Home Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
Article Open Access

Existence of normalized peak solutions for a coupled nonlinear Schrödinger system

  • Jing Yang EMAIL logo
Published/Copyright: January 27, 2024
Download Article (PDF)
Advances in Nonlinear Analysis
From the journal Advances in Nonlinear Analysis Volume 13 Issue 1

Abstract

In this article, we study the following nonlinear Schrödinger system

− Δ u 1 + V 1 ( x ) u 1 = α u 1 u 2 + μ u 1 , x ∈ R 4 , − Δ u 2 + V 2 ( x ) u 2 = α 2 u 1 2 + β u 2 2 + μ u 2 , x ∈ R 4 ,

with the constraint ∫ R 4 ( u 1 2 + u 2 2 ) d x = 1 , where α > 0 and α > β , μ ∈ R , V 1 ( x ) , and V 2 ( x ) are bounded functions. Under some mild assumptions on V 1 ( x ) and V 2 ( x ) , we prove the existence of normalized peak solutions by using the finite dimensional reduction method, combined with the local Pohozaev identities. Because of the interspecies interaction between the components, we aim to obtain some new technical estimates.

Keywords: normalized peak solutions; local Pohozaev identities; reduction method
MSC 2010: 35A10; 35B99; 35J60

1 Introduction and main result

In this article, we consider the following Schrödinger system with coupled quadratic nonlinearities

(1.1) − Δ u 1 + V 1 ( x ) u 1 = α u 1 u 2 + μ u 1 , x ∈ R 4 , − Δ u 2 + V 2 ( x ) u 2 = α 2 u 1 2 + β u 2 2 + μ u 2 , x ∈ R 4 ,

under the constraint

(1.2) ∫ R 4 ( u 1 2 + u 2 2 ) d x = 1 ,

where α > 0 and α > β , μ ∈ R is a chemical potential, and V 1 ( x ) and V 2 ( x ) are bounded functions.

Such type of systems like (1.1) with quadratic interaction have wide applications in physics, such as Bose-Einstein condensates, plasma physics, and nonlinear optics. For example, the following coupled nonlinear Schrödinger equations

(1.3) i ∂ ω k ∂ t + Δ ω k + ∑ j = 1 K μ j , k ∣ ω j ∣ 2 ω k = 0

for some μ j , k ∈ R and j , k = 1 , 2 , … , K , can describe the propagation of solitons with χ ( 3 ) nonlinear fiber couplers in nonlinear optic theory, where the complex function ω k denotes the k th component of the light beam, and ∑ j = 1 K ∣ ω j ∣ 2 is the change in refractive index profile created by all the incoherent components in the light beam. This system has gotten a lot of attentions experimentally and theoretically. If we consider the standing wave solutions for (1.3) of the form ω k ( x , t ) = e i λ k t u k ( x ) with u k a function independent of time, then u k satisfies

(1.4) Δ u k − λ k u k + ∑ j = 1 K μ j , k u j 2 u k = 0

for k = 1 , 2 , … , K . The existence and multiplicity of standing wave solutions to (1.4) and its relates problem have been explored by many authors in recent years, see previous studies by [2–4,6,9,10,13,22,24,26,30] and the references therein. However, when nonlinear optical effects such as second harmonic generation are investigated in the optical material that has a χ ( 2 ) nonlinear response, it is led to the system (1.1) (c.f. [1,7]). For the existence, multiplicity and asymptotic behavior of solutions for (1.1), we can refer to [27–29,31].

We note that the following Gross-Pitaevskii (GP) equations proposed by Gross [15] and Pitaevskii [25] in the 1960s

(1.5) i ∂ ω ( x , t ) ∂ t = − Δ ω ( x , t ) + V ( x ) ω ( x , t ) − a ∣ ω ( x , t ) ∣ 2 ω ( x , t ) , x ∈ R N ,

with the constraint

∫ R N ∣ ω ( x , t ) ∣ 2 d x = 1 ,

has been investigated extensively due to some new experimental advances in quantum phenomena (c.f. [5,11,14]). Here, N ≥ 2 , V ( x ) ≥ 0 is a real-valued potential and a ∈ R is treated as an arbitrary dimensionless parameter. If we want to find a solution for (1.5) of the form ω ( x , t ) = e − i μ t u ( x ) , where μ represents the chemical potential of the condensate and u ( x ) is a function independent of time, then the unknown pair ( μ , u ) satisfies the following nonlinear eigenvalue equation

(1.6) − Δ u + V ( x ) u = a u 3 + μ u , in R N ,

with the constraint

(1.7) ∫ R N u 2 d x = 1 .

For ground states of equation (1.6), one can refer to the studies by Guo et al. [18–20] and the references therein, where the main tools are the energy comparison under various assumptions of trapping potential V ( x ) , which can be described equivalently by positive L 2 minimizers of the following functional:

I a = 1 2 ∫ R N ( ∣ ∇ u ∣ 2 + V ( x ) u 2 ) d x − a 4 ∫ R N u 4 d x .

Very recently the existence and the local uniqueness of excites states for a class of degenerated trapping potential with nonisolated critical points were given in the study by Luo et al. [23], where we call any eigenfunction of equations (1.6) and (1.7) whose energy is larger than that of the ground state the excited states in the physics literatures (c.f. [8]). As for two-component Bose-Einstein condensates with the mass constraint, the studies by Guo and Yang [16] showed the existence of the general exited states by using the reduction argument combined with the local Pohozaev identities, which generalize the existence of the ground state with trapping potentials given in the study by Guo et al. [17] to some extent.

To our best knowledge, there seems to be no results on the existence of peak solutions to (1.1) with the L 2 constraint (1.2). So in this article, we want to investigate this problem and suppose the following conditions hold.

( K 1 ): V 1 ( x ) and V 2 ( x ) have m common critical points a 1 , … , a m for m ≥ 1 with a 1 , … , a m the nondegenerated critical points of Γ 1 2 V 1 ( x ) + Γ 2 2 V 2 ( x ) , where Γ 1 and Γ 2 are given below.

( K 2 ): V j ( x ) < V j ( a 1 ) for x ∈ B d ( a 1 ) \ { a 1 } and V j ( x ) ∈ C θ ( B ¯ d ( a 1 ) ) for some θ ∈ ( 0 , 1 ) with some small d > 0 for j = 1 , 2 .

To state our results, we first introduce some notations. Let U be the unique positive solution of the following problem:

− Δ u + u = u 2 , in R 4 , u ( 0 ) = max x ∈ R 4 u ( x ) , u ( x ) ∈ H 1 ( R 4 ) ,

and we denote b * = ∫ R 4 U 2 d x . From [21], we know that U ( x ) = U ( ∣ x ∣ ) is strictly decreasing and satisfies for ∣ s ∣ ≤ 1

∣ D s U ( x ) ∣ e ∣ x ∣ ∣ x ∣ − 3 2 ≤ C .

Also ( U 1 , U 2 ) ≔ ( Γ 1 U , Γ 2 U ) is the ground state of

(1.8) − Δ u 1 + u 1 = α u 1 u 2 in R 4 , − Δ u 2 + u 2 = α 2 u 1 2 + β u 2 2 , in R 4 ,

provided that α > β with

Γ 1 = 1 α 2 ( α − β ) α , Γ 2 = 1 α .

For any t ∈ R + and y ∈ R 4 , we define for i = 1 , 2 , U i , t , y ≔ t U i ( t ( x − y ) ) , and for any δ > 0 , ‖ ( u , v ) ‖ δ 2 ≔ ‖ u ‖ 1 , δ 2 + ‖ v ‖ 2 , δ 2 with

‖ u ‖ i , δ 2 ≔ 1 δ ∫ R 4 ( ∣ ∇ u ∣ 2 + ( δ + V i ( x ) ) u 2 ) d x .

The first result of our article is as follows.

Theorem 1.1

Assume that ( K 1 ) holds and α > β and α > 0 , then there exists a small constant ε > 0 such that for any ( α , β ) satisfying ∣ 3 α − 2 β α 3 − 1 m b * ∣ ≤ ε , problems (1.1) and (1.2) have a peak solution ( u 1 , μ , u 2 , μ , μ ) depending on α and β with the form

( u 1 , μ ( x ) , u 2 , μ ( x ) ) = ∑ j = 1 m U 1 , μ , y μ , j ( x ) + φ 1 , μ ( x ) , ∑ j = 1 m U 2 , μ , y μ , j ( x ) + φ 2 , μ ( x ) .

Also as 3 α − 2 β α 3 → 1 m b * , there hold μ → − ∞ and

∣ y μ , j − a j ∣ = o 1 − μ a n d ‖ ( φ 1 , μ , φ 2 , μ ) ‖ − μ = O − 1 μ .

Theorem 1.1 tells that we construct a synchronized solution for problems (1.1) and (1.2), where the synchronized solution means that the two components of the solutions for system (1.1) concentrate at the same set of concentrated points. Otherwise, we call it a segregated solution if the components concentrate at two different set of points.

Remark 1.2

If V 1 ( x ) and V 2 ( x ) have no common nondegenerate critical points, that is, V 1 ( x ) has m critical points a 1 , … , a m , and while V 2 ( x ) has m ¯ critical points a ¯ 1 , … , a ¯ m ¯ with m , m ¯ ≥ 1 , then we also can construct a segregated peak solution ( u 1 , μ , u 2 , μ , μ ) with u 1 , μ and u 2 , μ concentrating at these points, respectively.

Now we want to consider the clustering peak solutions to problems (1.1) and (1.2), and the another result is as follows:

Theorem 1.3

Assume that ( K 2 ) holds and α > β and α > 0 , then there exists a small constant ε > 0 such that for any ( α , β ) satisfying ∣ 3 α − 2 β α 3 − 1 m b * ∣ ≤ ε , problems (1.1) and (1.2) have a peak solution ( u 1 , μ , u 2 , μ , μ ) depending on α and β with the form

( u 1 , μ ( x ) , u 2 , μ ( x ) ) = ∑ j = 1 m U 1 , μ , y μ , j ( x ) + φ 1 , μ ( x ) , ∑ j = 1 m U 2 , μ , y μ , j ( x ) + φ 2 , μ ( x ) .

Also as 3 α − 2 β α 3 → 1 m b * , there hold μ → − ∞ , ‖ ( φ 1 , μ , φ 2 , μ ) ‖ − μ = O − 1 μ and

y μ , j → a 1 a n d μ ∣ y μ , j − y μ , l ∣ → ∞ , f o r l ≠ j .

Remark 1.4

To our best knowledge, this seems to be the first time to consider the existence of the normalized peak (or bubbling) solutions for problems (1.1) and (1.2). Also we note that only one coefficient a in (1.6) need to be discussed with the constraint mass, while there are two parameters α and β in system (1.1) to be considered, corresponding to the interactions within and between the components, respectively, which makes the problems (1.1) and (1.2) more complicated than the single equation, and we have to analyze the mutual influence of the parameters and the constraint condition very carefully.

We will mainly use the finite dimensional reduction to prove our results, which is an effective way to construct solutions for perturbed elliptic problems. Since the singularly perturbed problem has a small parameter naturally, the approximate solutions can be constructed by the standard steps of the reduction argument. However, the constraint problems (1.1) and (1.2) itself do not provide any natural limiting process explicitly. So to apply the reduction method, we have to instigate the relationship between the constraint conditions and the parameters appeared in the system. This process involve some various Pohozaev identities, which play an important role and need some more accurate estimates.

The structure of this article is organized as follows. We carry out the finite dimensional reduction to study the corresponding problem without constraint in Section 2. In Section 3, we prove Theorems 1.1 and 1.3 is proved in Section 4.

2 The finite dimensional reduction

In this section, we consider the following problem without constraint

(2.1) − Δ u 1 + ( λ + V 1 ( x ) ) u 1 = α u 1 u 2 in R 4 , − Δ u 2 + ( λ + V 2 ( x ) ) u 2 = α 2 u 1 2 + β u 2 2 in R 4 ,

where λ > 0 is a large parameter.

We want to find a peak solution of equation (2.1) with the form

(2.2) u i , λ ( x ) = ∑ j = 1 m U i , λ , y λ , j ( x ) + φ i , λ ( x ) , i = 1 , 2 ,

where U i , λ , y ≔ λ U i ( λ ( x − y ) ) for some y ∈ R 4 . To this aim, we define L λ be the bounded linear operator from H 1 ( R 4 ) × H 1 ( R 4 ) to itself as follows:

⟨ L λ ( φ 1 , φ 2 ) , ( ψ 1 , ψ 2 ) ⟩ λ ≔ ∫ R 4 ∇ φ 1 ∇ ψ 1 + ( λ + V 1 ( x ) ) φ 1 ψ 1 − α ∑ j = 1 m U 1 , λ , y λ , j φ 2 ψ 1 − α ∑ j = 1 m U 2 , λ , y λ , j φ 1 ψ 1 + ∫ R 4 ∇ φ 2 ∇ ψ 2 + ( λ + V 2 ( x ) ) φ 2 ψ 2 − α ∑ j = 1 m U 1 , λ , y λ , j φ 1 ψ 2 − 2 β ∑ j = 1 m U 2 , λ , y λ , j φ 2 ψ 2

for any ( ψ 1 , ψ 2 ) , ( φ 1 , φ 2 ) ∈ H 1 ( R 4 ) × H 1 ( R 4 ) .

Then to obtain the solution ( u 1 , λ , u 2 , λ ) of equation (2.1) with (2.2) is to solve the following problem:

L λ ( φ 1 , λ , φ 2 , λ ) = l λ + R λ ( φ 1 , λ , φ 2 , λ ) ,

where l λ ≔ ( l 1 , λ , l 2 , λ ) with

l 1 , λ = − ∑ j = 1 m V 1 ( x ) U 1 , λ , y λ , j + α ∑ j = 1 m U 1 , λ , y λ , j ∑ j = 1 m U 2 , λ , y λ , j − ∑ j = 1 m U 1 , λ , y λ , j U 2 , λ , y λ , j , l 2 , λ = − ∑ j = 1 m V 2 ( x ) U 2 , λ , y λ , j + α 2 ∑ j = 1 m U 1 , λ , y λ , j 2 − ∑ j = 1 m U 1 , λ , y λ , j 2 + β ∑ j = 1 m U 2 , λ , y λ , j 2 − ∑ j = 1 m U 2 , λ , y λ , j 2 ,

and R λ ( φ 1 , λ , φ 2 , λ ) ≔ ( R 1 , λ ( φ 1 , λ , φ 2 , λ ) , R 2 , λ ( φ 1 , λ , φ 2 , λ ) ) with

R 1 , λ ( φ 1 , λ , φ 2 , λ ) = α φ 1 , λ φ 2 , λ , R 2 , λ ( φ 1 , λ , φ 2 , λ ) = α 2 φ 1 , λ 2 + β φ 2 , λ 2 .

To carry out the reduction argument, we first introduce the following result, which has been proved in [29].

Proposition 2.1

For any α > 0 and α > β , ( U 1 , U 2 ) is nondegenerate for the system (1.8) in H 1 ( R 4 ) × H 1 ( R 4 ) in the sense that the kernel is given by

η ( α , β ) ∂ U ∂ x i , ∂ U ∂ x i i = 1 , 2 , 3 , 4

in H 1 ( R 4 ) × H 1 ( R 4 ) , where η ( α , β ) ≠ 0 .

Now we define the set

E λ = ( u 1 , u 2 ) ∈ H 1 ( R 4 ) × H 1 ( R 4 ) : ( u 1 , u 2 ) , ∂ U 1 , λ , y λ , j ∂ x k , ∂ U 2 , λ , y λ , j ∂ x k λ = 0 , k = 1 , … , 4 , j = 1 , … , m ,

and the norm ‖ ( u , v ) ‖ λ 2 ≔ ‖ u ‖ 1 , λ 2 + ‖ v ‖ 2 , λ 2 with

‖ u ‖ i , λ 2 ≔ 1 λ ∫ R 4 ( ∣ ∇ u ∣ 2 + ( λ + V i ( x ) ) u 2 ) d x .

Also we set the projection F λ from H 1 ( R 4 ) × H 1 ( R 4 ) to E λ as follows:

F λ ( u 1 , u 2 ) = ( u 1 , u 2 ) − ∑ i = 1 4 ∑ j = 1 m Γ λ , i , j ∂ U 1 , λ , y λ , j ∂ x i , ∂ U 2 , λ , y λ , j ∂ x i .

Proposition 2.2

There exist λ 0 , ε 0 , ρ > 0 independent of a j ( j = 1 , … , m ) , such that for any λ ∈ [ λ 0 , + ∞ ) and y λ , j ∈ B ε 0 ( a j ) , F λ L λ is bijective in E λ . Also, for any ( u 1 , u 2 ) ∈ E λ , it holds

‖ F λ L λ ( u 1 , u 2 ) ‖ λ ≥ ρ λ ‖ ( u 1 , u 2 ) ‖ λ .

Proof

By contradiction, suppose that there exist λ n → + ∞ , y λ n , j → a j , ( u 1 , n , u 2 , n ) ∈ E λ n such that

(2.3) ‖ F λ n L λ n ( u 1 , n , u 2 , n ) ‖ λ n ≤ λ n n ‖ ( u 1 , n , u 2 , n ) ‖ λ n ,

with ‖ ( u 1 , n , u 2 , n ) ‖ λ n 2 = 1 λ n . By (2.3), for any ( ϕ 1 , ϕ 2 ) ∈ E λ n , we have

⟨ L λ n ( ( u 1 , n , u 2 , n ) , ( ϕ 1 , ϕ 2 ) ) ⟩ λ n = o ( λ n ) ‖ ( u 1 , n , u 2 , n ) ‖ λ n ‖ ( ϕ 1 , ϕ 2 ) ‖ λ n = o ( λ n 1 2 ) ‖ ( ϕ 1 , ϕ 2 ) ‖ λ n ,

and from this, it holds

⟨ L λ n ( ( u 1 , n , u 2 , n ) , ( u 1 , n , u 2 , n ) ) ⟩ λ n = o ( 1 ) .

Now choosing R sufficiently large such that

( α + 2 β ) ∑ j = 1 m U 2 , λ n , y λ n , j < 1 4 ( λ n + V i ( x ) ) , in R 4 \ ∪ j = 1 m B R λ n ( y λ n , j ) ,

we find

⟨ L λ n ( ( u 1 , n , u 2 , n ) , ( u 1 , n , u 2 , n ) ) ⟩ λ n ≥ λ n 2 ‖ ( u 1 , n , u 2 , n ) ‖ λ n 2 − C λ n ∫ ∪ j = 1 m B R λ n ( y λ n , j ) ( u 1 , n 2 + u 2 , n 2 ) ,

which gives that

∫ ∪ j = 1 m B R λ n ( y λ n , j ) ( u 1 , n 2 + u 2 , n 2 ) ≥ C λ n .

To obtain a contradiction, next we want to prove that

(2.4) ∫ ∪ j = 1 m B R λ n ( y λ n , j ) ( u 1 , n 2 + u 2 , n 2 ) = o 1 λ n .

To this end, we define u ¯ k , n , j ( x ) = 1 λ n u k , n ( x λ n + y λ n , j ) . Then

∑ k = 1 2 ∫ R 4 ( ∣ ∇ u ¯ k , n , j ∣ 2 + u ¯ k , n , j 2 ) = O ( λ n ) ‖ ( u 1 , n , u 2 , n ) ‖ λ n 2 ≤ C .

Thus, we can assume that as n → + ∞ , u ¯ k , n , j ⇀ u ¯ k , j weakly in H 1 ( R 4 ) × H 1 ( R 4 ) , and u ¯ k , n , j → u ¯ k , j strongly in L l o c 2 ( R 4 ) × L l o c 2 ( R 4 ) for k = 1 , 2 and j = 1 , … , m . Now we just need to prove that

(2.5) u ¯ k , j = 0 , k = 1 , 2 and j = 1 , … , k .

For any ( ϕ 1 , ϕ 2 ) ∈ H 1 ( R 2 ) × H 1 ( R 2 ) , we can decompose ( ϕ 1 , ϕ 2 ) as follows:

( ϕ 1 , ϕ 2 ) = F λ n ( ϕ 1 , ϕ 2 ) + ∑ i = 1 4 ∑ j = 1 m Γ λ n , i , j ∂ U 1 , λ n , y λ n , j ∂ x i , ∂ U 2 , λ n , y λ n , j ∂ x i ,

with Γ λ n , i , j = b λ n , i , j ( ϕ 1 , ϕ 2 ) , ∂ U 1 , λ n , y λ n , j ∂ x i , ∂ U 2 , λ n , y λ n , j ∂ x i λ n for some constants b λ n , i , j . Then it holds

⟨ L λ n ( u 1 , n , u 2 , n ) , ( ϕ 1 , ϕ 2 ) ⟩ λ n = ⟨ L λ n ( u 1 , n , u 2 , n ) , F λ n ( ϕ 1 , ϕ 2 ) ⟩ λ n + ∑ i = 1 4 ∑ j = 1 m Γ λ n , i , j κ λ n , i , j ,

where κ λ n , i , j = L λ n ( u 1 , n , u 2 , n ) , ∂ U 1 , λ n , y λ n , j ∂ x i , ∂ U 2 , λ n , y λ n , j ∂ x i λ n . Also, by (2.3),

⟨ L λ n ( u 1 , n , u 2 , n ) , F λ n ( ϕ 1 , ϕ 2 ) ⟩ λ n = ⟨ F λ n L λ n ( u 1 , n , u 2 , n ) , F λ n ( ϕ 1 , ϕ 2 ) ⟩ λ n = o λ n 1 2 ‖ ( ϕ 1 , ϕ 2 ) ‖ λ n ,

which gives that

(2.6) ⟨ L λ n ( u 1 , n , u 2 , n ) , ( ϕ 1 , ϕ 2 ) ⟩ λ n = o λ n 1 2 ‖ ( ϕ 1 , ϕ 2 ) ‖ λ n + ∑ i = 1 4 ∑ j = 1 m ϑ λ n , i , j ( ϕ 1 , ϕ 2 ) , ∂ U 1 , λ n , y λ n , j ∂ x i , ∂ U 2 , λ n , y λ n , j ∂ x i λ n

for some constants ϑ λ n , i , j . From this, we estimate ϑ λ n , i , j as follows:

∑ i = 1 4 ∑ j = 1 m ϑ λ n , i , j ∂ U 1 , λ n , y λ n , j ∂ x h , ∂ U 2 , λ n , y λ n , j ∂ x h , ∂ U 1 , λ n , y λ n , j ∂ x i , ∂ U 2 , λ n , y λ n , j ∂ x i λ n = L λ n ( u 1 , n , u 2 , n ) , ∂ U 1 , λ n , y λ n , j ∂ x h , ∂ U 2 , λ n , y λ n , j ∂ x h λ n + o ( λ n ) = λ n ( u 1 , n , u 2 , n ) , ∂ U 1 , λ n , y λ n , j ∂ x h , ∂ U 2 , λ n , y λ n , j ∂ x h λ n + Q n ( u 1 , n , u 2 , n ) + o ( λ n ) = Q n ( u 1 , n , u 2 , n ) + o ( λ n 1 + γ ) ,

where γ > 0 is a small constant and

Q n ( u 1 , n , u 2 , n ) = − α ∫ R 4 U 1 , λ n , y λ n , j u 2 , n ∂ U 1 , λ n , y λ n , j ∂ x h − α ∫ R 4 U 2 , λ n , y λ n , j u 1 , n ∂ U 1 , λ n , y λ n , j ∂ x h − α ∫ R 4 U 1 , λ n , y λ n , j u 1 , n ∂ U 2 , λ n , y λ n , j ∂ x h − 2 β ∫ R 4 U 2 , λ n , y λ n , j u 2 , n ∂ U 2 , λ n , y λ n , j ∂ x h .

Noting that ( U 1 , λ n , y λ n , j , U 2 , λ n , y λ n , j ) satisfies that

− Δ U 1 , λ n , y λ n , j + λ n U 1 , λ n , y λ n , j = α U 1 , λ n , y λ n , j U 2 , λ n , y λ n , j in R 4 , − Δ U 2 , λ n , y λ n , j + λ n U 2 , λ n , y λ n , j = α 2 U 1 , λ n , y λ n , j 2 + β U 2 , λ n , y λ n , j 2 in R 4 ,

we find

− Δ ∂ U 1 , λ n , y λ n , j ∂ x h + λ n ∂ U 1 , λ n , y λ n , j ∂ x h = α U 2 , λ n , y λ n , j ∂ U 1 , λ n , y λ n , j ∂ x h + α U 1 , λ n , y λ n , j ∂ U 2 , λ n , y λ n , j ∂ x h in R 4 , − Δ ∂ U 2 , λ n , y λ n , j ∂ x h + λ n ∂ U 2 , λ n , y λ n , j ∂ x h = α U 1 , λ n , j ∂ U 1 , λ n , y λ n , j ∂ x h + 2 β U 2 , λ n , y λ n , j ∂ U 2 , λ n , y λ n , j ∂ x h in R 4 .

It follows from the definition of E λ n that

Q n ( u 1 , n , u 2 , n ) = o ( λ n 1 + γ ) .

Hence, we find

∑ i = 1 4 ∑ j = 1 m ϑ λ n , i , j ∂ U 1 , λ n , y λ n , j ∂ x i , ∂ U 2 , λ n , y λ n , j ∂ x i , ∂ U 1 , λ n , y λ n , j ∂ x h , ∂ U 2 , λ n , y λ n , j ∂ x h λ n = o ( λ n 1 + γ ) ,

which gives ϑ λ n , i , j = o ( 1 ) . So (2.6) gives that

(2.7) ⟨ L λ n ( u 1 , n , u 2 , n ) , ( ϕ 1 , ϕ 2 ) ⟩ λ n = o λ n 1 2 ‖ ( ϕ 1 , ϕ 2 ) ‖ λ n .

Now setting ϕ ¯ k , j ( x ) = λ n ϕ k ( λ n ( x − y λ n , j ) ) and using (2.7), we obtain

⟨ L λ n ( u 1 , n , u 2 , n ) , ( ϕ ¯ 1 , j , ϕ ¯ 2 , j ) ⟩ λ n = ∑ i = 1 2 ∫ R 4 ∇ u ¯ i , n , j ∇ ϕ i + 1 + 1 λ n V i z λ n + y λ n , j u ¯ i , n , j ϕ i − 2 β ∫ R 4 U 2 u ¯ 2 , n , j ϕ 2 − α ∫ R 4 ( U 1 u ¯ 2 , n , j ϕ 1 + U 1 u ¯ 1 , n , j ϕ 2 + U 2 u ¯ 1 , n , j ϕ 1 ) = o λ n 1 2 ‖ ( ϕ ¯ 1 , j , ϕ ¯ 2 , j ) ‖ λ n = o ( 1 ) .

Then, ( u ¯ 1 , j , u ¯ 2 , j ) solves

− Δ u ¯ 1 , j + u ¯ 1 , j = α u ¯ 1 , j U 2 + α u ¯ 2 , j U 1 in R 4 , − Δ u ¯ 2 , j + u ¯ 2 , j = α u ¯ 1 , j U 1 + 2 β u ¯ 2 , j U 2 in R 4 .

From this and the fact that ( U 1 , U 2 ) is nondegenerate, we have

(2.8) ( u ¯ 1 , j , u ¯ 2 , j ) = ∑ k = 1 4 τ k , j ∂ U 1 ∂ x k , ∂ U 2 ∂ x k .

On the other hand, from ( u 1 , n , u 2 , n ) ∈ E λ n and (2.8), it holds

0 = ( u 1 , n , u 2 , n ) , ∂ U 1 , λ n , y λ n , j ∂ x k , ∂ U 2 , λ n , y λ n , j ∂ x k λ n = ∫ R 4 1 λ n ∇ u 1 , n ∇ ∂ U 1 , λ n , y λ n , j ∂ x k + ∇ u 2 , n ∇ ∂ U 2 , λ n , y λ n , j ∂ x k + ∫ R 4 1 + 1 λ n V 1 ( x ) u 1 , n ∂ U 1 , λ n , y λ n , j ∂ x k + 1 + 1 λ n V 2 ( x ) u 2 , n ∂ U 2 , λ n , y λ n , j ∂ x k = ∫ R 4 ∇ u ¯ 1 , n , j ∇ ∂ U 1 ∂ x k + ∇ u ¯ 2 , n , j ∇ ∂ U 2 ∂ x k + ∫ R 4 1 + 1 λ n V 1 ( x ) u ¯ 1 , n , j ∂ U 1 ∂ x k + 1 + 1 λ n V 2 ( x ) u ¯ 2 , n , j ∂ U 2 ∂ x k = ∫ R 4 ∇ u ¯ 1 , j ∇ ∂ U 1 ∂ x k + ∇ u ¯ 2 , j ∇ ∂ U 2 ∂ x k + ∫ R 4 u ¯ 1 , j ∂ U 1 ∂ x k + u ¯ 2 , j ∂ U 2 ∂ x k + o ( 1 ) = α ∫ R 4 U 2 u ¯ 1 , j ∂ U 1 ∂ x k + U 1 u ¯ 2 , j ∂ U 1 ∂ x k + U 1 u ¯ 1 , j ∂ U 2 ∂ x k + 2 β ∫ R 4 U 2 u ¯ 2 , j ∂ U 2 ∂ x k + o ( 1 ) .

By the property of ( U 1 , U 2 ) , all the constants τ k , j = 0 in (2.8). This gives that (2.5) holds and then (2.4) follows.□

Similar to Proposition 2.2, we have

Proposition 2.3

There exist λ 1 , ε 1 , ρ 1 > 0 independent of a 1 , such that for any λ ∈ [ λ 1 , + ∞ ) and y λ , k close to a 1 with

(2.9) ∣ y λ , k − y λ , j ∣ ≥ ε 1 ln λ λ , f o r k ≠ j ,

F λ L λ is bijective in E λ . Moreover, it holds

‖ F λ L λ ( u 1 , u 2 ) ‖ λ ≥ ρ 1 λ ‖ ( u 1 , u 2 ) ‖ λ .

Lemma 2.4

We have

(2.10) ‖ l λ ‖ λ = O ( 1 ) .

Proof

Recall that l λ = ( l 1 , λ , l 2 , λ ) and

⟨ ( l 1 , λ , l 2 , λ ) , ( ϕ 1 , ϕ 2 ) ⟩ λ = ⟨ l 1 , λ , ϕ 1 ⟩ λ + ⟨ l 2 , λ , ϕ 2 ⟩ λ .

Using Hölder inequality, we find

(2.11) ⟨ l 1 , λ , ϕ 1 ⟩ λ = − ∫ R 4 ∑ j = 1 m V 1 ( x ) U 1 , λ , y λ , j ϕ 1 + α ∫ R 4 ∑ j = 1 m U 1 , λ , y λ , j ∑ j = 1 m U 2 , λ , y λ , j − ∑ j = 1 m U 1 , λ , y λ , j U 2 , λ , y λ , j ϕ 1 = O ( ‖ ϕ 1 ‖ 1 , λ ) ,

and

(2.12) ⟨ l 2 , λ , ϕ 2 ⟩ λ = − ∫ R 4 ∑ j = 1 m V 2 ( x ) U 2 , λ , y λ , j ϕ 2 + α 2 ∫ R 4 ∑ j = 1 m U 1 , λ , y λ , j 2 − ∑ j = 1 m U 1 , λ , y λ , j 2 ϕ 2 + β ∫ R 4 ∑ j = 1 m U 2 , λ , y λ , j 2 − ∑ j = 1 m U 2 , λ , y λ , j 2 ϕ 2 = O ( ‖ ϕ 2 ‖ 2 , λ ) .

Hence, (2.12) and (2.11) give (2.10).□

Lemma 2.5

It holds

(2.13) ∥ R λ ( φ 1 , λ , φ 2 , λ ) ∥ λ = O λ 1 2 ‖ ( φ 1 , λ , φ 2 , λ ) ‖ λ 2 .

Proof

Note that R λ ( φ 1 , λ , φ 2 , λ ) = ( R 1 , λ ( φ 1 , λ , φ 2 , λ ) , R 2 , λ ( φ 1 , λ , φ 2 , λ ) ) . By the direct computations, we have

(2.14) ⟨ R 1 , λ ( φ 1 , λ , φ 2 , λ ) , ϕ 1 ⟩ λ = α ∫ R 4 φ 1 , λ φ 2 , λ ϕ 1 ≤ C ‖ φ 1 , λ ‖ L 4 ‖ φ 2 , λ ‖ L 4 ‖ ϕ 1 ‖ L 2 = O λ 1 2 ‖ φ 1 , λ ‖ λ ‖ φ 2 , λ ‖ λ ‖ ϕ 1 ‖ λ ,

where we use the following classical Gagliardo-Nirenberg inequality:

(2.15) ∫ R 2 ∣ ϕ ∣ 4 1 4 = O ‖ ∇ ϕ ‖ L 2 1 2 ‖ ϕ ‖ L 2 1 2 = O λ 1 4 ‖ ϕ ‖ i , λ , for any ϕ ∈ H 1 ( R 4 ) .

With the same argument, it holds

(2.16) ⟨ R 2 , λ ( φ 1 , λ , φ 2 , λ ) , ϕ 2 ⟩ λ = O λ 1 2 ‖ φ 1 , λ ‖ λ ‖ φ 2 , λ ‖ λ ‖ ϕ 2 ‖ λ .

Hence, (2.13) follows by (2.16) and (2.14).□

Proposition 2.6

Let λ 0 be as in Proposition 2.2. Then for any λ ∈ [ λ 0 , + ∞ ) , and y λ , k close to a k for k = 1 , … , m , there exists unique φ i , λ ∈ E λ , such that

(2.17) ∫ R 4 ( ∇ u 1 , λ ∇ ϕ 1 + ∇ u 2 , λ ∇ ϕ 2 + ( λ + V 1 ( x ) ) u 1 , λ ϕ 1 + ( λ + V 2 ( x ) ) u 2 , λ ϕ 2 ) = ∫ R 4 α u 1 , λ u 2 , λ ϕ 1 + α 2 u 1 , λ 2 ϕ 2 + β u 2 , λ 2 ϕ 2

for any ( ϕ 1 , ϕ 2 ) ∈ E λ , where u i , λ ( x ) = ∑ k = 1 m U i , λ , y λ , k ( x ) + φ i , λ ( x ) . Moreover, it holds

(2.18) ‖ ( φ 1 , λ , φ 2 , λ ) ‖ λ ≤ C λ .

Proof

To obtain (2.17), it is equivalent to consider the following problem:

(2.19) F λ L λ ( φ 1 , φ 2 ) = F λ l λ + F λ R λ ( φ 1 , φ 2 ) , for ( φ 1 , φ 2 ) ∈ E λ .

By Proposition 2.2, we can rewrite (2.19) as follows:

(2.20) ( φ 1 , φ 2 ) = B ( φ 1 , φ 2 ) ≔ ( F λ L λ ) − 1 F λ l λ + ( F λ L λ ) − 1 F λ R λ ( φ 1 , φ 2 ) .

Now we define

M ≔ ( φ 1 , φ 2 ) ∈ E λ : ‖ ( φ 1 , φ 2 ) ‖ λ ≤ 1 λ 1 − γ ,

where γ > 0 is a fixed small constant.

Next we will prove B is a contraction map from M to M . First, from Proposition 2.2 and Lemmas 2.4 and 2.5, we have

‖ B ( φ 1 , φ 2 ) ‖ λ ≤ C 1 λ ‖ l λ ‖ λ + 1 λ ‖ R λ ( φ 1 , φ 2 ) ‖ λ ≤ 1 λ 1 − γ .

On the other hand, for ( φ 1 , φ 2 ) , ( φ ¯ 1 , φ ¯ 2 ) ∈ E λ , it holds

‖ B ( φ 1 , φ 2 ) − B ( φ ¯ 1 , φ ¯ 2 ) ‖ λ ≤ C λ ( ‖ R λ ( φ 1 , φ 2 ) ‖ λ + ‖ R λ ( φ ¯ 1 , φ ¯ 2 ) ‖ λ ) ≤ 1 λ 1 − γ .

By applying the contraction mapping theorem, we find that for any λ ∈ [ λ 0 , + ∞ ) , y λ , k close to a k for k = 1 , … , m , there exists unique ( φ 1 , λ , φ 2 , λ ) ∈ E λ depending on a k and λ solving (2.20) and then (2.17) follows. Finally, (2.18) follows from (2.20).□

Similar to Proposition 2.6, using Proposition 2.3 and Lemmas 2.4 and 2.5, we have the following result.

Proposition 2.7

Let λ 1 be as in Proposition 2.3. Then for any λ ∈ [ λ 1 , + ∞ ) , and y λ , k close to a 1 for k = 1 , … , m , satisfying (2.9), (2.17) and (2.18) hold with u i , λ = ∑ k = 1 m U i , λ , y λ , k + φ i , λ ( x ) .

3 Proof of Theorem 1.1

From Proposition 2.6, for any λ ∈ [ λ 0 , + ∞ ) , y λ , j → a j for j = 1 , … , m , there exist u i , λ and Γ λ , i , j such that

− Δ u 1 , λ + ( λ + V 1 ( x ) ) u 1 , λ = α u 1 , λ u 2 , λ + ∑ i = 1 4 ∑ j = 1 m Γ λ , i , j ∂ U 1 , λ , y λ , j ∂ x i , in R 4 , − Δ u 2 , λ + ( λ + V 2 ( x ) ) u 2 , λ = α 2 u 1 , λ 2 + β u 2 , λ 2 + ∑ i = 1 4 ∑ j = 1 m Γ λ , i , j ∂ U 2 , λ , y λ , j ∂ x i , in R 4 ,

where u i , λ ( x ) = ∑ j = 1 m U i , λ , y λ , j ( x ) + φ i , λ ( x ) .

Now to obtain a true solution for (2.1), we just need to choose suitable y λ , j such that

(3.1) Γ λ , i , j = 0 , for i = 1 , 2 , 3 , 4 and j = 1 , … , m .

To this end, we want to choose ( y λ , 1 , … , y λ , m ) solving for any d > 0 ,

(3.2) ∫ B d ( y λ , j ) − Δ u 1 , λ ∂ u 1 , λ ∂ x i + ( λ + V 1 ( x ) ) u 1 , λ ∂ u 1 , λ ∂ x i − α u 1 , λ u 2 , λ ∂ u 1 , λ ∂ x i + ∫ B d ( y λ , j ) − Δ u 2 , λ ∂ u 2 , λ ∂ x i + ( λ + V 2 ( x ) ) u 2 , λ ∂ u 2 , λ ∂ x i − α 2 u 1 , λ 2 ∂ u 2 , λ ∂ x i − β u 2 , λ 2 ∂ u 2 , λ ∂ x i = 0

with i = 1 , 2 , 3 , 4 and j = 1 , … , m .

Lemma 3.1

Assume that ( K 1 ) holds. Then there exists some large λ 0 such that for any λ ∈ [ λ 0 , + ∞ ) , problem (2.1) has a peak solution ( u 1 , λ , u 2 , λ ) with the form

(3.3) u i , λ ( x ) = ∑ j = 1 m U i , λ , y λ , j ( x ) + φ i , λ ( x ) ,

satisfying for j = 1 , … , m ,

(3.4) ∣ y λ , j − a j ∣ = o 1 λ a n d ‖ ( φ 1 , λ , φ 2 , λ ) ‖ λ = O 1 λ .

Proof

From the aforementioned discussions, we have to choose suitable y λ , j ∈ R 4 for j = 1 , … , m satisfying (3.2). Integrating on B d ( y λ , j ) , we find that (3.2) is equivalent to

∫ B d ( y λ , j ) ∂ V 1 ( x ) ∂ x i u 1 , λ 2 + ∂ V 2 ( x ) ∂ x i u 2 , λ 2 = ∫ ∂ B d ( y λ , j ) ∣ ∇ u 1 , λ ∣ 2 ν i − 2 ∫ ∂ B d ( y λ , j ) ∂ u 1 , λ ∂ x i ∂ u 1 , λ ∂ ν − α ∫ B d ( y λ , j ) u 2 , λ ∂ u 1 , λ 2 ∂ x i + ∫ ∂ B d ( y λ , j ) ( λ + V 1 ( x ) ) u 1 , λ 2 ν i + ∫ ∂ B d ( y λ , j ) ∣ ∇ u 2 , λ ∣ 2 ν i − 2 ∫ ∂ B d ( y λ , j ) ∂ u 2 , λ ∂ x i ∂ u 1 , λ ∂ ν + ∫ ∂ B d ( y λ , j ) ( λ + V 2 ( x ) ) u 2 , λ 2 ν i − α ∫ B d ( y λ , j ) u 1 , λ 2 ∂ u 2 , λ ∂ x i − 2 3 β ∫ ∂ B d ( y λ , j ) u 2 , λ 3 ν i ,

where ν = ( ν 1 , … , ν 4 ) is the outward unit normal of ∂ B d ( y λ , j ) . Also it holds

∫ B d ( y λ , j ) u 1 , λ 2 ∂ u 2 , λ ∂ x i + u 2 , λ ∂ u 1 , λ 2 ∂ x i = ∫ ∂ B d ( y λ , j ) u 1 , λ 2 u 2 , λ ν i = O ( e − θ λ )

for some θ > 0 . Then we have

(3.5) ∫ B d ( y λ , j ) ∂ V 1 ( x ) ∂ x i u 1 , λ 2 + ∂ V 2 ( x ) ∂ x i u 2 , λ 2 = O ( e − θ λ ) .

On the other hand,

(3.6) ∫ B d ( y λ , j ) ∂ V 1 ( x ) ∂ x i u 1 , λ 2 = ∫ B d ( y λ , j ) ∑ l = 1 4 ∂ 2 V 1 ( a j ) ∂ x i x l ( x l − a j , l ) u 1 , λ 2 + O ∫ B d ( y λ , j ) ∣ x − a j ∣ 2 u 1 , λ 2 = m ∑ l = 1 4 ∂ 2 V 1 ( a j ) ∂ x i x l ( y λ , j , l − a j , l ) ∫ R 4 U 1 2 ( x ) + O ( e − θ λ + ‖ φ 1 , λ ‖ 1 , λ 2 ) + o ( ∣ y λ , j − a j ∣ ) = m ∑ l = 1 4 ∂ 2 V 1 ( a j ) ∂ x i x l ( y λ , j , l − a j , l ) ∫ R 4 U 1 2 ( x ) + o 1 λ + ∣ y λ , j − a j ∣ .

Similarly, we have

(3.7) ∫ B d ( y λ , j ) ∂ V 2 ( x ) ∂ x i u 2 , λ 2 = m ∑ l = 1 4 ∂ 2 V 1 ( a j ) ∂ x i x l ( y λ , j , l − a j , l ) ∫ R 4 U 2 2 ( x ) + o 1 λ + ∣ y λ , j − a j ∣ .

Then (3.5), (3.6), and (3.7) give us that

(3.8) ∑ l = 1 4 ∂ 2 V 1 ( a j ) ∂ x i x l Γ 1 2 + ∂ 2 V 2 ( a j ) ∂ x i x l Γ 2 2 ( y λ , j , l − a j , l ) = o 1 λ + ∣ y λ , j − a j ∣ .

By the assumption ( K 1 ) , we find

□ ∣ y λ , j − a j ∣ = o 1 λ for j = 1 , … , m .

Lemma 3.2

It holds

(3.9) ∫ R 4 U 3 = 3 ∫ R 4 U 2 = 3 2 ∫ R 4 ∣ ∇ U ∣ 2 .

Proof

Note that U satisfies

(3.10) − Δ U + U = U 2 , in R 4 .

By direct computations, we have

∫ R 4 ( ∣ ∇ U ∣ 2 + U 2 ) = ∫ R 4 U 3 .

Also multiplying ( x ⋅ ∇ U ) on both sides of equation (3.10) and integrating on R 4 , it holds

∫ R 4 ∣ ∇ U ∣ 2 + 2 ∫ R 4 U 2 = 4 3 ∫ R 4 U 3 .

So the aforementioned two equalities give (3.9).□

Proposition 3.3

If ( u 1 , λ , u 2 , λ ) is a solution of (2.1) with the form (3.3) satisfying (3.4), then it holds

(3.11) λ m b * 3 2 α Γ 2 Γ 1 2 + β Γ 2 3 − ∫ R 4 ( u 1 , λ 2 + u 2 , λ 2 ) = b * ∑ i = 1 2 ∑ j = 1 m Γ i 2 V i ( a j ) + O 1 λ .

Proof

Let ( u 1 , λ , u 2 , λ ) be a solution of equation (2.1), multiplying ⟨ x − y λ , j , ∇ u i , λ ⟩ on both sides of the i th equation of equation (2.1) and integrating on B d ( y λ , j ) , we find

(3.12) ∫ B d ( y λ , j ) ∑ i = 1 2 ( ( 2 V i ( x ) + ⟨ ∇ V i ( x ) , x − y λ , j ⟩ ) u i , λ 2 + 2 λ u i , λ 2 ) − α u 2 , λ u 1 , λ 2 − 2 3 β u 2 , λ 3 = ∫ ∂ B d ( y λ , j ) W ( x ) d σ ,

where

W ( x ) ≔ ∑ i = 1 2 − 2 ∂ u i , λ ∂ ν ⟨ x − y λ , j , ∇ u i , λ ⟩ + ⟨ x − y λ , j , ν ⟩ [ ( λ + V i ( x ) ) u i , λ 2 ] − ⟨ x − y λ , j , ν ⟩ α u 2 , λ u 1 , λ 2 + 2 3 β u 2 , λ 3 .

Now, being u i , λ ( x ) = ∑ j = 1 m U i , λ , y λ , j ( x ) + φ i , λ ( x ) with ‖ ( φ 1 , λ , φ 2 , λ ) ‖ λ = O 1 λ , it holds

∫ B d ( y λ , j ) ( 2 V i ( x ) + ⟨ ∇ V i ( x ) , x − y λ , j ⟩ ) u i , λ 2 = ∫ B d ( y λ , j ) ( ⟨ ∇ V i ( x ) , x − x λ , j ⟩ + 2 ( V i ( x ) − V i ( a j ) ) ) U i , λ , y λ , j 2 ( x ) + 2 V i ( a j ) ∫ B d ( y λ , j ) u i , λ 2 + O ∫ B d ( y λ , j ) ∣ x − a j ∣ 2 U i , λ , y λ , j φ i , λ + ‖ φ i , λ ‖ λ 2 = 2 V i ( a j ) ∫ B d ( y λ , j ) u i , λ 2 + O 1 λ = 2 Γ i 2 b * V i ( a j ) + O 1 λ ,

where b * = ∫ R 4 U 2 . On the other hand, using (3.9), we find

∫ B d ( y λ , j ) α u 2 , λ u 1 , λ 2 + 2 3 β u 2 , λ 3 = λ ∫ R 4 α U 1 2 U 2 + 2 3 β U 2 3 + O 1 λ = 2 λ 3 2 α Γ 2 Γ 1 2 + β Γ 2 3 b * + O 1 λ .

So summing (3.12) from j = 1 to j = m , we find (3.11).□

Now we are in a position to prove Theorem 1.1.

Proof of Theorem 1.1

Now we set b λ = ∫ R 4 ( u 1 , λ 2 + u 2 , λ 2 ) and v i , λ = u i , λ ∕ b λ , then ( v 1 , λ , v 2 , λ ) satisfies that

− Δ v 1 , λ + ( λ + V 1 ( x ) ) v 1 , λ = α b λ 1 2 v 1 , λ v 2 , λ , in R 4 , − Δ v 2 , λ + ( λ + V 2 ( x ) ) v 2 , λ = α 2 b λ 1 2 v 1 , λ 2 + β b λ 1 2 v 2 , λ 2 , in R 4 ,

with

∫ R 4 ( v 1 , λ 2 + v 2 , λ 2 ) = 1 .

Therefore, from Proposition 3.3, we obtain that

(3.13) λ ( m D ( α , β ) b * − b λ ) = b * ∑ i = 1 2 ∑ j = 1 m Γ i 2 V i ( a j ) + o ( 1 ) ,

where D ( α , β ) = 3 2 α Γ 2 Γ 1 2 + β Γ 2 3 = 3 α − 2 β α 3 . Take ( α * , β * ) is a root of D ( α , β ) = 1 m b * .

If D ( α , β ) ↓ 1 m b * as ( α , β ) → ( α * , β * ) , letting N 1 ∈ R such that N 1 + V i ( a j ) > 0 and λ ¯ = λ − N 1 , we have

− Δ u 1 , λ + ( λ ¯ + ( N 1 + V 1 ( x ) ) ) u 1 , λ = α u 1 , λ u 2 , λ , in R 4 , − Δ u 2 , λ + ( λ ¯ + ( N 1 + V 2 ( x ) ) ) u 2 , λ = α 2 u 1 , λ 2 + β u 2 , λ 2 , in R 4 .

Similar to (3.13), it holds

(3.14) λ ¯ ( m D ( α , β ) b * − b λ ¯ ) = b * ∑ i = 1 2 ∑ j = 1 m Γ i 2 ( N 1 + V i ( a j ) ) + o ( 1 ) > 0 ,

which gives that there exists some λ ¯ 0 large enough such that b λ ¯ 0 ≤ 1 . In fact, if b λ ¯ 0 > 1 , then from the fact that D ( α , β ) ↓ 1 m b * as ( α , β ) → ( α * , β * ) , we can take ( α , β ) such that b λ ¯ 0 > m D ( α , β ) b * , which contradicts to (3.14).

Now we define T ( λ ) ≔ ( 1 − b λ ) λ 2 . If b λ ¯ 0 = 1 , then we have got the solution. Now we consider b λ ¯ 0 < 1 and T ( λ ¯ 0 ) > 0 for ( α , β ) → ( α * , β * ) . On the other hand, by (3.13),

lim λ → + ∞ T ( λ ) = lim λ → + ∞ ( ( m D ( α , β ) b * − b λ ) λ + ( 1 − m D ( α , β ) b * ) λ ) λ = lim λ → + ∞ ( 1 − m D ( α , β ) b * + o ( 1 ) ) λ 2 = − ∞ .

Thus, there exists some λ α , β ∈ [ λ ¯ 0 , + ∞ ) such that T ( λ α , β ) = 0 , which means that b λ α , β = 1 . So we have derived the existence for (1.1) with μ = − λ α , β and ∫ R 4 ( u 1 2 + u 2 2 ) = 1 if D ( α , β ) ∈ ( 1 m b * , 1 m b * + ε ] for small ε .

When D ( α , β ) ↑ 1 m b * as ( α , β ) → ( α * , β * ) , letting N 2 ∈ R such that N 2 + V i ( a j ) < 0 and λ ˜ = λ − N 2 , we have

− Δ u 1 , λ + ( λ ˜ + ( N 2 + V 1 ( x ) ) ) u 1 , λ = α u 1 , λ u 2 , λ , in R 4 , − Δ u 2 , λ + ( λ ˜ + ( N 2 + V 2 ( x ) ) ) u 2 , λ = α 2 u 1 , λ 2 + β u 2 , λ 2 , in R 4 .

With the same argument as mentioned earlier, we can also obtain the existence of a concentrated solution to (1.1)–(1.2) if D ( α , β ) ∈ [ 1 m b * − ε , 1 m b * ) .□

4 Proof of Theorem 1.3

In this section, we come to prove Theorem 1.3. First from Proposition 2.7, for any λ ∈ [ λ 1 , + ∞ ) , y λ , j → a 1 , λ ∣ y λ , j − y λ , l ∣ → ∞ , there exist u i , λ and Γ λ , i , j such that

− Δ u 1 , λ + ( λ + V 1 ( x ) ) u 1 , λ = α u 1 , λ u 2 , λ + ∑ i = 1 4 ∑ j = 1 m Γ λ , i , j ∂ U 1 , λ , y λ , j ∂ x i , in R 4 , − Δ u 2 , λ + ( λ + V 2 ( x ) ) u 2 , λ = α 2 u 1 , λ 2 + β u 2 , λ 2 + ∑ i = 1 4 ∑ j = 1 m Γ λ , i , j ∂ U 2 , λ , y λ , j ∂ x i , in R 4

with u i , λ ( x ) = ∑ j = 1 m U i , λ , y λ , j ( x ) + φ i , λ ( x ) satisfying (2.18).

Now we define

I λ ( u 1 , u 2 ) = 1 2 ∑ i = 1 2 ∫ R 4 ( ∣ ∇ u i ∣ 2 + ( λ + V 1 ( x ) ) u i 2 ) − α 2 ∫ R 4 u 1 2 u 2 − β 3 ∫ R 4 u 2 3 ,

and Q ( y ) ≔ I λ ∑ j = 1 m U 1 , λ , y j + φ 1 , λ , ∑ j = 1 m U 2 , λ , y j + φ 2 , λ with y = ( y 1 , … , y m ) ∈ R 4 m .

Then to obtain a true solution of (2.1), we need to choose suitable y λ , j such that

(4.1) Γ λ , i , j = 0 , for i = 1 , 2 , 3 , 4 and j = 1 , … , m .

To this aim, similar to Lemma 2.2.13 in [12], we will find the critical points of Q ( y ) because that if y λ = ( y λ , 1 , … , y λ , m ) is the critical point of Q ( y ) , then (4.1) holds.

Lemma 4.1

It holds

(4.2) I λ ∑ j = 1 m U 1 , λ , y λ , j , ∑ j = 1 m U 2 , λ , y λ , j = m b * λ 2 ∑ i = 1 2 Γ i 2 + b * 2 ∑ i = 1 2 ∑ j = 1 m Γ i 2 V i ( y λ , j ) − ∑ l ≠ j ( Γ 0 + o ( 1 ) ) λ 2 e − λ ∣ y λ , j − y λ , l ∣ 1 λ ∣ y λ , j − y λ , l ∣ 3 2 + O 1 λ θ 2 + λ ∑ l ≠ j e − 3 2 λ ∣ y λ , j − y λ , l ∣ 1 λ ∣ y λ , j − y λ , l ∣ 3 2 ,

for some constant Γ 0 > 0 .

Proof

First, by the definition of I λ , we have

I λ ∑ j = 1 m U 1 , λ , y λ , j , ∑ j = 1 m U 2 , λ , y λ , j = λ 2 2 ∑ i = 1 2 Γ i 2 ∫ R 4 ∑ j = 1 m ∇ U ( λ ( x − y λ , j ) ) 2 + λ 2 2 ∑ i = 1 2 Γ i 2 ∫ R 4 ( λ + V i ( x ) ) ∑ j = 1 m U ( λ ( x − y λ , j ) ) 2 − λ 3 3 ∑ i = 1 2 Γ i 2 ∫ R 4 ∑ j = 1 m U ( λ ( x − y λ , j ) ) 3 ,

where we use the fact that 3 α − 2 β α 3 = Γ 1 2 + Γ 2 2 . Now, we find

∫ R 4 ∑ j = 1 m ∇ U ( λ ( x − y λ , j ) ) 2 = 2 m λ b * + ∑ l ≠ j ∫ R 4 ∇ U ( λ ( x − y λ , j ) ) ⋅ ∇ U ( λ ( x − y λ , l ) ) ︸ ≔ A 1 .

And from the assumption ( K 2 ) and ∣ y λ , j − y λ , l ∣ → 0 , we compute that

∫ R 4 ( λ + V i ( x ) ) ∑ j = 1 m U ( λ ( x − y λ , j ) ) 2 = ∑ j = 1 m ( λ + V i ( y λ , j ) ) ∫ R 4 U 2 ( λ ( x − y λ , j ) ) + ∑ j = 1 m ∫ R 4 ( V i ( x ) − V i ( y λ , j ) ) U 2 ( λ ( x − y λ , j ) ) + ∑ l ≠ j m ∫ R 4 ( λ + V i ( x ) ) U ( λ ( x − y λ , j ) ) U ( λ ( x − y λ , l ) ) = 1 λ m b * + 1 λ 2 ∑ j = 1 m V i ( y λ , j ) b * + O 1 λ 2 + 1 2 θ + ∑ l ≠ j m ∫ R 4 λ U ( λ ( x − y λ , j ) ) U ( λ ( x − y λ , l ) ) ︸ ≔ A 2 + o 1 λ e − λ ∣ y λ , j − y λ , l ∣ 1 λ ∣ y λ , j − y λ , l ∣ 3 2 .

On the other hand, it holds

∫ R 4 ∑ j = 1 m U ( λ ( x − y λ , j ) ) 3 = 3 k b * λ 2 + 3 ∫ R 4 ∑ l ≠ j U 2 ( λ ( x − y λ , j ) ) U ( λ ( x − y λ , l ) ) ︸ ≔ A 3 + O ∑ l ≠ j ∫ R 4 U 3 2 ( λ ( x − y λ , j ) ) U 3 2 ( λ ( x − y λ , l ) ) = 3 k b * λ 2 + 3 A 3 + O 1 λ 2 e − 3 2 λ ∣ y λ , j − y λ , l ∣ 1 λ ∣ y λ , j − y λ , l ∣ 3 2 .

Finally, by using the equation satisfied by U , A 1 + A 2 − λ A 3 = 0 . So the aforementioned estimates give (4.2).□

Now we have the following energy expansion.

Lemma 4.2

It holds

(4.3) I λ ( u 1 , λ , u 2 , λ ) = I λ ∑ j = 1 m U 1 , λ , y λ , j , ∑ j = 1 m U 2 , λ , y λ , j + O 1 λ .

Proof

First, by definition, we have

(4.4) I λ ∑ j = 1 m U 1 , λ , y λ , j ( x ) + φ 1 , λ , ∑ j = 1 m U 2 , λ , y λ , j ( x ) + φ 2 , λ = I λ ∑ j = 1 m U 1 , λ , y λ , j ( x ) , ∑ j = 1 m U 2 , λ , y λ , j ( x ) + ∑ j = 1 m ∑ i = 1 2 ∫ R 4 ( ∇ U i , λ , y λ , j ∇ φ i , λ + ( λ + V i ( x ) ) U i , λ , y λ , j φ i , λ ) − α 2 ∫ R 4 ∑ j = 1 m U 1 , λ , y λ , j 2 φ 2 , λ + 2 ∑ j = 1 m U 1 , λ , y λ , j ∑ j = 1 m U 1 , λ , y λ , j φ 1 , λ + 2 ∑ j = 1 m U 1 , λ , y λ , j φ 1 , λ φ 2 , λ + ∑ j = 1 m ( U 2 , λ , y λ , j ) φ 1 , λ 2 + φ 1 , λ 2 φ 2 , λ − β 3 ∫ R 4 3 ∑ j = 1 m U 2 , λ , y λ , j 2 φ 2 , λ + 3 ∑ j = 1 m U 2 , λ , y λ , j φ 2 , λ + φ 2 , λ 3 + O ( λ ‖ ( φ 1 , λ , φ 2 , λ ) ‖ λ 2 ) .

By using the classical Gagliardo-Nirenberg inequality (2.15), similar to (2.14), we have

∫ R 4 ∑ j = 1 m U 1 , λ , y λ , j φ 1 , λ φ 2 , λ + ∑ j = 1 m ( U 2 , λ , y λ , j ) φ 1 , λ 2 + φ 1 , λ 2 φ 2 , λ + φ 2 , λ 3 = O λ 1 2 ‖ ( φ 1 , λ , φ 2 , λ ) ‖ λ 2 .

On the other hand, from ( φ 1 , λ , φ 2 , λ ) ∈ E λ ,

∫ R 4 ( ∇ U i , λ , y λ , j ∇ φ i , λ + ( λ + V i ( x ) ) U i , λ , y λ , j φ i , λ ) = 0 .

So, (4.4) gives that

(4.5) I λ ∑ j = 1 m U 1 , λ , y λ , j ( x ) + φ 1 , λ , ∑ j = 1 m U 2 , λ , y λ , j ( x ) + φ 2 , λ = I λ ∑ j = 1 m U 1 , λ , y λ , j ( x ) , ∑ j = 1 m U 2 , λ , y λ , j ( x ) − λ 2 ∫ R 4 α 2 Γ 1 2 + β Γ 2 2 ∑ j = 1 m U ( λ ( x − y λ , j ) ) 2 φ 2 , λ + α Γ 1 Γ 2 ∑ j = 1 m U ( λ ( x − y λ , j ) ) 2 φ 1 , λ + O ( λ ‖ ( φ 1 , λ , φ 2 , λ ) ‖ λ 2 ) .

Now using the classical Gagliardo-Nirenberg inequality (2.15) again, we have

∫ R 4 ∑ j = 1 m U ( λ ( x − y λ , j ) ) 2 φ i , λ = ∑ j = 1 m ∫ R 4 U 2 ( λ ( x − y λ , j ) ) φ i , λ + ∑ l ≠ j ∫ R 4 U ( λ ( x − y λ , j ) ) U ( λ ( x − y λ , l ) ) φ i , λ = O 1 λ 5 4 ‖ ( φ 1 , λ , φ 2 , λ ) ‖ λ .

Then (4.3) follows by the aforementioned estimates.□

Proposition 4.3

Assume that ( K 2 ) holds. Then there exist some large λ 1 such that for any λ ∈ [ λ 1 , + ∞ ) , problem (2.1) has a peak solution ( u 1 , λ , u 2 , λ ) with the form

(4.6) u i , λ ( x ) = ∑ j = 1 m U i , λ , y λ , j ( x ) + φ i , λ

satisfying

(4.7) y λ , j → a 1 , λ ∣ y λ , j − y λ , l ∣ → ∞ , for l ≠ j and ‖ ( φ 1 , λ , φ 2 , λ ) ‖ λ = O 1 λ .

Proof

First, it follows from Lemmas 4.1 and 4.2 that

(4.8) Q ( y ) = m b * λ 2 ∑ i = 1 2 Γ i 2 + b * 2 ∑ i = 1 2 ∑ j = 1 m Γ i 2 V i ( y λ , j ) − ∑ l ≠ j ( Γ 0 + o ( 1 ) ) λ 2 e − λ ∣ y λ , j − y λ , l ∣ 1 λ ∣ y λ , j − y λ , l ∣ 3 2 + O 1 λ m i n { θ 2 , 1 } + λ ∑ l ≠ j e − 3 2 λ ∣ y λ , j − y λ , l ∣ 1 λ ∣ y λ , j − y λ , l ∣ 3 2 .

Now we define

ℋ = y = ( y 1 , … , y m ) : y j ∈ B ε ¯ ( a 1 ) , ∣ y l − y j ∣ ≥ ε ln λ λ , j = 1 , … , m , l ≠ j ,

where ε is a small fixed constant. Consider the following problem

(4.9) max y ∈ ℋ Q ( y ) .

Suppose that it is achieved by y λ ∈ ℋ . To prove this y λ is a critical point of Q ( y ) , we just need to prove that y λ is the interior point of ℋ .

Let y ¯ λ , j , j = 1 , … , m , satisfy

∣ y ¯ λ , j − a 1 ∣ ≤ 1 λ γ and ∣ y ¯ λ , j − y ¯ λ , l ∣ ≥ 1 λ 1 − γ if l ≠ j ,

where 0 < ε < γ < < 1 is a small fixed constant. Then for y ¯ λ = ( y ¯ λ , 1 , … , y ¯ λ , m ) ∈ ℋ , it holds

Q ( y ¯ λ ) = λ m b * 2 ∑ i = 1 2 Γ i 2 + m b * 2 ∑ i = 1 2 Γ i 2 V i ( a 1 ) + O 1 λ γ θ 2 .

Suppose that there exists y λ , j 0 such that ∣ y λ , j 0 − a 1 ∣ = ε . Then by (4.8) and the assumption ( K 2 ) , we have

Q ( y λ ) ≤ λ m b * 2 ∑ i = 1 2 Γ i 2 + ( m − 1 ) b * 2 ∑ i = 1 2 Γ i 2 V i ( a 1 ) + b * 2 ∑ i = 1 2 Γ i 2 V i ( y λ , j 0 ) + o ( 1 ) < Q ( y ¯ λ ) ,

which contradicts the fact that y λ is a maximum point of Q ( y ) in ℋ .

Now suppose that there exist y λ , j 0 and y λ , m 0 such that ∣ y λ , j 0 − y λ , m 0 ∣ = ε ln λ λ for m 0 ≠ j 0 . Then by using (4.8), we find

Q ( y λ ) ≤ λ m b * 2 ∑ i = 1 2 Γ i 2 − κ ˜ 0 λ 1 − ε ( ln λ ) − 3 2 < Q ( y ¯ λ )

for some κ ˜ 0 > 0 , which gives a contradiction again. Thus, we have proved that y λ is the interior point of ℋ , and thus, it is a critical point of Q ( y ) .□

Finally, we conclude our second main result.

Proof of Theorem 1.3

Now we set b λ = ∫ R 4 ( u 1 , λ 2 + u 2 , λ 2 ) and v i , λ = u i , λ ∕ b λ , then ( v 1 , λ , v 2 , λ ) satisfies that

− Δ v 1 , λ + ( λ + V 1 ( x ) ) v 1 , λ = α b λ 1 2 v 1 , λ v 2 , λ , in R 4 , − Δ v 2 , λ + ( λ + V 2 ( x ) ) v 2 , λ = α 2 b λ 1 2 v 1 , λ 2 + β b λ 1 2 v 2 , λ 2 , in R 4 ,

with

∫ R 4 ( v 1 , λ 2 + v 2 , λ 2 ) = 1 .

Therefore, similar to Proposition 3.3, we obtain that

λ ( m D ( α , β ) b * − b λ ) = b * ∑ i = 1 2 ∑ j = 1 m Γ i 2 V i ( a 1 ) + o ( 1 ) .

Next, similar to the proofs of Theorem 1.1, we can finish the proof.□

  1. Funding information: This study was partially supported by NSFC (No. 12226324, 11961043).

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

  3. Data availability statement: Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.

References

[1] B. Alexander and K. Yuri, Solitons due to second harmonic generation, Phys. Lett. A 197 (1995), 407–412. 10.1016/0375-9601(94)00989-3Search in Google Scholar

[2] A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris 342 (2006), 453–458. 10.1016/j.crma.2006.01.024Search in Google Scholar

[3] A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. London Math. Soc. 75 (2007), no. 2, 67–82. 10.1112/jlms/jdl020Search in Google Scholar

[4] A. Ambrosetti, E. Colorado, and D. Ruiz, Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations 30 (2007), 85–112. 10.1007/s00526-006-0079-0Search in Google Scholar

[5] M. Anderson, J. Ensher, M. Matthews, C. Wieman, and E. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor, Science 269 (1995), 198–201. 10.1126/science.269.5221.198Search in Google Scholar PubMed

[6] W. Ao and J. Wei, Infinitely many positive solutions for nonlinear equations with non-symmetric potential, Calc. Var. Partial Differential Equations 51 (2014), 761–798. 10.1007/s00526-013-0694-5Search in Google Scholar

[7] O. Bang, Y.-S. Kivshar, A.-V. Buryak, A.-D. Rossi, and S. Trillo, Two-dimensional solitary waves in media with quadratic and cubic nonlinearity, Phys. Rev. E 58 (1998), 5057–5069. 10.1103/PhysRevE.58.5057Search in Google Scholar

[8] W. Bao and Y. Cai, Mathmatical theory and numerical methods for Bose-Einstein condensation, Kinetic Related Models 6 (2013), 1–135. 10.3934/krm.2013.6.1Search in Google Scholar

[9] T. Bartsch, N. Dancer, and Z.-Q. Wang, A Liouville theorem, a priori bounds, and bifurcating branches of positive solutions for a linear elliptic system, Calc. Var. Partial Differential Equations 37 (2010), 345–361. 10.1007/s00526-009-0265-ySearch in Google Scholar

[10] T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Differ. Equations 19 (2006), 200–207. Search in Google Scholar

[11] I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80 (2008), 885. 10.1103/RevModPhys.80.885Search in Google Scholar

[12] D. Cao, S. Peng, and S. Yan, Singularly Perturbed Methods for Nonlinear Elliptic Problems, Cambridge University Press, Cambridge, 202110.1017/9781108872638Search in Google Scholar

[13] N. Dancer, J. Wei, and T. Weth, A priori bounds versus multiple existence of positive solutions for a linear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), 953–969. 10.1016/j.anihpc.2010.01.009Search in Google Scholar

[14] K. Davis, M. Mewes, M. Andrews, N. van Druten, D. Durfee, D. Kurn, et al., Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett. 75 (1995), 3969–3973. 10.1103/PhysRevLett.75.3969Search in Google Scholar PubMed

[15] E. Gross, Structure of a quantized vortex in boson systems. Nuovo Cimento 20 (1961), 454–466. 10.1007/BF02731494Search in Google Scholar

[16] Q. Guo and J. Yang, Excited states for two-component Bose-Estein condenstates in dimension two, J. Differential Equations 343 (2023), 658–686. 10.1016/j.jde.2022.10.034Search in Google Scholar

[17] Y. Guo, S. Li, J. Wei, and X. Zeng, Ground states of two-component attractive Bose-Einstein condensates I: Existence and uniqueness, J. Funct. Anal. 276 (2019), no. 1, 183–230. 10.1016/j.jfa.2018.09.015Search in Google Scholar

[18] Y. Guo and R. Seiringer, On the mass concentration for Bose-Einstein condensates with attractive interactions. Lett. Math. Phys. 104 (2014), 141–156. 10.1007/s11005-013-0667-9Search in Google Scholar

[19] Y. Guo, Z. Wang, X. Zeng, and H. Zhou, Properties of ground states of attractive Gross-Pitaevskii equations with multi-well potentials. Nonlinearity 31 (2018), 957–979. 10.1088/1361-6544/aa99a8Search in Google Scholar

[20] Y. Guo, X. Zeng, and H. Zhou, Energy estimates and symmetry breaking in attractive Bose-Einsteincondensates with ring-shaped potentials, Ann. Inst. H. Poincaré Anal. Non Linéeaire 33 (2016), 809–828. 10.1016/j.anihpc.2015.01.005Search in Google Scholar

[21] M. Kwong, Uniqueness of positive solutions of Δu−u+up=0 in RN, Arch. Rational Mech. Anal. 105 (1989), no. 3, 243–266. 10.1007/BF00251502Search in Google Scholar

[22] T.C. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in RN,N≤3, Commun. Math. Phys. 255 (2005), 629–653. 10.1007/s00220-005-1313-xSearch in Google Scholar

[23] P. Luo, S. Peng, J. Wei, and S. Yan, Excited states on Bose-Einstein condesates with attactive interactions, Calc. Var. Partial Differential Equations 60 (2021), 155. 10.1007/s00526-021-02046-xSearch in Google Scholar

[24] S. Peng and Z.-Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Ration. Mech. Anal. 208 (2013), 305–339. 10.1007/s00205-012-0598-0Search in Google Scholar

[25] L. Pitaevskii, Vortex lines in an imperfect Bose gas, Sov. Phys. JETP 13 (1961), 451–454. Search in Google Scholar

[26] B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equation in RN, Comm. Math. Phys. 271 (2007), no. 1, 199–221. 10.1007/s00220-006-0179-xSearch in Google Scholar

[27] Z. Tang and H. Xie, Muti-spikes solutions for a system of coupled elliptic equations with quadratic nonlinearity, Commun. Pure Appl. Anal 19 (2020), 311–328. 10.3934/cpaa.2020017Search in Google Scholar

[28] R. Tian, Z.-Q. Wang, and L. Zhao, Schrödinger systems with quadratic interactions, Commun. Contemp. Math. 21 (2019), no. 8, 1850077, 20pp. 10.1142/S0219199718500773Search in Google Scholar

[29] C. Wang and J. Zhou, Infinitely many solitary waves due to the second-harmonic generation in quadratic media, Acta Math. Sci. Ser. B (Engl. Ed.) 40 (2020), no. 1, 16–34. 10.1007/s10473-020-0102-3Search in Google Scholar

[30] J. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schrödinger equations, Arch. Ration. Mech. Anal. 190 (2008), no. 1, 83–106. 10.1007/s00205-008-0121-9Search in Google Scholar

[31] J. Yang and T. Zhou, Existence of single peak solutions for a nonlinear Schróodinger system with coupled quadratic nonlinearity, Adv. Nonlinear Anal. 11 (2022), 417–431. 10.1515/anona-2021-0201Search in Google Scholar
Received: 2022-12-26
Accepted: 2023-10-11
Published Online: 2024-01-27

© 2024 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. Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
  3. Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation
  4. Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
  5. Existence and uniqueness of solution for a singular elliptic differential equation
  6. The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
  7. Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
  8. Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
  9. Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
  10. Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions
  11. Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator
  12. On a nonlinear Robin problem with an absorption term on the boundary and L1 data
  13. Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
  14. Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
  15. Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
  16. Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
  17. Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
  18. A minimization problem with free boundary for p-Laplacian weakly coupled system
  19. Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
  20. k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
  21. Infinitely many solutions for Hamiltonian system with critical growth
  22. On optimal control in a nonlinear interface problem described by hemivariational inequalities
  23. Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
  24. Potential and monotone homeomorphisms in Banach spaces
  25. Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
  26. Supercritical Hénon-type equation with a forcing term
  27. Concentration of blow-up solutions for the Gross-Pitaveskii equation
  28. Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
  29. Regularity for critical fractional Choquard equation with singular potential and its applications
  30. Double phase anisotropic variational problems involving critical growth
  31. Normalized solutions of NLS equations with mixed nonlocal nonlinearities
  32. Nontrivial solutions for resonance quasilinear elliptic systems
  33. Standing waves for Choquard equation with noncritical rotation
  34. Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
  35. Modified quasilinear equations with strongly singular and critical exponential nonlinearity
  36. Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
  37. Nests of limit cycles in quadratic systems
  38. Regularity of minimizers for double phase functionals of borderline case with variable exponents
  39. Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
  40. Ground state solutions for magnetic Schrödinger equations with polynomial growth
  41. Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
  42. Dynamics for wave equations connected in parallel with nonlinear localized damping
  43. Boussinesq's equation for water waves: Asymptotics in Sector I
  44. Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
  45. Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
  46. Normalized solutions for the double-phase problem with nonlocal reaction
  47. Normalized solutions for Sobolev critical fractional Schrödinger equation
  48. Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
  49. Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
  50. Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
  51. Quasiconvex bulk and surface energies: C1,α regularity
  52. Time decay estimates of solutions to a two-phase flow model in the whole space
  53. Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
  54. The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
  55. A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
  56. Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
  57. Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
  58. The second Yamabe invariant with boundary
  59. Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
  60. Normalized solutions for the Choquard equations with critical nonlinearities
  61. Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
  62. Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
  63. Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
  64. Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
  65. Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
  66. Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
  67. Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
  68. On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
  69. Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
  70. Uniqueness and nondegeneracy of ground states for − Δ u + u = ( I α ⋆ u 2 ) u in R 3 when α is close to 2
  71. Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
  72. Weighted Hardy-Adams inequality on unit ball of any even dimension
  73. Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
  74. Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
  75. Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
  76. The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
  77. Nonlocal heat equations with generalized fractional Laplacian
  78. Choquard equations with recurrent potentials
  79. Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
  80. The Riemann problem for two-layer shallow water equations with bottom topography
  81. Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
  82. On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
  83. Energy-variational solutions for viscoelastic fluid models
  84. Stability on 3D Boussinesq system with mixed partial dissipation
  85. Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
  86. Erratum
  87. Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
https://doi.org/10.1515/anona-2023-0113
Keywords for this article
normalized peak solutions; local Pohozaev identities; reduction method
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
  3. Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation
  4. Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
  5. Existence and uniqueness of solution for a singular elliptic differential equation
  6. The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
  7. Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
  8. Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
  9. Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
  10. Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions
  11. Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator
  12. On a nonlinear Robin problem with an absorption term on the boundary and L1 data
  13. Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
  14. Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
  15. Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
  16. Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
  17. Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
  18. A minimization problem with free boundary for p-Laplacian weakly coupled system
  19. Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
  20. k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
  21. Infinitely many solutions for Hamiltonian system with critical growth
  22. On optimal control in a nonlinear interface problem described by hemivariational inequalities
  23. Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
  24. Potential and monotone homeomorphisms in Banach spaces
  25. Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
  26. Supercritical Hénon-type equation with a forcing term
  27. Concentration of blow-up solutions for the Gross-Pitaveskii equation
  28. Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
  29. Regularity for critical fractional Choquard equation with singular potential and its applications
  30. Double phase anisotropic variational problems involving critical growth
  31. Normalized solutions of NLS equations with mixed nonlocal nonlinearities
  32. Nontrivial solutions for resonance quasilinear elliptic systems
  33. Standing waves for Choquard equation with noncritical rotation
  34. Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
  35. Modified quasilinear equations with strongly singular and critical exponential nonlinearity
  36. Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
  37. Nests of limit cycles in quadratic systems
  38. Regularity of minimizers for double phase functionals of borderline case with variable exponents
  39. Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
  40. Ground state solutions for magnetic Schrödinger equations with polynomial growth
  41. Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
  42. Dynamics for wave equations connected in parallel with nonlinear localized damping
  43. Boussinesq's equation for water waves: Asymptotics in Sector I
  44. Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
  45. Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
  46. Normalized solutions for the double-phase problem with nonlocal reaction
  47. Normalized solutions for Sobolev critical fractional Schrödinger equation
  48. Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
  49. Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
  50. Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
  51. Quasiconvex bulk and surface energies: C1,α regularity
  52. Time decay estimates of solutions to a two-phase flow model in the whole space
  53. Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
  54. The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
  55. A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
  56. Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
  57. Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
  58. The second Yamabe invariant with boundary
  59. Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
  60. Normalized solutions for the Choquard equations with critical nonlinearities
  61. Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
  62. Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
  63. Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
  64. Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
  65. Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
  66. Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
  67. Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
  68. On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
  69. Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
  70. Uniqueness and nondegeneracy of ground states for − Δ u + u = ( I α ⋆ u 2 ) u in R 3 when α is close to 2
  71. Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
  72. Weighted Hardy-Adams inequality on unit ball of any even dimension
  73. Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
  74. Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
  75. Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
  76. The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
  77. Nonlocal heat equations with generalized fractional Laplacian
  78. Choquard equations with recurrent potentials
  79. Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
  80. The Riemann problem for two-layer shallow water equations with bottom topography
  81. Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
  82. On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
  83. Energy-variational solutions for viscoelastic fluid models
  84. Stability on 3D Boussinesq system with mixed partial dissipation
  85. Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
  86. Erratum
  87. Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 5.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/anona-2023-0113/html
Scroll to top button