a unique solution λ1, which is also the first eigenvalue of the operator 𝒯= with

= diag ( ( - Δ + μ 1 ) - 1 , ( - Δ + μ 2 ) - 1 , , ( - Δ + μ k ) - 1 ) ,
= ( 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 λ1 given by Proposition A.1. Then, by Proposition A.1 once more, we can choose ei>0 for all i=1,2,,k. Now, multiplying system (1.1) with 𝐯1 and integrating by parts, we have from λλ1 that

2 λ 1 Ω i , j = 1 , i < j k e j u i φ 1 = Ω i = 1 k e i ( u i φ 1 + μ i u i φ 1 ) = Ω i = 1 k e i ( ν i u i 2 * - 1 + β j = 1 , j i k u j 2 * 2 u i 2 * 2 - 1 ) φ 1 + 2 λ Ω i , j = 1 , i < j k e j u i φ 1 > 2 λ 1 Ω i , j = 1 , i < j k e j u i φ 1 ,

which is impossible.

(3) By Proposition A.2,

j = 1 k λ μ j + 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<λλ0. Then, by the classical regularity theories, we know that the ui are all of class C2. Now, without loss of generality, we assume that Ω is star-shaped for 0. Then, by the Pohozaev identity (cf. [4]), it can be seen that

N - 2 2 N j = 1 k Ω | u j | 2 + 1 2 N j = 1 k Ω ( x , n ) | u j | 2 = - 1 2 Ω ( j = 1 k μ j u j 2 - 2 λ i , l = 1 ; i < l k u i u l ) + N - 2 2 N Ω ( j = 1 k ν j u j 2 * + 2 β i , j = 1 , i < j k u j 2 * 2 u i 2 * 2 ) ,

where n is the unit outer normal vector of Ω. It follows from 𝐮 being a solution to system (1.1) that

1 2 N j = 1 k Ω ( x , n ) | u j | 2 = - 1 N Ω ( j = 1 k μ j | u j | 2 - 2 λ i , l = 1 ; i l k u i u l ) .

Since min{μj}>0 and 0<λλ0, by Proposition A.2, we known that the quadratic form

Ω ( j = 1 k μ j | u j | 2 - 2 λ i , l = 1 ; i l k u i u l ) 0 ,

which contradicts the fact that Ω is star-shaped for 0. ∎

3 Existence Results

Recall that, without loss of generality, we assume that μ1μ2μk. Thus, owing to the nonexistence results given by Theorem 1.1, we always consider the case -α1<μ1μ2μk and λ0<λ<λ1 in this section, where α1 is the first eigenvalue of -Δ in H01(Ω) and λ0,λ1 are 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+α1k-1 and

λ 0 = { μ k - 1 for  0 < μ , 0 for  0 μ .

Since λ0<λ<λ1, for the sake of clarity, we re-denote λ=μk-1+α1k-1λ~, where

(3.1) λ ~ { ( 0 , 1 ) for μ 0 , ( - μ α 1 , 1 ) for μ < 0 .

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=ν is 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 v is a function satisfying some equations and ti>0 for all i=1,2,,k. Then we must have

(3.2) t j ( - Δ v + μ v ) = ( ν j t j 2 * - 1 + β i = 1 , i j k t i 2 * 2 t j 2 * 2 - 1 ) v 2 * - 1 + ( μ k - 1 + α 1 k - 1 λ ~ ) i = 1 , i j k t i v in Ω

for all j=1,2,,k. It follows that

- Δ v = j = 1 k ( ν j t j 2 * - 1 + β i = 1 , i j k t i 2 * 2 t j 2 * 2 - 1 ) j = 1 k t j v 2 * - 1 + α 1 λ ~ v in Ω .

Thus v=sw, where s>0 and w is a positive solution of

(3.3) - Δ w = w 2 * - 1 + α 1 λ ~ w in Ω , w H 0 1 ( Ω ) .

Recall that λ~ is 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

t j w 2 * - 1 + α 1 λ ~ t j w = t j ( - Δ w ) = s 2 * - 2 ( ν j t j 2 * - 1 + β i = 1 , i j k t i 2 * 2 t j 2 * 2 - 1 ) w 2 * - 1 - t j μ w + ( μ k - 1 + α 1 k - 1 λ ~ ) i = 1 , i j k t i w

for all j=1,2,,k. Thus we must have from μk-1+α1k-1λ~>0 and s>0 that

0 = i = 1 , i j k ( t i - t j ) and ν j t j 2 * 2 + β i = 1 , i j k t i 2 * 2 = t j 2 - 2 * 2 s 2 - 2 *

for all j=1,2,,k, which implies t1=t2==tk and ν1=ν2==νk. ∎

3.2 The General Case μ1μ2μk

In this section, we will use the Nehari manifold given by (1.10) to prove the existence of a ground state solution of (1.1). Since the functional

i = 1 k 1 2 ( u i 2 2 + μ i u i 2 2 ) - λ Ω 𝒬 ( 𝐮 )

is positive definite for -α1<μ1μ2μk and λ0<λ<λ1 by Proposition A.1, it is standard (cf. [15, Lemma 2.3]) to drive the following result, where 𝒬(𝐮) is given by (1.8), α1 is the first eigenvalue of -Δ in H01(Ω) and λ0,λ1 are respectively given by (1.6) and Theorem 1.1.

Lemma 3.1.

Let -α1<μ1μ2μk and λ0<λ<λ1, where α1 is the first eigenvalue of -Δ in H01(Ω) and λ0,λ1 are respectively given by (1.6) and Theorem 1.1. Then, for every uH\{0}, there exists a unique t>0 such that tu=(tu1,tu2,,tuk)M. Moreover, we also have C>0, where C=infuMJ(u).

To use given 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<μ1μ2μk and λ0<λ<λ1, where α1 is the first eigenvalue of -Δ in H01(Ω) and λ0,λ1 are 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 . Since ||vi|||vi| a.e. in N and

i , j = 1 , i < j k v i v j i , j = 1 , i < j k | v i | | v j | ,

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 , where wi=|vi| for all i=1,2,,k. Clearly, is a C1 manifold. Moreover, since 2*>2, it is also standard (cf. [24, Proposition 5.2]) to show that is a natural constraint. Thus 𝐰 is a critical point of 𝒥(𝐮) by the method of the Lagrange multiplier. It follows that 𝐰 satisfies the system

(3.4) { - Δ w i + μ i w i = ν i w i 2 * - 1 + β j = 1 , j i k w j 2 * 2 w i 2 * 2 - 1 + λ j = 1 , j i k w j in Ω , w i 0 in Ω , w i = 0 on Ω , i = 1 , 2 , , k .

It follows from the maximum principle that, for every i=1,2,,k, we have either wi>0 or wi0. Suppose that 𝐰 is not a solution of system (1.1). Then there exists at least one j{1,2,,k} such that wj0. Without loss of generality, we may assume that wj>0 for j=1,2,,i0 and wj0 for j=i0+1,,k with i0{1,2,,k-1}. Then (3.4) is equivalent to

{ - Δ w i + μ i w i = ν i w i 2 * - 1 + β j = 1 , j i i 0 w j 2 * 2 w i 2 * 2 - 1 + λ j = 1 , j i i 0 w j in Ω , i = 1 i 0 w i = 0 in Ω , w i > 0 in Ω , w i = 0 on Ω , i = 1 , 2 , , i 0 ,

which is impossible. Thus 𝐰=(|v1|,|v2|,,|vk|) must be a ground state of system (1.1). ∎

By Lemma 3.1, the Nehari manifold is a natural constraint. It follows from the Ekeland variational principle that there exists a (PS) sequence {𝐮n} at the least energy level . Here is given by Lemma 3.1. Note that the embedding map from H01(Ω) to L2*(Ω) is not compact. Thus we shall use the Brezis–Nirenberg argument (cf. [2]) to recover the compactness of {𝐮n}, which leads us to first study the minimizing problem

(3.5) c k = inf 𝒩 k k ( 𝐮 ) .

Here

k ( 𝐮 ) = i = 1 k 1 2 ( u i 2 , N 2 - ν i 2 * u i 2 * , N 2 * ) - 2 β 2 * N ( 𝐮 )

is a functional defined in 𝒟=(D1,2(N))k, and 𝒟 is a Hilbert space equipped with the inner product

𝐮 , 𝐯 N = i = 1 k N u i v i ,

( 𝐮 ) is given by (1.7), up,N=(N|u|p)1p is the usual norm in Lp(N) for all p2 and

(3.6) 𝒩 k = { 𝐮 𝒟 \ { 𝟎 } k ( 𝐮 ) 𝐮 = 0 } .

Proposition 3.2.

Let

(3.7) d k = inf 𝒫 k 𝒢 k ( 𝐭 ) ,

where

𝒢 k ( 𝐭 ) = i = 1 k ( t i 2 2 - ν i | t i | 2 * 2 * ) - 2 β 2 * i , j = 1 , i < j k | t j | 2 * 2 | t i | 2 * 2 ,
𝒫 k = { 𝐭 k \ { 𝟎 } | i = 1 k ( t i 2 - ν i | t i | 2 * ) - 2 β i , j = 1 , i < j k | t j | 2 * 2 | t i | 2 * 2 = 0 } .

Then ck=dkSN2 is attained by U if and only if

𝐔 = ( t ~ 1 U ε , z , t ~ 2 U ε , z , , t ~ k U ε , z ) ,

where S is the best Sobolev embedding constant from D1,2(RN) to L2*(RN),

(3.8) U ε , z ( x ) = [ N ( N - 2 ) ε 2 ] N - 2 4 ( ε 2 + | x - z | 2 ) N - 2 2

is the Talanti function that satisfies -ΔU=U2*-1 in RN and t~=(t~1,t~2,,t~k) satisfies (3.11). Moreover, if

  1. either N = 4 and β > max { ν j } ,

  2. or N 5 ,

then t~i>0 for all i=1,2,,k.

Proof.

By a standard argument (cf. [17]), we can see that

(3.9) c k = inf 𝐮 ( D 1 , 2 ( N ) ) k \ { 𝟎 } ( i = 1 k u i 2 , N 2 ) N 2 N ( i = 1 k ν i u i 2 * , N 2 * + 2 β 4 ( 𝐮 ) ) N - 2 2 ,

which, together with the Hölder and Sobolev inequalities, implies

(3.10) c k inf 𝐮 𝒟 \ { 𝟎 } ( i = 1 k u i 2 * , N 2 ) N 2 N ( i = 1 k ν i u i 2 * , N 2 * + 2 β i < j k u i 2 * , N 2 * 2 u j 2 * , N 2 * 2 ) N - 2 2 𝒮 N 2 .

Here (𝐮) is given by (1.7). Clearly, we also have

d k = inf 𝐭 k \ { 𝟎 } ( i = 1 k t i 2 ) N 2 N ( i = 1 k ν i | t i | 2 * + 2 β i , j = 1 , i < j k | t j | 2 * 2 | t i | 2 * 2 ) N - 2 2 ,

which can be attained by some 𝐭~ with t~i0 for all i=1,2,,k and t~i>0 for some i. By the method of Lagrange’s multiplier, 𝐭~ also satisfies the system

(3.11) { t ~ i = ν i t ~ i 2 * - 1 + β j = 1 , j i k t ~ j 2 * 2 t ~ i 2 * 2 - 1 for all i = 1 , 2 , , k , t ~ i 0 and i = 1 k t ~ i > 0 for all i = 1 , 2 , , k .

Thus ck=dk𝒮N2 can be attained by

𝐔 = ( t ~ 1 U ε , z , t ~ 2 U ε , z , , t ~ k U ε , z ) ,

where Uε,z is given by (3.8). Suppose now that ck is attained by some nonzero 𝐯. Then, by the Hölder and Sobolev inequalities, we must have from (3.10) and ck=dk𝒮N2 that vi2,N2=𝒮vi2*,N2 for all i=1,2,,k, which implies either vi=Uε,z for some ε>0 and zN or vi=0. Moreover, we also have that

𝐬 = ( v 1 2 * , N , v 2 2 * , N , , v k 2 * , N )

attains dk. Thus ck=dk𝒮N2 is attained by 𝐔 if and only if

𝐔 = ( t ~ 1 U ε , z , t ~ 2 U ε , z , , t ~ k U ε , z ) ,

where Uε,z is given by (3.8) and 𝐭~=(t~1,t~2,,t~k) satisfies (3.11). In what follows, we shall borrow some ideas from [1] to show that t~i>0 for all i=1,2,,k in one of the following cases:

  1. N = 4 and β>max{νj},

  2. N 5 .

We set m=1,2,,k-1 and 𝐥m={l1,l2,,lm}{1,2,,k} with l1<l2<<lm. We also define

c 𝐥 m , m = inf 𝒩 𝐥 m , m 𝐥 m , m ( 𝐮 ) .

Here

𝐥 m , m ( 𝐮 ) = i = 1 m 1 2 ( u l i 2 , N 2 - ν l i 2 * u l i 2 * , N 2 * ) - 2 β 2 * N 𝐥 m , m ( 𝐮 ) ,
𝐥 m , m ( 𝐮 ) = i , j = 1 , i < j m | u l i | 2 * 2 | u l j | 2 * 2 ,

and

𝒩 𝐥 m , m = { 𝐮 𝒟 \ { 𝟎 } 𝐥 m , m ( 𝐮 ) 𝐮 = 0 } .

If ck<c𝐥m,m for all m=2,3,,k-1 and 𝐥m={l1,l2,,lm}{1,2,,k} with l1<l2<<lm, then we can see that ti>0 for 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, Theorem 2.2], by considering 𝐰¯+sϕ, we can show that there exists a unique

(3.12) t ( s ) = 1 - ( 1 + o ( 1 ) ) 2 β s 2 * 2 N i = 1 k - 1 | w i | 2 * 2 | ϕ k | 2 * 2 - s 2 ϕ 2 , N 2 ( 2 * - 2 ) i = 1 k - 1 w i 2 , N 2

for s>0 small enough such that

t ( s ) ( 𝐰 ¯ + s ϕ ) = ( t ( s ) w 1 , , t ( s ) w k - 1 , t ( s ) s ϕ ) 𝒩 k ,

where ϕ=(0,0,,0,ϕ). Using the fact that 2*<4 for N5, we have from (3.12) that

𝒥 k ( t ( s ) ( 𝐰 ¯ + s ϕ ) ) = [ t ( s ) ] 2 N ( i = 1 k - 1 w i 2 , N 2 + s 2 ϕ 2 , N 2 ) = 1 N i = 1 k - 1 w i 2 , N 2 - 4 β s 2 * 2 N i = 1 k - 1 | w i | 2 * 2 | ϕ k | 2 * 2 ( 2 * - 2 ) i = 1 k - 1 w i 2 , N 2 + O ( s 2 ) < 1 N i = 1 k - 1 w i 2 , N 2 = 𝒥 k - 1 ( 𝐰 )

for s>0 small enough. In the case N=4, we see from the fact that 𝐰=(w1,w2,,wk-1) is a critical point of k-1(𝐮) that

w i 2 , 4 2 < β 4 j = 1 k - 1 | w j | 2 | w i | 2 for all i = 1 , 2 , , k - 1

if β>max{νj}. It follows that β>βk*, where βk* is given by

β k * = inf { ϕ 2 , 4 2 | ϕ H 1 ( 4 ) , 4 i = 1 k - 1 | w i | 2 | ϕ | 2 = 1 } .

Since wiD1,2(4), the eigenvalue βk* can be attained by some ϕk*. Thus we have

t ( s ) = 1 - ( 1 + o ( 1 ) ) ( 2 β - β k * ) s 2 4 i = 1 k - 1 | w i | 2 | ϕ k * | 2 d x 2 i = 1 k - 1 w i 2 , N 2 as s 0 ,

by taking ϕ=ϕk* in (3.12). It follows that

𝒥 k ( t ( s ) ( 𝐰 ¯ + s ϕ ) ) = t ( s ) 2 4 ( i = 1 k - 1 w i 2 , 4 2 + s 2 ϕ k * 2 , 4 2 ) = 1 4 i = 1 k - 1 w i 2 , 4 2 - 2 ( β - β k * ) 4 i = 1 k - 1 | w i | 2 | ϕ k * | 2 s 2 + o ( s 2 ) < 𝒥 k - 1 ( 𝐰 )

for β>max{νj} and s>0 small enough. This yields ck<min{c𝐥m,m} if

  1. either N=4 and β>max{νj},

  2. or N5,

which completes the proof. ∎

We re-denote 𝐭~=(t~1,t~2,,t~k), which is given by Proposition 3.2, by 𝐭~β=(t~1β,t~2β,,t~kβ).

Proposition 3.3.

Let N4. Then there exists βk>0 such that

𝐭 ~ β = ( t ~ 1 β , t ~ 2 β , , t ~ k β )

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=ti2, system (3.11) is equivalent to the linear system

(3.13) { 1 = ν i s i + β j = 1 , j i k s j for all i = 1 , 2 , , k , s i > 0 for all i = 1 , 2 , , k .

By the Cramer rule, linear system (3.13) has a unique solution 𝐬=(s1,s2,,sk) with

s i = 1 ( ν i - β ) ( 1 + j = 1 k β ν i - β ) for all i = 1 , 2 , , k ,

for β>max{νj}. In what follows, let us consider the case N5. Since 𝐭~β=(t~1β,t~2β,,t~kβ) is a solution of system (3.11), we have

(3.14) ( t ~ i β ) 2 = ν i ( t ~ i β ) 2 * + β j = 1 , j i k ( t ~ j β ) 2 * 2 ( t ~ i β ) 2 * 2 for all i = 1 , 2 , , k .

This yields

i = 1 k ( t ~ i β ) 4 - 2 * 2 ( min { ν j } + ( k - 1 ) β ) i = 1 k ( t ~ i β ) 2 * 2 ,

which, together with the fact that 2<2*<4 and the Young inequality, implies

i = 1 k ( t ~ i β ) 2 * 2 C ( 1 min { ν j } + ( k - 1 ) β ) N 4 .

It follows that

(3.15) i = 1 k ( t ~ i β ) 2 = O ( β - N - 2 2 )

for β>0 large enough. Let s~iβ=βN-24t~iβ for all i=1,2,,k. Then, by (3.14) and (3.15), {𝐬~β} is bounded for β large enough in k, and they satisfy

(3.16) s ~ i β = ν i β ( s ~ i β ) 2 * - 1 + j = 1 , j i k ( s ~ j β ) 2 * 2 ( s ~ i β ) 2 * 2 - 1 for all i = 1 , 2 , , k .

Without loss of generality, we assume that 𝐬~β𝐬~0 in k as β+ up to a subsequence. Note that t~iβ>0 for all i=1,2,,k and β>0 by Proposition 3.2. Thus, by (3.14), we can see that 𝐬~0 is a solution of the system

(3.17) { ( s ~ i 0 ) 2 = j = 1 , j i k ( s ~ j 0 ) 2 * 2 ( s ~ i 0 ) 2 * 2 , s ~ i 0 0 for all i = 1 , 2 , , k ,

which is equivalent to

(3.18) { ( s ~ i 0 ) 4 - 2 * 2 = j = 1 , j i k ( s ~ j 0 ) 2 * 2 , s ~ i 0 0 for all i = 1 , 2 , , k .

System (3.18) yields (s~i0)4-2*2-(s~l0)4-2*2=(s~l0)2*2-(s~i0)2*2 for all i,l=1,2,,k with il. Since 2<2*<4 for N5, we must have s~i0=s~l0 for all i,l=1,2,,k with il, which, together with (3.18), implies s~i0=(k-1)-12*-2 for all i,l=1,2,,k. Let

𝐬 0 = ( ( k - 1 ) - 1 2 * - 2 , ( k - 1 ) - 1 2 * - 2 , , ( k - 1 ) - 1 2 * - 2 ) .

Then we also have that 𝐬0 is the unique solution of (3.17). Since 𝐬~β𝐬0 in k as β+ for every subsequence, we have 𝐬~β𝐬0 in k as β+. Let

𝚪 ( 𝐬 , σ ) = ( Γ 1 ( 𝐬 , σ ) , , Γ k ( 𝐬 , σ ) )

with

Γ i ( 𝐬 , σ ) = s i - σ ν i s i 2 * - 1 - j = 1 , j i k s j 2 * 2 s i 2 * 2 - 1 for all i = 1 , 2 , , k .

Since 𝐬0 is the unique solution of (3.17), we have 𝚪(𝐬0,0)=𝟎. Moreover,

Γ j s j ( 𝐬 0 , 0 ) = 4 - 2 * 2 and Γ j s i ( 𝐬 0 , 0 ) = - 2 * 2 for all i , j = 1 , 2 , , k with i j .

It follows from a direct calculation that

det ( [ Γ j s i ( 𝐬 0 , 0 ) ] i , j = 1 , 2 , , k ) = 2 k - 2 ( 4 - k 2 * ) ,

which, together with k2 and 2<2*, implies det([Γjsi(𝐬0,0)]i,j=1,2,,k)0. By the implicit function theorem, we can see from (3.16) that (𝐬~β,1β) is the unique curve bifurcated from (𝐬0,0). Let 𝐭^β be any solution of (3.11). Then, repeating the above argument as used for 𝐭~β, we can show that s^iβ=βN-24t^iβ(k-1)-12*-2 as β+ for all i=1,2,,k. Thus there exists βk>0 such that 𝐭~β=(t~1β,t~2β,,t~kβ) is the unique solution of (3.11) for β>βk. ∎

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 , which is given by Lemma 3.1 as follows.

Lemma 3.3.

Let -α1<μ1μ2μk and λ0<λ<λ1, where α1 is the first eigenvalue of -Δ in H01(Ω) and λ0,λ1 are respectively given by (1.6) and Theorem 1.1. Then C<ck in one of the following cases:

  1. N = 3 and μ k < 0 with μ k + α 1 > 0 small enough,

  2. N 4 and μ k < 0 ,

  3. N 4 and β > β k , where β k is given by Proposition 3.3.

Here ck is given by (3.5).

Proof.

Without loss of generality, we assume 0Ω. Choose ρ>0 such that 𝔹2ρ(0)Ω, and let ψC02(𝔹2ρ(0)) be a radial symmetric cut-off function satisfying 0ψ(x)1 and ψ(x)1 in 𝔹ρ(0). Furthermore, we define Vε(x)=ψ(x)Uε,0(x) with Uε,0 given by (3.8). Then it is well known (cf. [2]) that

(3.19) V ε 2 2 = 𝒮 N 2 + O ( ε N - 2 ) ,