Home Mathematics Symmetric results of a Hénon-type elliptic system with coupled linear part
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Symmetric results of a Hénon-type elliptic system with coupled linear part

  • Zhenluo Lou EMAIL logo , Huimin Li and Ping Zhang
Published/Copyright: December 31, 2022
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Open Mathematics
From the journal Open Mathematics Volume 20 Issue 1

Abstract

In this article, we study the elliptic system:

− Δ u + μ 1 u = ∣ x ∣ α u 3 + λ v , x ∈ Ω − Δ v + μ 2 v = ∣ x ∣ α v 3 + λ u , x ∈ Ω u , v > 0 , x ∈ Ω , u = v = 0 , x ∈ ∂ Ω ,

where Ω ⊂ R 3 is the unit ball. By the variational method, we prove that if α is sufficiently small, the ground state solutions of the system are radial symmetric, and if α > 0 is sufficiently large, the ground state solutions are nonradial; however, the solutions are Schwarz symmetry.

Keywords: elliptic system; variational method; ground state solutions; symmetry
MSC 2010: 35A15; 35B06; 35B09; 35B38; 35J57

1 Introduction

In this article, we study the following elliptic system:

(1.1) − Δ u + μ 1 u = ∣ x ∣ α u 3 + λ v , x ∈ Ω − Δ v + μ 2 v = ∣ x ∣ α v 3 + λ u , x ∈ Ω u , v > 0 , x ∈ Ω , u = v = 0 , x ∈ ∂ Ω ,

where Ω ⊂ R 3 is the unit ball, μ 1 , μ 2 > 0 , α > 0 . The parameter λ > 0 satisfies several conditions.

System (1.1) is the nonautonomous case of the following elliptic system:

(1.2) − Δ u + μ 1 u = u 3 + λ v , x ∈ Ω − Δ v + μ 2 v = v 3 + λ u , x ∈ Ω u , v > 0 , x ∈ Ω , u = v = 0 , x ∈ ∂ Ω ,

where Ω ⊂ R N ( N = 2 , 3 ) is an open domain. (1.2) arises in a binary mixture of the Bose-Einstein condensates with a coupled linear part. In recent years, many researchers have been concerned with system (1.2). They have proved the existence, multiplicity, and limit problems of the solutions of system (1.2), and see [1–4] and the references therein.

Gidas et al. [5] studied the elliptic equation:

(1.3) − Δ u = f ( ∣ x ∣ , u ) , u > 0 , x ∈ Ω , u = 0 , x ∈ ∂ Ω .

They proved that if Ω is a ball and f ( ∣ x ∣ , u ) = f ( u ) ∈ C 1 , then the solutions are radially symmetric. After that, many researchers studied the symmetric properties of elliptic equations, and see [6–9] and the references therein. In [7], they pointed out that if the domain cannot be convex or f ( ∣ x ∣ , u ) is increasing by ∣ x ∣ , one can have the nonradial positive solutions. Smets et al. [9] studied the Hénon equation:

(1.4) − Δ u = ∣ x ∣ α u p , u > 0 , x ∈ Ω , u = 0 x ∈ ∂ Ω ,

where Ω is the unit ball and α > 0 , 1 < p < 2 ∗ − 1 . They proved that if α > 0 is large enough, the ground state solutions of (1.4) is nonradial symmetry, and Boheure et al. [6] studied the symmetry-breaking results of the Hénon-Lane-Emden system, and see also [7] and the references therein.

Following these ideas, we prove the following.

Theorem 1.1

Let λ > 0 be small. Then there exists α ¯ > 0 such that for all α > α ¯ the ground state solution is nonradial. However, the solutions are foliated Schwarz symmetric.

In [8,10,11], Kajikiya obtained the multiple nonradial solutions of equation (1.4) by the group critical theory. By the unique and nondegenerate results, they proved, if α is sufficiently small, the ground state solutions of (1.4) is radially symmetric. It is an interesting result; since for the Hénon equation, the moving plane method cannot be used, one also prove the radial results. In [12,13], the authors also considered the nonradial solutions. In this article, we first obtain a result of unicity, then we prove the following:

Theorem 1.2

For all λ > 0 be sufficiently small, the system (1.2) has a unique positive radial solution. If this solution is nondegenerate, then for α small, the ground state solution of system (1.1) should be radially symmetric.

This article is organized as follows. In Section 2, we present several preliminary results, and in Sections 3 and 4, we present the proof of the main results.

2 Some preliminaries and lemmas

Let H 0 1 ( Ω ) be the classical Sobolev space and ℋ = H 0 1 ( Ω ) × H 0 1 ( Ω ) with the norm:

‖ ( u , v ) ‖ = ∫ Ω ∣ ∇ u ∣ 2 + μ 1 u 2 + ∣ ∇ v ∣ 2 + μ 2 v 2 d x 1 2 .

We assume 0 < λ < min { μ 1 , μ 2 } .

The energy functionals of (1.1) and (1.2) are

I α ( u , v ) = 1 2 ‖ ( u , v ) ‖ 2 − 1 4 ∫ Ω ∣ x ∣ α ( u 4 + v 4 ) d x − λ ∫ Ω u v d x

and

I ( u , v ) = 1 2 ‖ ( u , v ) ‖ 2 − 1 4 ∫ Ω ( u 4 + v 4 ) d x − λ ∫ Ω u v d x .

The Nehari manifolds are

N α = { ( u , v ) ∈ ℋ \ { ( 0 , 0 ) } : I α ′ ( u , v ) ( u , v ) = 0 }

and

N = { ( u , v ) ∈ ℋ \ { ( 0 , 0 ) } : I ′ ( u , v ) ( u , v ) = 0 } .

Definition 2.1

( u ¯ , v ¯ ) is the ground state solution of (1.1), if it satisfies

c α = I α ( u ¯ , v ¯ ) = inf N α I α ( u , v ) .

If the infimum is obtained from the radial functions, we denote it as c α r .

By the moving plane method, one can prove that the solutions of system (1.2), denoted ( ω , h ) , are radial. If

(2.1) − Δ ϕ + μ 1 ϕ = 3 ω 2 ϕ + λ ψ , − Δ ψ + μ 2 ψ = 3 h 2 ψ + λ ϕ

has only trivial solutions, in this case ( ω , h ) is nondegenerate. One can check Damascelli and Pacella [14] and Troy [15] for the radially symmetric results, see also Dancer et al. [16], Sato and Wang [17], and Tavares and Weth [18]. For unique and nondegenerate results of the elliptic system, one can see [19–21].

3 The Proof of Theorem 1.1

Lemma 3.1

For any ( u , v ) ∈ ℋ , there exists a unique t > 0 such that ( t u , t v ) ∈ N α .

Proof

For any ( u , v ) ∈ H , by calculating directly, we have

(3.1) I α ′ ( t u , t v ) ( t u , t v ) = t 2 ‖ ( u , v ) ‖ 2 − 2 t 2 λ ∫ Ω u v d x − t 4 ∫ Ω ∣ x ∣ α ( u 4 + v 4 ) d x ,

and thus,

t 2 = ‖ ( u , v ) ‖ 2 − 2 λ ∫ Ω u v d x ∫ Ω ∣ x ∣ α ( u 4 + v 4 ) d x .

We have completed the proof.□

Lemma 3.2

c α is achieved, and if α > 0 is large enough, we have that

c α ≤ C α ,

where C is a constant independent of α .

Proof

By the variational method, we can prove that c α and c α r are achieved, and c α ≤ c α r . Now we give the estimate of c α .

Let φ , ψ ∈ C 0 ∞ ( Ω ) and x 0 = 1 − 1 α , 0 , 0 , and

φ α ( x ) = φ ( α ( x − x 0 ) ) , ψ α ( x ) = ψ ( α ( x − x 0 ) ) .

When α is large, one can prove supp φ α ⊆ Ω , supp ψ α ⊆ Ω . Since C 0 ∞ ( Ω ) ⊆ H 0 1 ( Ω ) , then

c α ≤ I α ( t φ , t ψ ) ,

where t satisfies ( t φ , t ψ ) ∈ N α . Then we have

I α ( t φ α , t ψ α ) = 1 4 t 4 ∫ Ω ∣ x ∣ α ( φ α 4 + ψ α 4 ) d x

and

t 2 = ‖ ( φ α , ψ α ) ‖ 2 − 2 λ ∫ Ω φ α ψ α d x ∫ Ω ∣ x ∣ 4 ( φ α 4 + ψ α 4 ) d x .

By combining these results, we have that

I α ( t φ α , t ψ α ) = 1 4 ‖ ( φ α , ψ α ) ‖ 2 − 2 λ ∫ Ω φ α ψ α d x 2 ∫ Ω ∣ x ∣ 4 ( φ α 4 + ψ α 4 ) d x .

Let y = α ( x − x 0 ) , then d x = 1 α 3 d y . Because of supp φ α ⊂ Ω , then we have y ∈ Ω , so x − x α ∈ Ω 1 α , which is a ball with r = 1 α , and then we have that

(3.2) ∫ Ω ∣ x ∣ α ( φ α 4 + ψ α 4 ) d x ≥ 1 − 2 α α ∫ Ω ( φ α 4 + ψ α 4 ) d x = 1 − 2 α α ∫ Ω ( φ 4 + ψ 4 ) d x = 1 − 2 α α ∫ Ω 1 α 3 ( φ 4 + ψ 4 ) d y .

On the other hand, we have

(3.3) ∫ Ω ( ∣ ∇ φ α ∣ 2 + φ α 2 ) d x = ∫ Ω α − 3 ( ∣ ∇ φ ∣ 2 + φ 2 ) d y = α − 1 ∫ Ω ( α − 2 ∣ ∇ φ ∣ 2 + α − 2 φ 2 ) d y ≤ α − 1 ∫ Ω ( ∣ ∇ φ ∣ 2 + α − 2 φ 2 ) d y ,

and similarly, we have that

(3.4) ∫ Ω ( ∣ ∇ ψ α ∣ 2 + ψ α 2 ) d x ≤ α − 1 ∫ Ω ( ∣ ∇ ψ ∣ 2 + α − 2 ψ 2 ) d y .

Combining the previous inequalities (3.2)–(3.4), we have that

c α ≤ C α ,

where C is a positive constant independent of α .□

Lemma 3.3

Let ( u α , v α ) be a radial ground state solution of (1.1) and c α r is the energy value. Then we have that

c α r ≥ C α 3 ,

where C is a constant independent of α .

Proof

For convenience, we replace ( u , v ) of ( u α , v α ) . Define

Y ( x ) = ( ∇ u , x ) ∇ u + ( ∇ v , x ) ∇ v − 1 2 ( ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) x + 1 4 ∣ x ∣ α ( u 4 + v 4 ) x + λ u v x − 1 2 ( μ 1 u 2 + μ 2 v 2 ) x ,

where x ∈ R 3 . Let

r = x 1 2 + x 2 2 + x 3 2 ,

then we have

( ∇ u , x ) ∇ u = ( u r ′ ) 2 ( x 1 , x 2 , x 3 ) , ( ∇ v , x ) ∇ v = ( v r ′ ) 2 ( x 1 , x 2 , x 3 ) ,

and

(3.5) div ( ∇ u , x ) ∇ u − 1 2 ∣ ∇ u ∣ 2 x + 1 4 ∣ x ∣ α u 4 x − 1 2 u 2 x = 3 2 ∣ ∇ u ∣ 2 + 3 + α 4 ∣ x ∣ α u 4 − 3 2 u 2 + r u r ′ ( u r ″ + u 3 ∣ x ∣ α − u ) ,

and

(3.6) div ( ∇ v , x ) ∇ v − 1 2 ∣ ∇ v ∣ 2 x + 1 4 ∣ x ∣ α v 4 x − 1 2 v 2 x = 3 2 ∣ ∇ v ∣ 2 + 3 + α 4 ∣ x ∣ α v 4 − 3 2 v 2 + r v r ′ ( v r ″ + v 3 ∣ x ∣ α − v ) ,

and

(3.7) div ( λ u v x ) = 3 λ u v + λ r u v r ′ + λ r v u r ′ .

By combining the previous equalities, we obtain the following:

(3.8) div Y ( x ) = 3 2 ( ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) + 3 + α 4 ∣ x ∣ α ( u 4 + v 4 ) + 3 λ u v − 3 2 ( μ 1 u 2 + μ 2 v 2 ) + r u r ′ ( u r ″ + u 3 ∣ x ∣ α − μ 1 u + λ v ) + r v r ′ ( v r ″ + v 3 ∣ x ∣ α − μ 2 v + λ u ) .

Since ( u , v ) is a radial solution,

u r ″ = Δ u − 2 u r ′ r , v r ″ = Δ v − 2 v r ′ r ,

r u r ′ ( u r ″ + u 3 ∣ x ∣ α − μ 1 u + λ v ) = r u r ′ Δ u − 2 u r ′ r + ∣ x ∣ α u 3 − μ 1 u + λ v = r u r ′ ⋅ − 2 u r ′ r = − 2 ( u r ′ ) 2 = − 2 ∣ ∇ u ∣ 2 ,

− 2 ∣ ∇ v ∣ 2 = r v r ′ ( v r ″ + v 3 ∣ x ∣ α − μ 2 v + λ u ) .

Thus,

div Y ( x ) = 3 + α 4 ∣ x ∣ α ( u 4 + v 4 ) + 3 λ u v − 1 2 ( ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) − 3 2 ( μ 1 u 2 + μ 2 v 2 ) .

By divergence theorem, we obtain that , i.e.,

∫ Ω div Y ( x ) d x = ∫ ∂ Ω Y ( x ) n d S

∫ ∂ Ω ( ∇ u , x ) ∇ u ⋅ n + ( ∇ v , x ) ∇ v ⋅ n − 1 2 ( ∣ ∇ u ∣ 2 ⋅ x + ∣ ∇ v ∣ ⋅ x ) ⋅ n d S = ∫ Ω 3 + α 4 ∣ x ∣ α ( u 4 + v 4 ) + 3 λ ( ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) − 3 2 ( μ 1 u 2 + μ 2 v 2 ) d x .

Again since ( u , v ) is a solution, we have that

∫ ∂ Ω ( ∇ u , x ) ∂ u ∂ n + ( ∇ v , x ) ∂ v ∂ n − 1 2 ( ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) ( x ⋅ n ) d S = α − 3 4 ∫ Ω ∣ x ∣ α ( u 4 + v 4 ) d x + ∫ Ω ( ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) d x ≥ α − 3 4 ∫ Ω ∣ x ∣ α ( u 4 + v 4 ) d x .

Since u and v are radial functions, following pp. 72 of [6], we have

(3.9) ( ∇ u , x ) ( ∇ v , x ) = ( ∇ u , ∇ v ) ∣ x ∣ 2

and

∫ ∂ Ω ∂ u ∂ n 2 d S = C ∫ ∂ Ω ∂ u ∂ n d S 2 , ∫ ∂ Ω ∂ v ∂ n 2 d S = C ∫ ∂ Ω ∂ v ∂ n d S 2 .

Since

c α = 1 4 ∫ Ω ∣ x ∣ α ( u 4 + v 4 ) d x

and

∫ ∂ Ω ( ∇ u , x ) ∂ u ∂ n + ( ∇ v , x ) ∂ v ∂ n − 1 2 ( ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) ( x ⋅ n ) d S = 1 2 ∫ ∂ Ω ∂ u ∂ n 2 + ∂ v ∂ n 2 d S ,

and then we obtain

(3.10) ( α − 3 ) c α ≤ 1 2 ∫ ∂ Ω ∂ u ∂ n 2 + ∂ v ∂ n 2 d S .

By the divergence theorem, we have ∫ Ω Δ u d x = ∫ ∂ Ω ∂ u ∂ n d S . Notice that

− Δ u + μ 1 u = ∣ x ∣ α u 3 + λ v ,

and

(3.11) ∫ Ω ∣ x ∣ α u 3 d x = ∫ Ω ∣ x ∣ α 4 ⋅ ∣ x ∣ 3 α 4 u 3 d x ≤ ∫ Ω ∣ x ∣ α d x 1 4 ∫ Ω ∣ x ∣ α u 4 d x 3 4 = C ∫ 0 1 r α + 2 d r 1 4 ∫ Ω ∣ x ∣ α u 4 d x 3 4 = C ( α + 3 ) 1 4 ( ∫ Ω ∣ x ∣ α u 4 d x ) 3 4 ,

and

(3.12) ∫ Ω λ v + μ 1 u d x ≤ C ∫ Ω ∣ ∇ u ∣ 2 d x 1 2 + ∫ Ω ∣ ∇ v ∣ 2 d x 1 2 ≤ 2 C ∫ Ω ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 d x 1 2 .

Thus, we obtain that

(3.13) ∫ Ω Δ u d x ≤ 1 ( α + 3 ) 1 4 ∫ Ω ∣ x ∣ α u 4 d x 3 4 + 2 C ∫ Ω ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 d x 1 2 ≤ 1 ( α + 3 ) 1 4 c α 3 4 + C c α 1 2

and

(3.14) ∫ Ω Δ v d x ≤ 1 ( α + 3 ) 1 4 c α 3 4 + C ( λ ) c α 1 2 .

We also obtain that

∫ ∂ Ω ∂ u ∂ n 2 + ∂ v ∂ n 2 d S = C ∫ Ω Δ u d x 2 + ∫ Ω Δ v d x 2 ,

and by the previous estimations, one can obtain that

(3.15) ∫ ∂ Ω ∂ u ∂ n 2 + ∂ v ∂ n 2 d S ≤ C 1 ( α + 3 ) 1 4 c α 3 4 + C c α 1 2 2 .

From (3.10), we obtain

( α − 3 ) c α ≤ 1 ( α + 3 ) 1 4 c α 3 4 + C c α 1 2 2 ,

and then we have

( α − 3 ) 1 ∕ 2 c α 1 ∕ 2 ≤ C 1 ( α + 3 ) 1 4 c α 3 4 + C 2 c α 1 2

and

( α − 3 ) 1 ∕ 2 ≤ C 1 ( α + 3 ) 1 4 c α 1 4 + C 2 .

If α > 0 is large enough, we obtain α 3 4 ≤ C c α 1 4 , i.e.,

c α ≥ C α 3 ,

where C , C 1 , and C 2 are positive constants independently of α . In conclusion, we complete the proof.□

Now we list the proof of Theorem 1.1.

The proof of Theorem 1.1

By Lemmas 3.2 and 3.3, we can prove that if α > 0 is large enough, the ground state solutions cannot be radial. Let

f 1 ( ∣ x ∣ , u , v ) = ∣ x ∣ α u 3 + λ v , f 2 ( ∣ x ∣ , u , v ) = ∣ x ∣ α v 3 + λ u ,

then f 1 ( ∣ x ∣ , u , v ) and f 2 ( ∣ x ∣ , u , v ) are convex with respect to u and v . Since ( u , v ) is a ground state solution, then the Morse index M ( u , v ) ≤ 1 < N = 3 , and then by [14], we have that ( u , v ) is foliated Schwarz symmetry, and one can also follow the idea of Wang and Willem [23].

4 The Proof of Theorem 1.2

By using the idea of Wei and Yao [20], we obtain the following results.

Lemma 4.1

Let λ > 0 be small enough. Then the positive solution of system (1.2) is radially symmetric, and it is unique.

Proof

Following the idea of Gidas et al. [5], Damascelli and Pacella [14], or Troy [15], we prove that the positive solutions are radially symmetric.

Let ω i be the unique positive solution of the following scalar equation with the subcritical exponent:

− Δ u + μ i u = u 3 x ∈ Ω , u = 0 x ∈ ∂ Ω ,

where i = 1 , 2 .

Let Φ ( λ , u , v ) = I α ′ ( u , v ) , one can prove that Φ ( 0 , ω 1 , ω 2 ) = 0 , and we have

Φ ′ ( 0 , ω 1 , ω 2 ) ( ϕ , ψ ) = ∫ Ω ∣ ∇ ϕ ∣ 2 + μ 1 ϕ 2 + ∣ ∇ ψ ∣ 2 + ψ 2 d x − 3 ∫ Ω ω 1 2 ϕ 2 + ω 2 ψ 2 d x .

By the compact theorem, one can prove that Φ ′ ( 0 , ω 1 , ω 2 ) is invertible, and then by implicity function theorem, if λ > 0 is small enough, the solution of system is uniqueness.□

Next we present the proof of Theorem 1.2.

By contradiction, the ground state solutions of (1.1) are nonradial. By the variational method, we prove that (1.1) has a nontrivial radial solution, one can see [22], denoting ( u ˜ α , v ˜ α ) , and another nonradial solution, denoting ( u α , v α ) . Now we give the L ∞ estimate of ( u α , v α ) and ( u ˜ α , v ˜ α ) , for convenience, we replace ( u α , v α ) by ( u , v ) . Since ( u , v ) is a solution of (1.1), let u 2 q − 1 be a test function, then we have

∫ Ω − Δ u u 2 q − 1 + μ 1 u 2 q d x = ∫ Ω ∣ x ∣ α u 2 q + 2 d x + λ ∫ Ω u 2 q − 1 v d x ,

where 2 q − 1 > 1 . By Green’s formula, we obtain that

LHS = ∫ Ω ∇ u ∇ ( u 2 q − 1 ) + μ 1 u 2 q d x = ( 2 q − 1 ) ∫ Ω ∣ ∇ u ∣ 2 u 2 q − 2 d x + μ 1 ∫ Ω u 2 q d x = 2 q − 1 q 2 ∫ Ω ∣ ∇ u q ∣ 2 d x + μ 1 ∫ Ω ∣ u q ∣ 2 d x .

By Hölder’s and Young’s inequality, we have that

∫ Ω v u 2 q − 1 d x ≤ ∫ Ω u 2 q + 2 d x 2 q − 1 2 q + 2 ∫ Ω v 2 q + 2 3 d x 3 2 q + 2 ≤ 2 q − 1 2 q + 2 ∫ Ω u 2 q + 2 d x + 3 2 q + 2 ∫ Ω v 2 q + 2 3 d x ≤ 2 q − 1 2 q + 2 ∫ Ω u 2 q + 2 d x + 3 2 q + 2 ∫ Ω v 2 q + 2 1 ∕ 3 ∫ Ω 1 d x 2 ∕ 3 ≤ ∫ Ω u 2 q + 2 d x + C ≤ C ∫ Ω u 2 q + 2 d x ,

where C is a constant independent of q . On the other hand,

∫ Ω ∣ x ∣ α u 2 q + 2 d x ≤ ∫ Ω u 2 q + 2 d x .

Then we obtain that

(4.1) ∫ Ω ∣ ∇ u q ∣ 2 + ∣ u q ∣ 2 d x ≤ C q 2 2 q − 1 ∫ Ω u 2 q + 2 d x .

Again by Hölder and Sobolev embedding inequality, we obtain

∫ Ω u 2 q + 2 d x ≤ ∫ Ω u 3 q d x 2 ∕ 3 ( u 6 d x ) 1 ∕ 3 ≤ C ∫ Ω u 3 q d x 2 ∕ 3

and

∫ Ω u 4 q d x 1 4 ≤ ∫ Ω ∣ ∇ u q ∣ 2 + ∣ u q ∣ 2 d x 1 2 .

Thus, we obtain that

∫ Ω u 4 q d x 1 4 q ≤ c q 2 2 q − 1 1 2 q ∫ Ω u 3 q d x 1 3 q .

Now let q ( 1 ) = 1 and q ( n + 1 ) = 4 3 q ( n ) , and one can prove that q ( n ) → + ∞ as n → + ∞ . By the proceeding estimate, we have that

‖ u ‖ 4 q ( n + 1 ) ≤ C ∏ i = 2 n + 1 q 2 ( i ) 2 q ( i ) − 1 1 2 q ( i ) ‖ u ‖ 3 q ( 1 ) .

Notice that q ( n + 1 ) = 4 3 n , and thus, we can prove that

ln ∏ i = 2 n + 1 q 2 ( i ) 2 q ( i ) − 1 1 2 q ( i ) = ∑ i = 2 n + 1 2 ln q ( i ) − ln ( 2 q ( i ) − 1 ) 2 q ( i ) < ∞ ,

and thus, ∏ i = 2 n + 1 q 2 ( i ) 2 q ( i ) − 1 1 2 q ( i ) < ∞ , then we complete the L ∞ estimate.

Let λ > 0 satisfy the previous conditions, and { α n } satisfy α n → 0 as n → + ∞ . We denote that { ( u n , v n ) } is the ground state solutions of system (1) and { ( u ˜ n , v ˜ n ) } is the radial ground state solutions of system (1).

Following the idea of [8], by contradiction, u n ≠ u ˜ n , v n ≠ v ˜ n , let

ω n = u n − u ˜ n ‖ u n − u ˜ n ‖ ∞ , h n = v n − v ˜ n ‖ v n − v ˜ n ‖ L ∞ ,

then ω n ≠ 0 , h n ≠ 0 and ω n , h n ∈ W 2 , t ( Ω ) , which is classical Sobolev space, for any 1 < t < + ∞ . Let

f 1 ( u , v ) = ∣ x ∣ α u 3 + λ v and f 2 ( u , v ) = ∣ x ∣ α v 3 + λ u ,

then

(4.2) − Δ ω n + μ 1 ω n = f 1 ( u n , v n ) − f 1 ( u ˜ n , v ˜ n ) u n − u ˜ n ω n , − Δ h n + μ 2 h n = f 2 ( u n , v n ) − f 2 ( u ˜ n , v ˜ n ) v n − v ˜ n h n .

By the L ∞ estimate, we have ω n → ω , h n → h in C 1 ( Ω ) , and ω and h satisfy

(4.3) − Δ ω + μ 1 ω = 3 u 0 2 ω + λ c 1 h , − Δ h + μ 2 h = 3 v 0 2 h + λ c 2 ω ,

where c 1 = lim n → + ∞ ‖ v n − v ˜ n ‖ ∞ ‖ u n − u ˜ n ‖ ∞ and c 2 = 1 c 1 are positive constants, and ( u o , v 0 ) is the uniqueness solution of (1.2).

Now let ω ¯ = ω c 1 and h ¯ = h , we have that

(4.4) − Δ ω ¯ + μ 1 ω ¯ = 3 u 0 2 ω ¯ + λ h ¯ , − Δ h ¯ + μ 2 h ¯ = 3 λ 2 v 0 2 h ¯ + λ ω ¯ .

It is a contradiction, then we complete the proof.

Acknowledgement

The authors thank the referees for valuable comments and suggestions, which improved the presentation of the manuscript.

  1. Funding information: Lou is supported by NSFC 11571339 and 12101192 and Key Scientific Research Projects of Higher Education Institutions in Henan Province 20B110004.

  2. Conflict of interest: The authors state no conflict of interest.

  3. Data availability statement: No data, models, or code are generated or used during the study.

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Received: 2022-02-13
Revised: 2022-11-28
Accepted: 2022-12-03
Published Online: 2022-12-31

© 2022 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
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  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
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  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
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  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
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  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
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  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
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  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
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  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
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  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
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  109. Some new characterizations of finite p-nilpotent groups
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  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
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  127. Review Article
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  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
https://doi.org/10.1515/math-2022-0539
Keywords for this article
elliptic system; variational method; ground state solutions; symmetry
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Articles in the same Issue

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
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