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

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<λ≤λ0, 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 λ0→0 with λ0min⁡{μi}→+∞ as min⁡{μi}→0+. For the sake of simplicity, we re-define

Definition 1.1.

Let 𝐯→∈ℋ be a critical point of 𝒥⁢(𝐮→), that is, 𝒥′⁢(𝐯→)=𝟎→ in ℋ-1, where 𝒥′⁢(𝐮→) is the Fréchet derivative of 𝒥⁢(𝐮→) and ℋ-1 is the dual space of ℋ. Then 𝐯→ is called nontrivial if vi≠0 for all i=1,2,…,k; 𝐯→ is called nonzero if 𝐯→≠𝟎→ in ℋ; 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→)=ℭ.

Definition 1.2.

𝐯 → is called a ground state solution of (1.1) if 𝐯→ is a positive ground state of 𝒥⁢(𝐮→).

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

where

is the Talanti function that satisfies -Δ⁢U=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 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>0 and the ground state solution must be unique for β>0 large 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.

Theorem 1.3.

Let -α1<μ1≤μ2≤⋯≤μk 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

in ℋ 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).

Theorem 1.4.

Let -α1<μ1≤μ2≤⋯≤μk 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

where L⁢(u→) and Q⁢(u→) are given by (1.7) and (1.8), respectively. If we have -α1<μ1≤μ2≤⋯≤μk≤0, then v→λ,β→v→λ,∞ strongly in H as β→+∞ up to a subsequence, where viλ,β=βN-24⁢uiλ,β for all i=1,2,…,k and v→λ,∞ is a ground state solution of the system

Theorem 1.5.

Let -α1<μ1≤μ2≤⋯≤μk 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

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

is a constant that depends only on β for all u→∈N1*\{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 w→λ,β→w→0,β strongly in H as λ→λ1, where

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 𝐯→λ,β and 𝐰→λ,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 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.

Theorem 1.6.

Let -α1<μ1≤μ2≤⋯≤μk 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 ℋ as λ → 0 such that 𝒥 ˇ ⁢ ( 𝐮 ^ → 0 , β ) = ℭ 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 ℭ ˇ ⁢ ( β ) = inf ℳ 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 ℒ ⁢ ( 𝐮 → )   𝑎𝑛𝑑   ℳ ˇ = { 𝐮 → ∈ ℋ \ { 𝟎 → } ∣ 𝒥 ˇ ′ ⁢ ( 𝐮 → ) ⁢ 𝐮 → = 0 } ,

   where ℒ ⁢ ( 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 , N≥4 and β>βk, where βk is given by Proposition 3.3, then we have 𝒜λ→+∞ and viλ,β→t~i⁢Uε,z strongly in D1,2⁢(ℝN) for all i=1,2,…,k as λ→λ0, 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 ∥ ∞ , ℝ 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 λ→λ0, 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 𝐮→λ,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.

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

which is impossible.

(2) By Proposition A.1,

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

Now let us also suppose that system (1.1) has a solution 𝐮=(u1,u2,…,uk). Let 𝐯1=(e1⁢φ1,e2⁢φ1,…,ek⁢φ1) 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 λ≥λ1 that