Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 49, pp. 1–22.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.49

NORMALIZED SOLUTIONS FOR FRACTIONAL

SCHRÖDINGER-CHOQUARD SYSTEMS WITH

SOBOLEV CRITICAL COUPLED NONLINEARITY

ZILIN CHEN, YANG YANG

Abstract. We study the existence of normalized solutions for a system of fractional Schrödinger-

Choquard with Sobolev critical coupling term. We obtain the existence of the positive normal-
ized ground state solution, and in a special case we obtain a mountain pass type normalized

solution.

1. Introduction and main results

In this article, we are concerned with the fractional Schrödinger-Choquard system with Sobolev
critical coupling term,

(−∆)su+ λ1u = (Iα ∗ |u|p)|u|p−2u+
βr1
2∗s

|u|r1−2u|v|r2 , in RN ,

(−∆)sv + λ2v = (Iα ∗ |v|q)|v|q−2v +
βr2
2∗s

|u|r1 |v|r2−2v, in RN ,

(1.1)

having prescribed L2-norm ∫
RN

|u|2 dx = a2 and

∫
RN

|v|2 dx = b2, (1.2)

where N ≥ 3, s ∈ (0, 1), α ∈ (0, N), a, b > 0, β > 0; the exponents p, q, r1, r2 satisfy

r1 > 1, r2 > 1, r1 + r2 = 2∗s, 2α < p, q < 2∗α,s;

and λ1, λ2 ∈ R appear as Lagrange multipliers which are part of the unknowns. 2∗s := 2N
N−2s is the

fractional Sobolev critical exponent, 2α := N+α
N is the Hardy-Littlewood-Sobolev lower critical

exponent and 2∗α,s := N+α
N−2s is the fractional Hardy-Littlewood-Sobolev upper critical exponent.

Here, Iα(x) := |x|α−N , and the fractional Laplacian (−∆)s is defined for u ∈ S(RN ) by

(−∆)su(x) = CN,sP.V.

∫
RN

u(x)− u(y)

|x− y|N+2s
dy, x ∈ RN ,

where S(RN ) denotes the Schwartz space of rapidly decreasing smooth functions, P.V. stands for
the principle value of the integral and CN,s is the positive normalization constant. The fractional
Laplacian appears in many diverse domains, including optimization, phase transitions, conserva-
tion laws, anomalous diffusion, stratified materials, ultra-relativistic limits of quantum mechanics,
crystal dislocation, water waves and so on. We refer to [8, 12] for more applications.

2020 Mathematics Subject Classification. 35A15, 35J50, 35R11, 35B33.
Key words and phrases. Fractional Schrödinger-Choquard system; normalized solutions; critical exponent;

Pohozaev manifold.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted March 11, 2025. Published May 13, 2025.

1



2 Z. CHEN, Y. YANG EJDE-2025/49

System (1.1) is closely related to the physical model

(−∆)sΨ1 = i
∂Ψ1

∂t
+ (Iα ∗ |Ψ1|p)|Ψ1|p−2Ψ1 + βr1|Ψ1|r1−2Ψ1|Ψ2|r2 ,

(−∆)sΨ2 = i
∂Ψ2

∂t
+ (Iα ∗ |Ψ2|q)|Ψ2|q−2Ψ2 + βr2|Ψ1|r1 |Ψ2|r2−2Ψ2,

(1.3)

which describes several physical phenomenon, such as the Bose-Einstein condensates with multiple
states, or the propagation of mutually incoherent waves packets in nonlinear optics, see [38].
Physically, an important and well-known feature of (1.3) is conservation of mass: the L2-norms
|Ψ1(t, ·)|2, |Ψ2(t, ·)|2 of solutions are independent of t ∈ R. And these norms have a clear physical
meaning, for example, they represent the number of particles of each component in Bose-Einstein
condensates, or the power supply in the nonlinear optics framework. It is worth mentioning
that an important topic of (1.3) is to look for standing wave solutions Ψ1(t, x) = eiλ1tu(x) and
Ψ2(t, x) = eiλ2tv(x), and (Ψ1,Ψ2) solves (1.3) if and only if (u, v) is a solution of the system

(−∆)su+ λ1u = (Iα ∗ |u|p)|u|p−2u+ βr1|u|r1−2u|v|r2 , in RN ,

(−∆)sv + λ2v = (Iα ∗ |v|q)|v|q−2v + βr2|u|r1 |v|r2−2v, in RN .

There are two approaches to deal with this problem. On the one hand, we can regard the frequen-
cies λ1, λ2 as fixed. On the other hand, we regard the problem as having prescribed mass, and
λ1, λ2 appear as Lagrange multipliers under the mass constraint. The solution with prescribed
mass is called a normalized solution. In this article, we pay close attention to the latter.

In recent years, many scholars have increasingly focused on the study for normalized solutions
of the nonlinear Schrödinger equations or Choquard equations. Jeanjean [23] firstly studied L2-
supercritical case, and dealt with the existence of normalized solutions by using the mountain pass
lemma and a skillful compactness argument. Soave [34] considered the existence of normalized
solutions and orbitally stable for the particular case of a combined nonlinearity of power type.
Moreover, the Sobolev critical case was also studied by Soave [35], where he obtained the exis-
tence and nonexistence of normalized solutions. Recently, Lan et al. [27] studied the fractional
critical Choquard equation with a nonlocal perturbation. Under L2-subcritical, L2-critical and
L2-supercritical perturbation, they obtained the existence of normalized ground states and moun-
tain pass type solutions. Gao et al. [17] explored the critical Choquard equations with combined
nonlinearities. They got the normalized solution by the Pohozaev manifold and mountain pass
theorem, and they used the truncation technique and the genus theory to obtain the multiplicity
of normalized solutions. Moreover, Cai et al. [9] considered the double-phase problem with non-
local reaction. They used the Hardy-Littlewood Sobolev subcritical approximation to obtain the
existence and nonexistence of normalized solutions, and also studied the existence of normalized
solutions to the double-phase problem involving double Hardy-Littlewood-Sobolev critical expo-
nents. In addition, for more results with regard to nonlinear Schrödinger equation or Choquard
equation, we refer to [14, 21, 25, 36, 41, 42, 43] and the references therein.

For nonlinear systems with critical exponents, Li et al. [32] considered the following Schrödinger
systems with critical and subcritical nonlinearities:

−∆u+ λ1u = µ1|u|p−2u+ βr1|u|r1−2u|v|r2 , in RN ,

−∆v + λ2v = µ2|v|q−2v + βr2|u|r1 |v|r2−2v, in RN ,∫
RN

|u|2 dx = a21,

∫
RN

|v|2 dx = a22,

where p, r1 + r2 < 2∗ := 2N
N−2 and q ≤ 2∗. They studied the geometry of the Pohozaev manifold

and the associated minimization problem. Under some assumptions on a1, a2 and β, they showed



EJDE-2025/49 FRACTIONAL SCHRÖDINGER-CHOQUARD SYSTEMS 3

the existence of a positive normalized ground state solution. Bartsch et al. [5] explored the system

−∆u+ λ1u = |u|2
∗−2u+ να|u|α−2u|v|β , in RN ,

−∆v + λ2v = |v|2
∗−2v + νβ|u|α|v|β−2v, in RN ,∫

RN

|u|2 dx = a2,

∫
RN

|v|2 dx = b2,

where N = 3, 4 and 2 < α+β < 2∗. When ν > 0 and α+β ≤ 2+ 4
N , they proved that the system

has a normalized ground state solution for 0 < ν < ν0. When α + β ≥ 2 + 4
N , they showed the

existence of a threshold ν1 ≥ 0 such that a normalized ground state solution exists for ν > ν1.
Zhang et al. [44] studied the following Schrödinger system with a coupled critical nonlinearity:

−∆u = λ1u+ µ1|u|p−2u+ βr1|u|r1−2u|v|r2 , in RN ,

−∆v = λ2v + µ2|v|q−2v + βr2|u|r1 |v|r2−2v, in RN ,∫
RN

|u|2 dx = a,

∫
RN

|v|2 dx = b,

where N ≥ 3, a, b, µ1, µ2, β > 0, 2 ≤ p, q < 2 + 4
N and r1 + r2 = 2∗. They mainly focused on the

L2-subcritical case, and then obtained the existence of normalized positive ground state solution
for any 0 < β < β0. Moreover, Liu et al. [26] considered the following Sobolev critical Schrödinger
system:

−∆u+ λ1u = |u|2
∗−2u+ µ1|u|p−2u+ βr1|u|r1−2u|v|r2 , in RN ,

−∆v + λ2v = |v|2
∗−2v + µ2|v|q−2v + βr2|u|r1 |v|r2−2v, in RN ,∫

RN

|u|2 dx = a2,

∫
RN

|v|2 dx = b2.

When p, q, r1 + r2 ∈ (2 + 4
N , 2∗), by revealing the basic behavior of the mountain-pass energy,

they showed the existence of the positive normalized solution. When p = q = r1 + r2 = 2∗, they
obtained the nonexistence of the positive normalized solution.

When it comes to investigating normalized solutions for the fractional systems with critical
nonlinearity, Zuo et al. [46] studied the fractional Sobolev critical nonlinear Schrödinger coupled
systems. By scaling transformation and concentration-compactness principle, they obtained the
existence of a positive normalized solution under some suitable assumptions for the mass super-
critical case. Guo et al. [19] considered the following fractional Choquard system with a local
perturbation:

(−∆)su = λ1u+ (Iα ∗ |u|2
∗
α,s)|u|2

∗
α,s−2u+ µ1|u|p−2u+ βr1|u|r1−2u|v|r2 , in RN ,

(−∆)sv = λ2v + (Iα ∗ |v|2
∗
α,s)|v|2

∗
α,s−2v + µ2|v|q−2v + βr2|u|r1 |v|r2−2v, in RN ,∫

RN

|u|2 dx = a21,

∫
RN

|v|2 dx = a22.

For p, q and r1 + r2 ∈ (2 + 4s
N , 2∗s), they obtained the existence of the positive normalized solution

when β is big enough. Moreover, for the case of p = q = r1 + r2 = 2∗s, they obtained the
nonexistence of the positive normalized solution. Besides, Dou et al. [10] studied the fractional
Schrödinger-Poisson system with multiple competing potentials and a critical nonlocal term, and
obtained the multiplicity of positive solutions. Gao et al. [18] considered the nonlinear coupled
Kirchhoff system with purely Sobolev critical exponent, and got the existence of positive ground
states. For more results with regard to Schrödinger or Choquard systems, we refer to [45, 2, 47,
11, 20, 4, 3, 7, 1, 40] and the references therein.

Motivated by the aforementioned works, we shall investigate the normalized solutions for the
fractional Schrödinger-Choquard system with Sobolev critical coupling term. We mainly focus on
the problem (1.1)-(1.2) for two different scenarios: (i) L2-subcritical case: 2α < p, q < N+2s+α

N ;

(ii) L2-supercritical case: N+2s+α
N < p, q < 2∗α,s.



4 Z. CHEN, Y. YANG EJDE-2025/49

Before we state our main results, we first introduce some notation. Throughout this paper,
Lr(RN ) denotes the Lebesgue space with the norm ∥u∥r = (

∫
RN |u|r dx) 1

r for any 1 ≤ r < ∞.

Hs(RN ) is the fractional Hilbert space defined as

Hs(RN ) := {u ∈ L2(RN ) : (−∆)s/2u ∈ L2(RN )},

which is endowed with the standard inner product and norm, given respectively by

⟨u, v⟩ :=
∫
RN

(
(−∆)s/2u(−∆)s/2v + uv

)
dx, ∥u∥2Hs = ⟨u, u⟩ = ∥(−∆)s/2u∥22 + ∥u∥22,

where

∥(−∆)s/2u∥22 =

∫
RN

∫
RN

|u(x)− u(y)|2

|x− y|N+2s
dxdy.

Then, we define H := Hs(RN )×Hs(RN ), which is equipped with the norm ∥(u, v)∥2H := ∥u∥2Hs +
∥v∥2Hs .

The solutions of the system (1.1)-(1.2) can be found as critical points of the energy functional
E(u, v) : H 7→ R,

E(u, v) =
1

2

∫
RN

∣∣(−∆)s/2u
∣∣2 dx+

1

2

∫
RN

∣∣(−∆)s/2v
∣∣2 dx− 1

2p

∫
RN

(
Iα ∗ |u|p

)
|u|p dx

− 1

2q

∫
RN

(
Iα ∗ |v|q

)
|v|q dx− β

2∗s

∫
RN

|u|r1 |v|r2 dx.
(1.4)

It is standard to check that the energy functional is of class C1. The critical points of E(u, v) are
restricted on the constraint S(a, b) := S(a)× S(b), where

S(a) := {u ∈ Hs(RN ) : ∥u∥22 = a2}, S(b) := {v ∈ Hs(RN ) : ∥v∥22 = b2}.

Now, we recall the some definitions.

Definition 1.1. We say that (u, v) ∈ H is a weak solution to problem (1.1)-(1.2) if∫
RN

(−∆)s/2u(−∆)s/2φdx+

∫
RN

(−∆)s/2v(−∆)s/2ϕdx+ λ1

∫
RN

uφdx+ λ2

∫
RN

vϕdx

=

∫
RN

(
Iα ∗ |u|p

)
|u|p−2uφdx+

∫
RN

(
Iα ∗ |v|q

)
|v|q−2vϕdx

+
β

2∗s
r1

∫
RN

|u|r1−2|v|r2uφdx+
β

2∗s
r2

∫
RN

|u|r1 |v|r2−2vϕdx,

for any (φ, ϕ) ∈ H.

Definition 1.2. We say that (ua, vb) is a normalized ground state solution of problem (1.1)-(1.2) if
(ua, vb, λa, λb) is a solution to the problem (1.1)-(1.2), and (ua, vb) has the minimal energy among
all the solutions which belong to S(a, b), that is,

E′|S(a,b)(ua, vb) = 0 and E(ua, vb) = inf{E(u, v) : E′|S(a,b)(u, v) = 0 and (u, v) ∈ S(a, b)}.

Our main results are as follows.

Theorem 1.3. Let a, b > 0 be fixed and 2α < p, q < N+2s+α
N . Then there exists β0 = β0(a, b) > 0

such that for any 0 < β < β0, problem (1.1)-(1.2) has a positive normalized ground state solution

(u0, v0) with the corresponding Lagrange multipliers λ̂1 > 0, λ̂2 > 0.

Theorem 1.4. Let a, b > 0 be given. Suppose that

(i) N ≥ 4s, 0 < α < N , N+2s+α
N < p, q < 2∗α,s;

(ii) 2s < N < 4s, 0 < α < N , max{N+2s+α
N , 2∗α,s − 2s

4s−N } < p, q < 2∗α,s.

If (i) or (ii) holds, then there exists β̃ > 0 such that for every β > β̃, problem (1.1)-(1.2) possesses
a mountain pass type normalized solution (ũ, ṽ) which is positive and radially symmetric with the

corresponding Lagrange multipliers λ̃1, λ̃2 > 0.



EJDE-2025/49 FRACTIONAL SCHRÖDINGER-CHOQUARD SYSTEMS 5

Comparing this with [32] and [44], we extend the results to the fractional case and introduce
nonlocal perturbations, which make the problem much more complex. In the L2-subcritical case, a
solution is obtained by minimizing the functional on a component of the Pohozaey manifold where
the functional becomes negative. We establish an upper estimate of the normalized ground state
level, which allows us to recover the compactness of Palais-Smale sequence. In the L2-supercritical
case, we derive the existence of bounded (PSP) sequence. Moreover, to overcome the difficulty of
losing compactness, a precise threshold for the mountain pass level is required. Under the dual
influence of fractional Laplacian and nonlocal perturbation, the estimates and calculations become
more challenging, but we solve these difficulties. We comprehensively consider the combined effects
of fractional operators, nonlocal terms, and Sobolev critical coupled term, and then obtain new
and interesting results about the existence of normalized solutions for the fractional Schrödinger-
Choquard system.

This article is organized as follows. In Section 2, we give some preliminary results which will
be used later. In Section 3, we deal with the L2-subcritical case: 2α < p, q < N+2s+α

N and prove

Theorem 1.3. In Section 4, we consider the L2-supercritical case: N+2s+α
N < p, q < 2∗α,s, and give

the proof of Theorem 1.4.
Throughout this paper, we also use the following notation:

• A ∼ B represent C1B ≤ A ≤ C2B for some positive constants C1, C2 > 0.
• The letters C,C1, C2, C3 ... are universal positive constant (possibly different).

2. Preliminaries

In this section, we present various preliminary results that will be used later. Firstly, let us
recall the following fractional Sobolev embedding, see [12, Theorem 6.5].

Lemma 2.1. Let 0 < s < 1 and N > 2s. Then there exists a constant S = S(N, s) > 0 such that

S = inf
u∈Hs(RN )\{0}

∥(−∆)s/2u∥22
∥u∥22∗s

, (2.1)

where 2∗s = 2N
N−2s . Meanwhile, Hs(RN ) is continuously embedded into Lq(RN ) for any q ∈ [2, 2∗s]

and compactly embedded into Lq
loc(RN ) for every q ∈ [2, 2∗s). Furthermore, we define

Sr1,r2 := inf
u,v∈Hs(RN )\{0}

∥(−∆)s/2u∥22 + ∥(−∆)s/2v∥22( ∫
RN |u|r1 |v|r2 dx

) 2
2∗s

,

then from [31, Lemma 2.2] we know that

Sr1,r2 =
((r1

r2

)r2/2
∗
s

+
(r2
r1

) r1
2∗s
)
S.

The following Hardy-Littlewood-Sobolev inequality is of importance, see [28].

Lemma 2.2. Let α ∈ (0, N) and r, t > 1 with 1
r + 1

t = 1 + α
N . Let f ∈ Lr(RN ) and h ∈ Lt(RN ),

then there exists a constant C(r, t, α,N) such that∣∣∣ ∫
RN

∫
RN

f(x)h(y)

|x− y|N−α
dxdy

∣∣∣ ≤ C(r, t, α,N)∥f∥r∥h∥r.

In particular, if r = t = 2N
N+α , then

C(r, t, α,N) = Cα := π
N−α

2
Γ(α2 )

Γ(N+α
2 )

{Γ(N2 )

Γ(N)

}−α/N

.

Next, we introduce the fractional Gagliardo-Nirenberg inequality of Hartree type established in
[16].

Lemma 2.3. Let 0 < s < 1, N > 2s and p ∈ (2α, 2
∗
α,s). Then, for all u ∈ Hs(RN ),∫

RN

(
Iα ∗ |u|p

)
|u|p dx ≤ C̃∥(−∆)s/2u∥2pδp,s2 ∥u∥2p(1−δp,s)

2 ,



6 Z. CHEN, Y. YANG EJDE-2025/49

where δp,s :=
Np−N−α

2ps and the optimal constant C̃ is

C̃ =
2sp

2sp−Np+N + α

(2sp−Np+N + α

Np−N − α

)Np−N−α
2s ∥W∥2−2p

2 ,

where W is the ground state solution of (−∆)sW +W − (Iα ∗ |W |p)|W |p−2W = 0.

Next, we show the weak compactness result for the nonlocal nonlinearities, see [43, Lemma 2.7].

Lemma 2.4. Suppose that p ∈ (2α, 2
∗
α,s). If {un} is a sequence satisfying un ⇀ u weakly in

Hs(RN ), then, for any φ ∈ Hs(RN ), we have∫
RN

(
Iα ∗ |un|p

)
|un|p−2unφdx →

∫
RN

(
Iα ∗ |u|p

)
|u|p−2uφdx,

as n → ∞.

Then, we present the following Brezis-Lieb type lemmas, see [13, 22] and [29, Lemma 2.3],
respectively.

Lemma 2.5. Let p ∈ (2α, 2
∗
α,s) and {un} be a bounded sequence in Hs(RN ). If un → u a.e. in

RN as n → ∞, then∫
RN

(
Iα ∗ |un − u|p

)
|un − u|p dx =

∫
RN

(
Iα ∗ |un|p

)
|un|p dx−

∫
RN

(
Iα ∗ |u|p

)
|u|p dx+ on(1).

Lemma 2.6. Let {(un, vn)} ⊂ H be a bounded sequence, r1, r2 > 1 and 2 < r1 + r2 ≤ 2∗s. If
(un, un) → (u, v) a.e. in RN as n → ∞, then∫

RN

|un − u|r1 |vn − v|r2 dx =

∫
RN

|un|r1 |vn|r2 dx−
∫
RN

|u|r1 |v|r2 dx+ on(1).

Next, to recover the compactness of Palais-Smale sequences, we need to introduce some estab-
lished results for the equation

(−∆)su+ λu = (Iα ∗ |u|p)|u|p−2u in RN ,∫
RN

|u|2 dx = c2 > 0.
(2.2)

In a variational approach, normalized solutions of problem (2.2) are obtained as critical points of
the associated energy functional

Jp(u) =
1

2

∫
RN

∣∣(−∆)s/2u
∣∣2 dx− 1

2p

∫
RN

(
Iα ∗ |u|p

)
|u|p dx,

for u ∈ Hs(RN ), on the constraint S(c) = {u ∈ Hs(RN ) : ∥u∥22 = c2}. The following two results
are established in [13] and [30] respectively.

Lemma 2.7. Let 2α < p < 1 + α+2s
N . Then problem (2.2) has a normalized ground state solution

uc,p with the corresponding Lagrange multiplier λc,p > 0. Moreover, the corresponding energy level
satisfies Jp(uc,p) < 0.

Lemma 2.8. Let 1+ α+2s
N < p < 2∗α,s. Then problem (2.2) has a normalized ground state solution

ûc,p with the corresponding Lagrange multiplier λ̂c,p > 0. Besides, the corresponding energy level
satisfies Jp(ûc,p) > 0.

Now, we introduce the Pohozaev manifold

P(a, b) := {(u, v) ∈ S(a, b) : P (u, v) = 0},
where

P (u, v) = ∥(−∆)s/2u∥22 + ∥(−∆)s/2v∥22 − δp,s

∫
RN

(
Iα ∗ |u|p

)
|u|p dx

− δq,s

∫
RN

(
Iα ∗ |v|q

)
|v|q dx− β

∫
RN

|u|r1 |v|r2 dx.



EJDE-2025/49 FRACTIONAL SCHRÖDINGER-CHOQUARD SYSTEMS 7

Lemma 2.9. Let (u, v) ∈ S(a, b) be a weak solution of problem (1.1)-(1.2), then (u, v) ∈ P(a, b).

Proof. From [15], we find that the Pohozaev identity for problem (1.1)-(1.2) is

N − 2s

2

(
∥(−∆)s/2u∥22 + ∥(−∆)s/2v∥22

)
+

N

2

(
λ1∥u∥22 + λ2∥v∥22

)
=

N + α

2p

∫
RN

(
Iα ∗ |u|p

)
|u|p dx+

N + α

2q

∫
RN

(
Iα ∗ |v|q

)
|v|q dx+

β

2∗s
N

∫
RN

|u|r1 |v|r2 dx.

Since (u, v) is a weak solution, we have

∥(−∆)s/2u∥22 + ∥(−∆)s/2v∥22 + λ1∥u∥22 + λ2∥v∥22

=

∫
RN

(
Iα ∗ |u|p

)
|u|p dx+

∫
RN

(
Iα ∗ |v|q

)
|v|q dx+ β

∫
RN

|u|r1 |v|r2 dx.

Then, we obtain that

∥(−∆)s/2u∥22 + ∥(−∆)s/2v∥22

= δp,s

∫
RN

(
Iα ∗ |u|p

)
|u|p dx+ δq,s

∫
RN

(
Iα ∗ |v|q

)
|v|q dx+ β

∫
RN

|u|r1 |v|r2 dx.

Hence, we can draw the conclusion. □

For each (u, v) ∈ S(a, b) and t ∈ R, we introduce the L2-invariant scaling

(t ⋆ u, t ⋆ v) :=
(
e

N
2 tu(etx), e

N
2 tv(etx)

)
for a.e. x ∈ RN .

It results that (t ⋆ u, t ⋆ v) ∈ S(a, b). Then we define the fibering map

Φu,v(t) := E(t ⋆ u, t ⋆ v)

=
1

2
e2st

(
∥(−∆)s/2u∥22 + ∥(−∆)s/2v∥22

)
− 1

2p
e2pδp,sst

∫
RN

(
Iα ∗ |u|p

)
|u|p dx

− 1

2q
e2qδq,sst

∫
RN

(
Iα ∗ |v|q

)
|v|q dx− β

2∗s
e2

∗
sst

∫
RN

|u|r1 |v|r2 dx.

(2.3)

Lemma 2.10. Let (u, v) ∈ S(a, b), then t ∈ R is the critical point of Φu,v(t) if and only if
(t ⋆ u, t ⋆ v) ∈ S(a, b).

Proof. For (u, v) ∈ S(a, b) and t ∈ R, through direct calculation, it is easy to check that

Φ′
u,v(t) = se2st

(
∥(−∆)s/2u∥22 + ∥(−∆)s/2v∥22

)
− sδp,se

2pδp,sst

∫
RN

(
Iα ∗ |u|p

)
|u|p dx

− sδq,se
2qδq,sst

∫
RN

(
Iα ∗ |v|q

)
|v|q dx− βse2

∗
sst

∫
RN

|u|r1 |v|r2 dx

= sP (t ⋆ u, t ⋆ v).

Then, we can easily draw this conclusion. □

Then, it is natural to consider the decomposition of P(a, b) into the disjoint unions P(a, b) =
P+(a, b) ∪ P0(a, b) ∪ P−(a, b), where

P+(a, b) := {(u, v) ∈ P(a, b) : Φ′′
u,v(0) > 0},

P0(a, b) := {(u, v) ∈ P(a, b) : Φ′′
u,v(0) = 0},

P−(a, b) := {(u, v) ∈ P(a, b) : Φ′′
u,v(0) < 0}.



8 Z. CHEN, Y. YANG EJDE-2025/49

3. The case 2α < p, q < N+2s+α
N

Recalling Lemma 2.7, we define

ω1 := ua,p, ω2 := vb,q,

m1 := Jp(ω1) < 0, m2 := Jq(ω2) < 0.

Lemma 3.1. Let a, b > 0 be given. Then there exist β0 = β0(a, b) > 0 and h0 = h0(a, b) >(
∥(−∆)s/2ω1∥22 + ∥(−∆)s/2ω2∥22

)1/2

such that

E(u, v) > 0 on S(a, b) ∩A(2h0) \A(h0) for any 0 < β < β0,

where A(h) :=
{
(u, v) ∈ H : ∥(−∆)s/2u∥22 + ∥(−∆)s/2v∥22 < h2

}
.

Proof. For each (u, v) ∈ H, set h =
(
∥(−∆)s/2u∥22+∥(−∆)s/2v∥22

)1/2

. By Young’s inequality and

Lemma 2.1, we have∫
RN

|u|r1 |v|r2 dx ≤
∫
RN

r1
2∗s

|u|2
∗
s dx+

∫
RN

r2
2∗s

|v|2
∗
s dx

≤ S−2∗s/2
( r1
2∗s

∥(−∆)s/2u∥2
∗
s

2 +
r2
2∗s

∥(−∆)s/2v∥2
∗
s

2

)
≤ S−2∗s/2

(
∥(−∆)s/2u∥22 + ∥(−∆)s/2v∥22

)2∗s/2

.

(3.1)

Then, by Lemma 2.3, (1.4) and (3.1), it follows that

E(u, v) ≥ 1

2

(
∥(−∆)s/2u∥22 + ∥(−∆)s/2v∥22

)
− β

2∗s
S−2∗s/2

(
∥(−∆)s/2u∥22 + ∥(−∆)s/2v∥22

)2∗s/2

− 1

2p
C̃∥(−∆)s/2u∥2pδp,s2 ∥u∥2p(1−δp,s)

2 − 1

2q
C̃∥(−∆)s/2v∥2qδq,s2 ∥v∥2q(1−δq,s)

2

≥ 1

2
h2 − C̃

2p
a2p(1−δp,s)h2pδp,s − C̃

2q
b2q(1−δq,s)h2qδq,s − β

2∗s
S−2∗s/2h2∗s

= h2
(1
2
− C̃

2p
a2p(1−δp,s)h2pδp,s−2 − C̃

2q
b2q(1−δq,s)h2qδq,s−2 − β

2∗s
S−2∗s/2h2∗s−2

)
.

(3.2)

Considering that 2pδp,s−2 < 0 and 2qδq,s−2 < 0, we can take a large enough h0 >
(
∥(−∆)s/2ω1∥22+

∥(−∆)s/2ω2∥22
)1/2

such that

C̃

2p
a2p(1−δp,s)h

2pδp,s−2
0 +

C̃

2q
b2q(1−δq,s)h

2qδq,s−2
0 ≤ 1

4
. (3.3)

In view of 2∗s − 2 > 0, there exists β0 > 0 such that

β0

2∗s
S−2∗s/2(2h0)

2∗s−2 ≤ 1

8
. (3.4)

Then, according to (3.2), (3.3) and (3.4), for 0 < β < β0 and (u, v) ∈ S(a, b) ∩ A(2h0) \ A(h0), it
holds E(u, v) > 0. □

Next, we define

m(a, b) := inf
(u,v)∈S(a,b)∩A(2h0)

E(u, v),

where h0 is determined in Lemma 3.1.

Lemma 3.2. Let a, b > 0 be given. Then, for any 0 < β < β0, the following statements are true.

(i) m(a, b) < m1 +m2 < 0;
(ii) m(a, b) ≤ m(a1, b1), for any 0 < a1 ≤ a, 0 < b1 ≤ b.



EJDE-2025/49 FRACTIONAL SCHRÖDINGER-CHOQUARD SYSTEMS 9

Proof. (i) By Lemma 3.1 we have (ω1, ω2) ∈ A(h0), which implies (ω1, ω2) ∈ S(a, b) ∩ A(2h0).
Then by Lemma 2.7 we obtain that

m(a, b) ≤ E(ω1, ω2) = Jp(ω1) + Jq(ω2)−
β

2∗s

∫
RN

|ω1|r1 |ω2|r2 dx < m1 +m2 < 0.

(ii) It is sufficient to prove that for any ε > 0, one has

m(a, b) ≤ m(a1, b1) + ε, for any 0 < a1 ≤ a, 0 < b1 ≤ b.

By the definition of m(a1, b1) and Lemma 3.1, there exists (u, v) ∈ S(a, b) ∩A(2h0) such that

E(u, v) ≤ m(a1, b1) +
1

2
ε. (3.5)

Let κ(x) ∈ C∞
0 (RN ) be a cut-off function such that

0 ≤ κ(x) ≤ 1 and κ(x) =

{
0, |x| ≥ 2;

1, |x| ≤ 1.

For any ξ > 0, we define uξ(x) = u(x)κ(ξx) and vξ(x) = v(x)κ(ξx). Then, we can obtain that
(uξ, vξ) → (u, v) in H as ξ → 0. Hence, we can fix a ξ > 0 small enough such that

E(uξ, vξ) ≤ E(u, v) +
ε

4
, (3.6)(

∥(−∆)s/2uξ∥22 + ∥(−∆)s/2vξ∥22
)1/2

≤ 2h0 − η, (3.7)

for a small η > 0. Now let ϑ(x) ∈ C∞
0 (RN ) such that supp(ϑ) ⊂ {x ∈ RN : 4

ξ ≤ |x| ≤ 1 + 4
ξ}, and

set

ϑa =

√
a2 − ∥uξ∥22
∥ϑ∥2

ϑ and ϑb =

√
b2 − ∥vξ∥22
∥ϑ∥2

ϑ.

According to [24, Lemma 3.2] or [5, Lemma 2.3], for any λ ≤ 0, we have

supp(uξ) ∩ supp(λ ⋆ ϑa) = ∅ and supp(vξ) ∩ supp(λ ⋆ ϑb) = ∅.

Then, for each λ ≤ 0,(
supp(uξ) ∪ supp(vξ)

)
∩
(
supp(λ ⋆ ϑa) ∪ supp(λ ⋆ ϑb)

)
= ∅, (3.8)

then we have (uξ + λ ⋆ ϑa, vξ + λ ⋆ ϑb) ∈ S(a, b). Since E(λ ⋆ ϑa, λ ⋆ ϑb) → 0 and
(
∥(−∆)s/2(λ ⋆

ϑa)∥22 + ∥(−∆)s/2(λ ⋆ ϑb)∥22
)1/2

→ 0 as λ → −∞. Then we infer that

E(λ ⋆ ϑa, λ ⋆ ϑb) ≤
ε

4
, (3.9)(

∥(−∆)s/2(λ ⋆ ϑa)∥22 + ∥(−∆)s/2(λ ⋆ ϑb)∥22
)1/2

≤ η

2
, (3.10)

as λ → −∞. It follows from (3.7), (3.8) and (3.10) that (uξ + λ ⋆ ϑa, vξ + λ ⋆ ϑb) ∈ A(2h0) for
λ ≪ 0. Therefore, by (3.5), (3.6), (3.8) and (3.9) we obtain that

m(a, b) ≤ E(uξ + λ ⋆ ϑa, vξ + λ ⋆ ϑb)

= E(uξ, vξ) + E(λ ⋆ ϑa, λ ⋆ ϑb)

≤ E(u, v) +
ε

4
+

ε

4
≤ m(a1, b1) + ε,

which completes the proof. □

Lemma 3.3. Let a, b > 0 be given. Then, for any 0 < β < β0 and (u, v) ∈ S(a, b), Φu,v(t) has
two critical points τu,v < tu,v ∈ R and two zeros cu,v < du,v ∈ R with τu,v < cu,v < du,v < tu,v.
Moreover,

(i) if (t ⋆ u, t ⋆ v) ∈ P(a, b), then either t = τu,v or t = tu,v;



10 Z. CHEN, Y. YANG EJDE-2025/49

(ii) (
∥(−∆)s/2(t ⋆ u)∥22 + ∥(−∆)s/2(t ⋆ v)∥22

)1/2

≤ h0 ∀t ≤ cu,v,

E(τu,v ⋆ u, τu,v ⋆ v) = min
{
E(t ⋆ u, t ⋆ v) : t ∈ RN and(
∥(−∆)s/2(t ⋆ u)∥22 + ∥(−∆)s/2(t ⋆ v)∥22

)1/2

≤ h0

}
< 0,

where h0 is defined in Lemma 3.1;
(iii) E(tu,v ⋆ u, tu,v ⋆ v) = max{E(t ⋆ u, t ⋆ v) : t ∈ R} > 0.

Proof. In view of 0 < pδp,s, qδq,s < 1 and 2∗s > 2, by (2.3) we have Φu,v(−∞) = 0− and
Φu,v(+∞) = −∞. Combining this with Lemma 3.1, we obtain that Φu,v(t) has at least two
critical points τu,v < tu,v, where τu,v is a local minimum point of Φu,v(t) at negative level and tu,v
is a global maximum point at positive level. On the other hand, it is standard to prove that Φu,v(t)
has at most two critical points as in [32, Remark 3.1] or [34, Lemma 5.2]. Hence, it has exactly
two critical points τu,v and tu,v. Then, it follows from Lemma 2.10 that (t ⋆ u, t ⋆ v) ∈ P(a, b) if
and only if t = τu,v or t = tu,v, which implies (i). Meanwhile, by the monotonicity, Φu,v(t) has
exactly two zeros cu,v and du,v with τu,v < cu,v < du,v < tu,v. Moreover, according to Lemma 3.1
and the above properties of τu,v and tu,v, we can conclude (ii) and (iii). □

Lemma 3.4. Let a, b > 0 be given. Then, for any 0 < β < β0, we have

−∞ < m(a, b) = inf
P(a,b)

E(u, v) < 0.

Proof. By Lemma 3.3, we obtain that P+(a, b) ⊂ S(a, b) ∩A(2h0) and

inf
P(a,b)

E(u, v) = inf
P+(a,b)

E(u, v) < 0.

Obviously, infP(a,b) E(u, v) ≥ m(a, b). Also, for any (u, v) ∈ S(a, b) ∩A(2h0), there exists

inf
P(a,b)

E(u, v) ≤ E(τu,v ⋆ u, τu,v ⋆ v) ≤ E(u, v).

Hence, we have m(a, b) = infP(a,b) E(u, v) < 0. □

Proof of Theorem 1.3. Let {(ûn, v̂n)} ⊂ S(a, b) be a minimizing sequence for E|S(a,b)∩A(2h0). By
the symmetric decreasing rearrangement, we may assume that (ûn, v̂n) are both radial. After
passing to (|ûn|, |v̂n|) we may also assume that (ûn, v̂n) are nonnegative. By Lemma 3.3, we have(

∥(−∆)s/2(τu,v ⋆ ûn)∥22 + ∥(−∆)s/2(τu,v ⋆ v̂n)∥22
)1/2

≤ h0,

and the sequence {τun,vn ⋆ ûn, τun,vn
⋆ v̂n} is still a minimizing sequence for E|S(a,b)∩A(2h0). By

Ekeland’s variational principle, there exists a radially symmetric Palais-Smale sequence {(un, vn)}
satisfying

∥un − τun,vn ⋆ ûn∥H + ∥vn − τun,vn ⋆ v̂n∥H → 0, as n → ∞;

E(un, vn) → m(a, b), as n → ∞;

P (un, vn) → 0, as n → ∞;

E′|S(a,b)(un, vn) → 0, as n → ∞.

(3.11)

Then, by the last property in (3.11) and the Lagrange multipliers rule, there exist two sequences
{λ1,n} ⊂ R and {λ2,n} ⊂ R such that∫

RN

(−∆)s/2un(−∆)s/2φdx+

∫
RN

(−∆)s/2vn(−∆)s/2ϕdx+ λ1,n

∫
RN

unφdx

+ λ2,n

∫
RN

vnϕdx−
∫
RN

(
Iα ∗ |un|p

)
|un|p−2unφdx−

∫
RN

(
Iα ∗ |vn|q

)
|vn|q−2vnϕdx

− β

2∗s
r1

∫
RN

|un|r1−2|vn|r2unφdx− β

2∗s
r2

∫
RN

|un|r1 |vn|r2−2vnϕ dx

= on(1)(∥φ∥+ |ϕ∥),

(3.12)



EJDE-2025/49 FRACTIONAL SCHRÖDINGER-CHOQUARD SYSTEMS 11

for any (φ, ϕ) ∈ H. Taking (un, 0) and (0, vn) as test functions, we obtain that

−λ1,na
2 = ∥(−∆)s/2un∥22 −

∫
RN

(
Iα ∗ |un|p

)
|un|p dx,

−λ2,nb
2 = ∥(−∆)s/2vn∥22 −

∫
RN

(
Iα ∗ |vn|q

)
|vn|q dx.

(3.13)

In view of (3.11), we know {(un, vn)} ∈ S(a, b) ∩ A(2h0), which implies {(un, vn)} is bounded.
Then by (3.13) and Lemma 2.2, it follows that {λ1,n} and {λ2,n} are also bounded. Therefore,

there exists λ̂1, λ̂2 ∈ R, and u0, v0 ∈ Hs(RN ) with u0 ≥ 0, v0 ≥ 0 such that

(un, vn) ⇀ (u0, v0) in Hs(RN )×Hs(RN );

(un, vn) → (u0, v0) in Lp(RN )× Lq(RN ) for 2 < p, q < 2∗s;

(un, vn) → (u0, v0) a.e. in RN ;

(λ1, λ2) → (λ̂1, λ̂2) in R2.

(3.14)

Hence, by (3.12), (3.14) and Lemma 2.4, (u0, v0) is a weak solution of problem (1.1)-(1.2). Then,
by Lemma 2.9, we have P (u0, v0) = 0.

Let (un, vn) = (un−u0, vn− v0). According to (3.14), Lemma 2.2, Lemma 2.5 and Lemma 2.6,
there exist ∫

RN

(
Iα ∗ |un|p

)
|un|p dx =

∫
RN

(
Iα ∗ |u0|p

)
|u0|p dx+ on(1), (3.15)∫

RN

(
Iα ∗ |vn|q

)
|vn|q dx =

∫
RN

(
Iα ∗ |v0|q

)
|v0|q dx+ on(1), (3.16)∫

RN

|un|r1 |vn|r2 dx =

∫
RN

|u0|r1 |v0|r2 dx+

∫
RN

|un|r1 |vn|r2 dx+ on(1). (3.17)

By (3.1), (3.15) - (3.17), combined with the fact that P (un, vn)−P (u0, v0) = on(1), we have that

∥(−∆)s/2un∥22 + ∥(−∆)s/2vn∥22 = β

∫
RN

|un|r1 |vn|r2 dx+ on(1)

≤ βS−2∗s/2
(
∥(−∆)s/2un∥22 + ∥(−∆)s/2vn∥22

)2∗s/2

+ on(1).

(3.18)

Assuming that ∥(−∆)s/2un∥22 + ∥(−∆)s/2vn∥22 → l ≥ 0, then we obtain

l = 0 or l ≥
( 1

β

)N−2s
2s

S
N
2s .

If l ≥
(
1
β

)N−2s
2s S

N
2s , by (3.11) and (3.15)-(3.18), we have

m(a, b) = lim
n→∞

E(un, vn)

= E(u0, v0) + lim
n→∞

E(un, vn)

≥ m(∥u0∥22, ∥v0∥22) + lim
n→∞

[1
2

(
∥(−∆)s/2un∥22 + ∥(−∆)s/2vn∥22

)
− β

2∗s

∫
RN

|un|r1 |vn|r2 dx
]

≥ m(∥u0∥22, ∥v0∥22) +
(1
2
− 1

2∗s

)
lim

n→∞

(
∥(−∆)s/2un∥22 + ∥(−∆)s/2vn∥22

)
= m(∥u0∥22, ∥v0∥22) +

s

N

( 1

β

)N−2s
2s

S
N
2s ,

which contradicts with Lemma 3.2(ii) by the fact that ∥u0∥22 ≤ a2, ∥v0∥22 ≤ b2. As a consequence,
we obtain ∥(−∆)s/2un∥22 + ∥(−∆)s/2vn∥22 → 0 as n → ∞.

Finally, we will prove (un, vn) → (u0, v0) in H. Choosing (un, vn) as the test function in (3.12),
there exists

E′(un, vn)(un, vn) = −λ1,na
2 − λ2,nb

2 + on(1).



12 Z. CHEN, Y. YANG EJDE-2025/49

Combined with limn→∞ E(un, vn) = m(a, b), we have

− λ1,na
2 − λ2,nb

2 + on(1)− 2m(a, b)

= (
1

p
− 1)

∫
RN

(
Iα ∗ |un|p

)
|un|p dx+ (

1

q
− 1)

∫
RN

(
Iα ∗ |vn|q

)
|vn|q dx− 2sβ

N

∫
RN

|un|r1 |vn|r2 dx,

which implies λ̂1a
2 + λ̂2b

2 > 0 combining the fact of p, q > 1 and m(a, b) < 0. It follows that at

least one of λ̂1 and λ̂2 is positive.

Case 1: λ̂1 > 0 and λ̂2 > 0. Using (un, 0) and (u0, 0) as the test functions in (3.12), it follows
that

∥(−∆)s/2un∥22 + λ̂1∥un∥22 =

∫
RN

(
Iα ∗ |un|p

)
|un|p dx+ on(1),

∥(−∆)s/2u0∥22 + λ̂1∥u0∥22 =

∫
RN

(
Iα ∗ |u0|p

)
|u0|p dx,

which implies

−λ̂1

(
∥un∥22 − ∥u0∥22

)
= ∥(−∆)s/2un∥22 − ∥(−∆)s/2u0∥22 + on(1)

+
[ ∫

RN

(
Iα ∗ |un|p

)
|un|p dx−

∫
RN

(
Iα ∗ |u0|p

)
|u0|p dx

]
.

Then, by (3.15) and the fact that ∥(−∆)s/2un∥22 → 0 as n → ∞, we obtain ∥un∥22 − ∥u0∥22 → 0 as

n → ∞. Consequently, we can conclude that un → u0 in Hs(RN ) as λ̂1 > 0. Similarly, we have

that vn → v0 in Hs(RN ) as λ̂2 > 0.

Case 2: λ̂1 > 0 and λ̂2 ≤ 0. Since (u0, v0) is a weak solution of (1.1) with u0 ≥ 0 and v0 ≥ 0,
then

(−∆)sv0 + λ̂2v0 = (Iα ∗ vq0)v
q−1
0 +

β

2∗s
r2u

r1
0 vr2−1

0 in RN ,

which yields that v0 ≡ 0 by [33, Proposition 2.17] and the maximum principle. It follows that u0

is a solution of the problem

(−∆)su+ λ1u = (Iα ∗ |u|p)|u|p−2u, u ∈ Hs(RN ),∫
RN

|u|2 dx = a2.

By Lemma 2.7, Jp(u0) ≥ m1. Then we have

m(a, b) = lim
n→∞

E(un, vn) = Jp(u0) + lim
n→∞

Jq(vn)−
β

2∗s
lim

n→∞

∫
RN

|un|r1 |vn|r2 dx ≥ m1,

which contradicts Lemma 3.2(i). Therefore, the Case 2 is impossible.

Case 3: λ̂1 ≤ 0 and λ̂2 > 0. It is similar to the arguments in Case 2 and It is also impossible.
Thus, we have (un, vn) → (u0, v0) in H. In addition, by Lemma 3.4, one has

E(u0, v0) = inf
(u,v)∈P(a,b)

E(u, v) = m(a, b) < 0,

which implies (u0, v0) is a ground state. By the maximum principle, we conclude that (u0, v0)
is a positive solution. To sum up, (u0, v0) is a positive normalized ground state solution for the

problem (1.1)-(1.2) with the corresponding Lagrange multipliers λ̂1 > 0 and λ̂2 > 0. □

4. The case N+2s+α
N < p, q < 2∗α,s

In this section, we first denote Hs
r (RN ) as the subspace of functions in Hs(RN ) which are

radially symmetric with respect to 0. Subsequently, we will work in Hr := Hs
r (RN ) × Hs

r (RN ).
In addition, we define Sr(a, b) := Sr(a) × Sr(b), where Sr(a) := S(a) ∩Hs

r (RN ), Sr(b) := S(b) ∩
Hs

r (RN ). Pr(a, b) := P(a, b) ∩Hr.

Lemma 4.1. Let (u, v) ∈ Sr(a, b). Then we have the following conclusions:

(i) ∥(−∆)s/2(t ⋆ u)∥22 + ∥(−∆)s/2(t ⋆ v)∥22 → 0 and E(t ⋆ u, t ⋆ v) → 0 as t → −∞;



EJDE-2025/49 FRACTIONAL SCHRÖDINGER-CHOQUARD SYSTEMS 13

(ii) ∥(−∆)s/2(t ⋆ u)∥22 + ∥(−∆)s/2(t ⋆ v)∥22 → +∞ and E(t ⋆ u, t ⋆ v) → −∞ as t → +∞;
(iii) P (t ⋆ u, t ⋆ v) → 0 as t → −∞, P (t ⋆ u, t ⋆ v) → −∞ as t → +∞.

Proof. By straightforward calculations, it follows that

∥(−∆)s/2(t ⋆ u)∥22 + ∥(−∆)s/2(t ⋆ v)∥22 = e2st
(
∥(−∆)s/2u∥22 + ∥(−∆)s/2v∥22

)
,

E(t ⋆ u, t ⋆ v) = Φu,v(t)

= e2st
[1
2

(
∥(−∆)s/2u∥22 + ∥(−∆)s/2v∥22

)
− e2st(pδp,s−1)

2p

∫
RN

(
Iα ∗ |u|p

)
|u|p dx

− e2st(qδq,s−1)

2q

∫
RN

(
Iα ∗ |v|q

)
|v|q dx− βest(2

∗
s−2)

2∗s

∫
RN

|u|r1 |v|r2 dx
]
,

P (t ⋆ u, t ⋆ v) = e2st
[
∥(−∆)s/2u∥22 + ∥(−∆)s/2v∥22 − pδp,se

2st(pδp,s−1)

∫
RN

(
Iα ∗ |u|p

)
|u|p dx

− qδq,se
2st(qδq,s−1)

∫
RN

(
Iα ∗ |v|q

)
|v|q dx− βest(2

∗
s−2)

∫
RN

|u|r1 |v|r2 dx
]
.

Since pδp,s − 1 > 0, qδq,s − 1 > 0 and 2∗s − 2 > 0, we have

∥(−∆)s/2(t ⋆ u)∥22 + ∥(−∆)s/2(t ⋆ v)∥22 → 0, E(t ⋆ u, t ⋆ v) → 0 as t → −∞,

∥(−∆)s/2(t ⋆ u)∥22 + ∥(−∆)s/2(t ⋆ v)∥22 → +∞, E(t ⋆ u, t ⋆ v) → −∞ as t → +∞,

P (t ⋆ u, t ⋆ v) → 0 as t → −∞, P (t ⋆ u, t ⋆ v) → −∞ as t → +∞.

□

Lemma 4.2. There exists Ka,b > 0 sufficiently small such that

0 < sup
(u,v)∈D1

E(u, v) < inf
(u,v)∈D2

E(u, v)

and P (u, v) > 0 for all (u, v) ∈ D1, where

D1 :=
{
(u, v) ∈ Sr(a, b) : ∥(−∆)s/2(t ⋆ u)∥22 + ∥(−∆)s/2(t ⋆ v)∥22 ≤ Ka,b

}
,

D2 :=
{
(u, v) ∈ Sr(a, b) : ∥(−∆)s/2(t ⋆ u)∥22 + ∥(−∆)s/2(t ⋆ v)∥22 = 2Ka,b

}
.

Proof. For any (u, v) ∈ Sr(a, b), by Lemma 2.3, we obtain that

1

2p

∫
RN

(
Iα ∗ |u|p

)
|u|p dx ≤ 1

2p
C̃∥(−∆)s/2u∥2pδp,s2 a2p(1−δp,s)

≤ C1

(
∥(−∆)s/2u∥22 + ∥(−∆)s/2v∥22

)pδp,s
,

(4.1)

1

2q

∫
RN

(
Iα ∗ |v|q

)
|v|q dx ≤ 1

2q
C̃∥(−∆)s/2v∥2qδq,s2 b2q(1−δq,s)

≤ C2

(
∥(−∆)s/2u∥22 + ∥(−∆)s/2v∥22

)qδq,s
.

(4.2)

In view of (3.1), we have

β

2∗s

∫
RN

|u|r1 |v|r2 dx ≤ C3

(
∥(−∆)s/2u∥22 + ∥(−∆)s/2v∥22

)2∗s/2

. (4.3)

Let K > 0 be arbitrary, and set (u1, v1), (u2, v2) ∈ Sr(a, b) satisfying l1 ≤ K, l2 = 2K, where
l1 := ∥(−∆)s/2u1∥22 + ∥(−∆)s/2v1∥22, l2 := ∥(−∆)s/2u2∥22 + ∥(−∆)s/2v2∥22. Then, for K > 0 small
enough and for any (u1, v1) ∈ Sr(a, b) satisfying l1 ≤ K, it follows from (4.1) - (4.3) that

E(u1, v1) ≥
1

2
l1 − C1l

pδp,s
1 − C2l

qδq,s
1 − C3l

2∗s/2
1 ≥ 1

8
l1 > 0,

where we used that pδp,s > 1, qδq,s > 1 and 2∗s > 2. Thus we have proved E(u, v) > 0 for
(u, v) ∈ D1, namely, sup(u,v)∈D1

E(u, v) > 0. Similarly, we can obtain

P (u1, v1) ≥ l1 − Cl
pδp,s
1 − Cl

qδq,s
1 − Cl

2∗s/2
1 > 0.



14 Z. CHEN, Y. YANG EJDE-2025/49

So, we have P (u, v) > 0 for any (u, v) ∈ D1.
On the other hand, for K = K(a, b) > 0 small enough, we have

E(u2, v2)− E(u1, v1) ≥
1

2
(l2 − l1)− C1l

pδp,s
2 − C2l

qδq,s
2 − C3l

2∗s/2
2

≥ 1

2
K − C1(2K)pδp,s − C2(2K)qδq,s − C3(2K)2

∗
s/2

≥ 1

8
K > 0,

for any (u1, v1) ∈ D1 and (u2, v2) ∈ D2. Then, we have E(u2, v2) ≥ 1
8K+sup(u,v)∈D1

E(u, v). Ac-

cording to the definition of the infimum, we obtain inf(u,v)∈D2
E(u, v) ≥ 1

8K+sup(u,v)∈D1
E(u, v).

Hence, for K = K(a, b) > 0 sufficiently small, we obtain inf(u,v)∈D2
E(u, v) > sup(u,v)∈D1

E(u, v).

To sum up, there exists K = K(a, b) > 0 sufficiently small such that

0 < sup
(u,v)∈D1

E(u, v) < inf
(u,v)∈D2

E(u, v). □

We have obtained the geometry of mountain pass, so we give the minimax picture: In view of
Lemma 2.8, we can define û is a ground state solution of problem (2.2) with respect to parameters
a, p, and v̂ is a ground state solution of problem (2.2) with respect to parameters b, q. Then, we
fix (û, v̂) ∈ Sr(a, b).

According to Lemmas 4.1 and 4.2, there exist two numbers t1 ≤ −1 < 0 < 1 ≤ t2 such that

∥(−∆)s/2(t1 ⋆ û)∥22 + ∥(−∆)s/2(t1 ⋆ v̂)∥22 <
Ka,b

2
, E(t1 ⋆ û, t1 ⋆ v̂) > 0;

∥(−∆)s/2(t2 ⋆ û)∥22 + ∥(−∆)s/2(t2 ⋆ v̂)∥22 > 2Ka,b, E(t2 ⋆ û, t2 ⋆ v̂) ≤ 0.

Moreover, for t1 small enough and t2 large enough, we also have

P (t1 ⋆ û, t1 ⋆ v̂) > 0 and P (t2 ⋆ û, t2 ⋆ v̂) < 0. (4.4)

Then, we can define the path

Γ :=
{
γ ∈ C([0, 1], Sr(a, b)) : γ(0) = (t1 ⋆ û, t1 ⋆ v̂), γ(1) = (t2 ⋆ û, t2 ⋆ v̂)

}
.

Let γ0(σ) :=
(
[(1 − σ)t1 + σt2] ⋆ û, [(1 − σ)t1 + σt2] ⋆ v̂

)
. Then we have γ0(0) = (t1 ⋆ û, t1 ⋆ v̂),

γ0(1) = (t2 ⋆ û, t2 ⋆ v̂), and it is clear that γ0(σ) is continuous. Then, we have γ0(σ) ∈ Sr(a, b) and
γ0(σ) ∈ Γ. Thus, Γ is not empty.

Based on the above discussion, now we define a minimax level

cβ(a, b) := inf
γ∈Γ

max
σ∈[0,1]

E(γ(σ)).

Obviously, cβ(a, b) > 0. Furthermore, according to the strategy introduced in [23] and the L2-

invariant scaling (t ⋆ u, t ⋆ u) = (e
N
2 tu(etx), e

N
2 tv(etx)), it is standard to check that E(u, v) and

E(t ⋆ u, t ⋆ v) = Φu,v(t) have the same mountain pass geometry structure and mountain pass level.
Consequently, similar to the argument in [23, Proposition 2.2 and Lemma 2.4], one can work

on a Hilbert space, corresponding an inner product structure, we can derive from Lemmas 4.1 and
4.2 to obtain the following result.

Lemma 4.3. There exists a nonnegative sequence {(un, vn)} ⊂ Sr(a, b) such that

E(un, vn) → cβ(a, b), E′|Sr(a,b)(un, vn) → 0, P (un, vn) → 0, as n → ∞.

Lemma 4.4. For each a, b > 0 fixed, it holds that

lim
β→+∞

cβ(a, b) = 0.

Proof. Fix (u, v) ∈ Sr(a, b) and let it follow the path

γ(σ) :=
(
[(1− σ)t1 + σt2] ⋆ u, [(1− σ)t1 + σt2] ⋆ v

)
∈ Γ.

Then, through direct calculation, we obtain that

cβ(a, b) ≤ max
σ∈[0,1]

E(γ(σ))



EJDE-2025/49 FRACTIONAL SCHRÖDINGER-CHOQUARD SYSTEMS 15

≤ max
k≥0

{1

2
k2

(
∥(−∆)s/2u∥22 + ∥(−∆)s/2v∥22

)
− β

2∗s
k2

∗
s

∫
RN

|u|r1 |v|r2 dx
}
,

where k = e[(1−σ)t1+σt2]s. Set d1 = ∥(−∆)s/2u∥22 + ∥(−∆)s/2v∥22, d2 =
∫
RN |u|r1 |v|r2 dx, then we

consider the function

g(k) =
1

2
d1k

2 − β

2∗s
d2k

2∗s , for all k ≥ 0.

Letting g′(k) = d1k − βd2k
2∗s−1 = 0, we can obtain the maximum point of g(k), that is kmax =

( d1

βd2
)

N−2s
4s . Then, we have

max
k≥0

{1

2
k2

(
∥(−∆)s/2u∥22 + ∥(−∆)s/2v∥22

)
− β

2∗s
k2

∗
s

∫
RN

|u|r1 |v|r2 dx
}

=
1

2
d1

( d1
βd2

) N
2s−1

− β

2∗s
d2

( d1
βd2

) N
2s ≤ 1

2
d1

( d1
βd2

) N
2s−1

.

Therefore, there exists d > 0 which do not depend on β such that

cβ(a, b) ≤ d
( 1

β

) N
2s−1

→ 0, as β → +∞,

with the fact of N > 2s. □

By recalling Lemma 2.8, we can define

cβ(a, 0) := Jp(ûa,p) > 0 and cβ(0, b) := Jq(v̂b,q) > 0.

Then combining this with Lemma 4.4, we have the following corollary.

Corollary 4.5. There exists β∗ > 0 such that for any β > β∗,

cβ(a, b) < min{cβ(a, 0), cβ(0, b)}.

Lemma 4.6. For every (u, v) ∈ Sr(a, b), Φu,v(t) has exactly one critical point t̃u,v. Also

(i) Pr(a, b) = P−
r (a, b);

(ii) (t ⋆ u, t ⋆ v) ∈ Pr(a, b) if and only if t = t̃u,v;
(iii) Φu,v(t̃u,v) = maxt∈R Φu,v(t).

Proof. Let (u, v) ∈ Sr(a, b), by (2.3) we have

Φ′
u,v(t) = se2st

[
η1 − δp,sη2e

2st(pδp,s−1) − δq,sη3e
2st(qδq,s−1) − βη4e

st(2∗s−2)
]
,

where η1 = ∥(−∆)s/2u∥22 + ∥(−∆)s/2v∥22, η2 =
∫
RN

(
Iα ∗ |u|p

)
|u|p dx, η3 =

∫
RN

(
Iα ∗ |v|q

)
|v|q dx

and η4 =
∫
RN |u|r1 |v|r2 dx. Let

W (t) = δp,sη2e
2st(pδp,s−1) + δq,sη3e

2st(qδq,s−1) + βη4e
st(2∗s−2).

In view of pδp,s, qδq,s > 1 and 2∗s > 2, we have W (t) is increasing on R, W (−∞) = 0+ and
W (+∞) = +∞. Hence, Φ′

u,v(t) has a unique zero point t̃u,v. By Lemma 2.10, we can deduce

that (t̃u,v ⋆ u, t̃u,v ⋆ v) ∈ P(a, b) if and only if t = t̃u,v. Moreover, Φ′
u,v(t) > 0 on (−∞, t̃u,v) and

Φ′
u,v(t) < 0 on (t̃u,v,+∞), which implies that Φ(t̃u,v) = maxt∈R Φu,v(t). On the other hand, for

any (u, v) ∈ P(a, b) \ P−(a, b), it holds

η1 = δp,sη2 + δq,sη3 + βη4,

2η1 ≥ 2pδp,s · δp,sη2 + 2qδq,s · δq,sη3 + 2∗s · βη4.
This implies

0 ≥ 2(pδp,s − 1)δp,sη2 + 2(qδq,s − 1)δq,sη3 + (2∗s − 2)βη4,

which is impossible because pδp,s > 1, qδq,s > 1 and 2∗s > 2. Hence, Pr(a, b) = P−
r (a, b). □

Lemma 4.7. Let a, b > 0 be given. Then, the following statements are true.

(i) cβ(a, b) = infPr(a,b) E(u, v);
(ii) cβ(a, b) ≤ cβ(a1, b1), for any 0 < a1 ≤ a, 0 < b1 ≤ b.



16 Z. CHEN, Y. YANG EJDE-2025/49

Proof. (i) Indeed, for any (u, v) ∈ Pr(a, b), By Lemmas 4.1 and 4.2, there exist two numbers
t3 ≤ −1 < 0 < 1 ≤ t4 such that

∥(−∆)s/2(t3 ⋆ u)∥22 + ∥(−∆)s/2(t3 ⋆ v)∥22 <
Ka,b

2
, E(t3 ⋆ u, t3 ⋆ v) > 0;

∥(−∆)s/2(t4 ⋆ u)∥22 + ∥(−∆)s/2(t4 ⋆ v)∥22 > 2Ka,b, E(t4 ⋆ u, t4 ⋆ v) ≤ 0;

P(t3 ⋆ u, t3 ⋆ v) > 0, P(t4 ⋆ u, t4 ⋆ v) < 0. Set

γ̄(σ) =
(
[(1− σ)t3 + σt4] ⋆ u, [(1− σ)t3 + σt4] ⋆ v

)
, σ ∈ [0, 1],

then γ̄(t) ∈ Γ. From Lemma 4.6 we have

cβ(a, b) ≤ max
σ∈[0,1]

E(γ(σ)) ≤ max
t∈R

Φu,v(t) = E(u, v).

Hence, we have cβ(a, b) ≤ infPr(a,b) E(u, v). On the other hand, for any γ(σ) ∈ Γ, by (4.4), we
know that P (γ(0)) > 0, P (γ(1)) < 0. Consequently, we can deduce that there exists σ̃ ∈ (0, 1)
such that P (γ(σ̃)) = 0, which implies γ(σ̃) ∈ Pr(a, b). Then

max
σ∈[0,1]

E(γ(σ)) ≥ E(γ(σ̃)) ≥ inf
Pr(a,b)

E(u, v),

which implies cβ(a, b) ≥ infPr(a,b) E(u, v). Therefore, cβ(a, b) = infPr(a,b) E(u, v).
(ii) The proof is similar to Lemma 3.2(ii) and [5, Lemma 3.2]; we omit it here. □

Lemma 4.8. If N ≥ 4s, 0 < α < N , N+2s+α
N < p, q < 2∗α,s or 2s < N < 4s, 0 < α < N ,

max{N+2s+α
N , 2∗α,s − 2s

4s−N } < p, q < 2∗α,s, then there exists a constant β∗ > 0 such that for every
β > β∗, it holds that

0 < cβ(a, b) <
s

N
β−N−2s

2s S
N
2s
r1,r2 .

Proof. By [6, Lemma 14.7], we know S in (2.1) is attained by

Uϵ(x) := C(N, s)
( ϵ

ϵ2 + |x|2
)N−2s

2

, ∀ϵ > 0,

where C(N, s) is some positive constant. Let ηϵ(x) = ς(x)Uϵ, where ς(x) ∈ C∞
0 (RN ) is a radial

cut-off function such that 0 ≤ ς ≤ 1, ς(x) = 1 when |x| ≤ 1, ς(x) = 0 when |x| ≥ 2. Then, by [37],
we have

∥(−∆)s/2ηϵ∥22 ≤ S
N
2s +O(ϵN−2s), ∥ηϵ∥

2∗s
2∗s

= S
N
2s +O(ϵN ). (4.5)

We take uϵ =
√
r1c

∥ηϵ∥2
ηϵ and vϵ =

√
r2c

∥ηϵ∥2
ηϵ for a small constant c > 0. Let tϵ := t̃uϵ,vϵ be given

by Lemma 4.6, then (tϵ ⋆ uϵ, tϵ ⋆ vϵ) ∈ Pr(
√
r1c,

√
r2c). Thus, for a proper c > 0 combined with

Lemma 4.7(ii) we have

cβ(a, b) ≤ cβ(
√
r1c,

√
r2c)

≤ E(tϵ ⋆ uϵ, tϵ ⋆ vϵ)

=
e2stϵ

2

(
∥(−∆)s/2uϵ∥22 + ∥(−∆)s/2vϵ∥22

)
− e2stϵ·pδp,s

2p

∫
RN

(
Iα ∗ |uϵ|p

)
|uϵ|p dx

− e2stϵ·qδq,s

2q

∫
RN

(
Iα ∗ |vϵ|q

)
|vϵ|q dx− β

2∗s
estϵ·2

∗
s

∫
RN

|uϵ|r1 |vϵ|r2 dx

=
r1 + r2

2
· e2stϵ

∥ηϵ∥22
c2 · ∥(−∆)s/2ηϵ∥22 −

rp1c
2pe2stϵpδp,s

2p∥ηϵ∥2p2

∫
RN

(
Iα ∗ |ηϵ|p

)
|ηϵ|p dx

− rq2c
2qe2stϵqδq,s

2q∥ηϵ∥2q2

∫
RN

(
Iα ∗ |ηϵ|q

)
|ηϵ|q dx− βr

r1
2
1 r

r2
2
2

2∗s
· e2

∗
sstϵ

∥ηϵ∥
2∗s
2

c2
∗
s · ∥ηϵ∥

2∗s
2∗s

≤ max
ρ>0

(2∗s
2
∥(−∆)s/2ηϵ∥22ρ2 −

βr
r1
2
1 r

r2
2
2

2∗s
∥ηϵ∥

2∗s
2∗s
ρ2

∗
s

)
− Ce2stϵpδp,s

∥ηϵ∥2p2

∫
RN

(
Iα ∗ |ηϵ|p

)
|ηϵ|p dx− Ce2stϵqδq,s

∥ηϵ∥2q2

∫
RN

(
Iα ∗ |ηϵ|q

)
|ηϵ|q dx,

(4.6)



EJDE-2025/49 FRACTIONAL SCHRÖDINGER-CHOQUARD SYSTEMS 17

where ρ = estϵ

∥ηϵ∥2
c. Then, for the function

f(ρ) =
2∗s
2
∥(−∆)s/2ηϵ∥22ρ2 −

βr
r1
2
1 r

r2
2
2

2∗s
∥ηϵ∥

2∗s
2∗s
ρ2

∗
s ,

by a direct calculation, we have that f(ρ) has a unique critical point

ρ0 =
(2∗s∥(−∆)s/2ηϵ∥22
βr

r1
2
1 r

r2
2
2 ∥ηϵ∥

2∗s
2∗s

)N−2s
4s

,

which is also a global maximum point. Thereby, we can define f(ρ)max := f(ρ0), which is the
maximum of f(ρ). Then, by (4.5) and Lemma 2.1, we have

f(ρ)max =
s

N

[ 2∗s∥(−∆)s/2ηϵ∥22(
βr

r1
2
1 r

r2
2
2 ∥ηϵ∥

2∗s
2∗s

)2/2∗s ] N
2s

=
s

N
β−N−2s

2s

((r1
r2

)r2/2
∗
s

+
(r2
r1

) r1
2∗s
)(

S +O(εN−2s)
) N

2s

=
s

N
β−N−2s

2s S
N
2s
r1,r2 +O(ϵN−2s).

(4.7)

On the other hand, in view of P (tϵ ⋆ uϵ, tϵ ⋆ vϵ) = 0, we have

e(2
∗
s−2)stϵβ

∫
RN

|uϵ|r1 |vϵ|r2 dx ≤ ∥(−∆)s/2uϵ∥22 + ∥(−∆)s/2vϵ∥22,

namely

estϵ ≤
(∥(−∆)s/2uϵ∥22 + ∥(−∆)s/2vϵ∥22

β
∫
RN |uϵ|r1 |vϵ|r2 dx

) 1
2∗s−2

=
(
C1

∥(−∆)s/2ηϵ∥22∥ηϵ∥
2∗s−2
2

β∥ηϵ∥
2∗s
2∗s

) 1
2∗s−2

. (4.8)

Combining this with P (tϵ ⋆ uϵ, tϵ ⋆ vϵ) = 0 and (4.8), it follows that

e(2
∗
s−2)stϵ =

∥(−∆)s/2uϵ∥22 + ∥(−∆)s/2vϵ∥22
β
∫
RN |uϵ|r1 |vϵ|r2 dx

−
δp,s

∫
RN

(
Iα ∗ |uϵ|p

)
|uϵ|p dx

β
∫
RN |uϵ|r1 |vϵ|r2 dx

e2stϵ(pδp,s−1)

−
δq,s

∫
RN

(
Iα ∗ |vϵ|q

)
|vϵ|q dx

β
∫
RN |uϵ|r1 |vϵ|r2 dx

e2stϵ(qδq,s−1)

≥ ∥(−∆)s/2uϵ∥22 + ∥(−∆)s/2vϵ∥22
β
∫
RN |uϵ|r1 |vϵ|r2 dx

−
δp,s

∫
RN

(
Iα ∗ |uϵ|p

)
|uϵ|p dx

β
∫
RN |uϵ|r1 |vϵ|r2 dx

(
C1

∥(−∆)s/2ηϵ∥22∥ηϵ∥
2∗s−2
2

β∥ηϵ∥
2∗s
2∗s

) 2(pδp,s−1)

2∗s−2

−
δq,s

∫
RN

(
Iα ∗ |vϵ|q

)
|vϵ|q dx

β
∫
RN |uϵ|r1 |vϵ|r2 dx

(
C1

∥(−∆)s/2ηϵ∥22∥ηϵ∥
2∗s−2
2

β∥ηϵ∥
2∗s
2∗s

) 2(qδq,s−1)

2∗s−2

=
∥(−∆)s/2ηϵ∥22∥ηϵ∥

2∗s−2
2

∥ηϵ∥
2∗s
2∗s

×
[C1

β
−

C2∥(−∆)s/2ηϵ∥
4(pδp,s−1)

2∗s−2
−2

2

∫
RN

(
Iα ∗ |ηϵ|p

)
|ηϵ|p dx

β
2(pδp,s−1)

2∗s−2
+1∥ηϵ∥

22∗s (pδp,s−1)

2∗s−2

2∗s
∥ηϵ∥

2p(1−δp,s)
2

−
C3∥(−∆)s/2ηϵ∥

4(qδq,s−1)

2∗s−2
−2

2

∫
RN

(
Iα ∗ |ηϵ|q

)
|ηϵ|q dx

β
2(qδq,s−1)

2∗s−2
+1∥ηϵ∥

22∗s (qδq,s−1)

2∗s−2

2∗s
∥ηϵ∥

2q(1−δq,s)
2

]
.



18 Z. CHEN, Y. YANG EJDE-2025/49

According to (4.5) and Lemma 2.3, we infer that ∥(−∆)s/2ηϵ∥22 ∼ 1, ∥ηϵ∥
2∗s
2∗s

∼ 1, and∫
RN

(
Iα ∗ |ηϵ|p

)
|ηϵ|p dx

∥ηϵ∥
2p(1−δp,s)
2

≤ C̃∥(−∆)s/2ηϵ∥
2pγp,s

2 ≤ C4,∫
RN

(
Iα ∗ |ηϵ|q

)
|ηϵ|q dx

∥ηϵ∥
2q(1−δq,s)
2

≤ C̃∥(−∆)s/2ηϵ∥
2qγq,s

2 ≤ C5.

Considering pδp,s > 1, qδq,s > 1, 2∗s > 2, we can take β∗ > 0 such that C1β
−1
∗ −C6β

− 2(pδp,s−1)

2∗s−2
−1

∗ −

C7β
− 2(qδq,s−1)

2∗s−2
−1

∗ > 0. As a consequence, for each β > β∗, we have that estϵ ≥ C∥ηϵ∥2.
Substituting estϵ ≥ C∥ηϵ∥2 into (4.6), and using (4.7), we obtain

cβ(a, b) ≤
s

N
β−N−2s

2s S
N
2s
r1,r2 +O(ϵN−2s)− C

∫
RN

(
Iα ∗ |ηϵ|p

)
|ηϵ|p dx

∥ηϵ∥
2p(1−δp,s)
2

− C

∫
RN

(
Iα ∗ |ηϵ|q

)
|ηϵ|q dx

∥ηϵ∥
2q(1−δq,s)
2

.

From [27, Lemma 4.10], we know that∫
RN

(
Iα ∗ |ηϵ|p

)
|ηϵ|p dx

∥ηϵ∥
2p(1−δp,s)
2

≥


O
(
ϵN+α+p(2sδp,s−N)

)
, if N > 4s;

O
(
| log ϵ|−p(1−δp,s), if N = 4s;

O
(
ϵN+α−p(N−2s)(2−δp,s)

)
, if 2s < N < 4s.

If N > 4s, it is easy to check that N + α + p(2sδp,s − N) < N − 2s; If N = 4s, we have

limϵ→0 ϵ
N−2s| log ϵ|p(1−δp,s) = 0; If 2s < N < 4s, let N + α − p(N − 2s)(2 − δp,s) < N − 2s, we

have

p >
N + α

N − 2s
− 2s

4s−N
.

We analyze q in the same way. In conclusion, if N ≥ 4s, 0 < α < N or 2s < N < 4s, 0 < α < N ,
max{N+2s+α

N , 2∗α,s − 2s
4s−N } < p, q < 2∗α,s, we conclude that

0 < cβ(a, b) <
s

N
β−N−2s

2s S
N
2s
r1,r2 .

□

Lemma 4.9. Let N ≥ 4s, 0 < α < N , N+2s+α
N < p, q < 2∗α,s or 2s < N < 4s, 0 < α < N ,

max{N+2s+α
N , 2∗α,s − 2s

4s−N } < p, q < 2∗α,s, and β > β∗. Moreover, assume that cβ(a, b) <

min{cβ(a, 0), cβ(0, b)}. If {(un, vn)} ⊂ Sr(a, b) is a nonnegative (PSP )cβ(a,b) sequence for E|Sr(a,b),
that is,

un ≥ 0, vn ≥ 0, E(un, vn) → cβ(a, b), E′|Sr(a,b)(un, vn) → 0, P (un, vn) → 0, as n → ∞.

Then, there exist ũ > 0, ṽ > 0 ∈ Hs
r (RN ), and λ̃1, λ̃2 > 0 such that up to a subsequence,

(un, vn) → (ũ, ṽ) in Hr and (λ1,n, λ2,n) → (λ̃1, λ̃2) in R2.

Proof. From P (un, vn) = on(1), we have

E(un, vn) =
1

2

(
∥(−∆)s/2un∥22 + ∥(−∆)s/2vn∥22

)
− 1

2p

∫
RN

(
Iα ∗ |un|p

)
|un|p dx

− 1

2q

∫
RN

(
Iα ∗ |vn|q

)
|vn|q dx− β

2∗s

∫
RN

|un|r1 |vn|r2 dx

=
1

2
(δp,s −

1

p
)

∫
RN

(
Iα ∗ |un|p

)
|un|p dx+

1

2
(δq,s −

1

q
)

∫
RN

(
Iα ∗ |vn|q

)
|vn|q dx

+ β(
1

2
− 1

2∗s
)

∫
RN

|un|r1 |vn|r2 dx+ on(1).

Considering E(un, vn) → cβ(a, b) as n → ∞, pδp,s > 1, qδq,s > 1 and 2∗s > 2, thus we conclude
that the sequences {

∫
RN

(
Iα ∗ |un|p

)
|un|p dx}, {

∫
RN

(
Iα ∗ |vn|q

)
|vn|q dx} and {

∫
RN |un|r1 |vn|r2 dx}



EJDE-2025/49 FRACTIONAL SCHRÖDINGER-CHOQUARD SYSTEMS 19

are bounded. Then, combining this with P (un, vn) = on(1), we infer that
{
∥(−∆)s/2un∥22

}
,{

∥(−∆)s/2vn∥22
}
are also bounded. Therefore, we have {(un, vn)} is bounded in Hr.

Moreover, by E′(un, vn) → 0 and the Lagrange multipliers rule, there exist sequences {λ1,n}
and {λ2,n} such that∫

RN

(−∆)s/2un(−∆)s/2φdx+

∫
RN

(−∆)s/2vn(−∆)s/2ϕ dx+ λ1,n

∫
RN

unφdx

+ λ2,n

∫
RN

vnϕ dx−
∫
RN

(
Iα ∗ |un|p

)
|un|p−2unφdx−

∫
RN

(
Iα ∗ |vn|q

)
|vn|q−2vnϕ dx

− β

2∗s
r1

∫
RN

|un|r1−2|vn|r2unφdx− β

2∗s
r2

∫
RN

|un|r1 |vn|r2−2vnϕdx

= on(1)(∥φ∥+ ∥ϕ∥),

(4.9)

for all (φ, ϕ) ∈ Hr. Taking (un, 0) and (0, vn) as test functions, we obtain that

−λ1,na
2 = ∥(−∆)s/2un∥22 −

∫
RN

(
Iα ∗ |un|p

)
|un|p dx,

−λ2,nb
2 = ∥(−∆)s/2vn∥22 −

∫
RN

(
Iα ∗ |vn|q

)
|vn|q dx.

In view of {(un, vn)} is bounded in Hr, we know that {λ1,n} and {λ2,n} are also bounded. As a

result, there exists λ̃1, λ̃2 ∈ R, and ũ, ṽ ∈ Hs
r (RN ) such that

(un, vn) ⇀ (ũ, ṽ) in Hs
r (RN )×Hs

r (RN );

(un, vn) → (ũ, ṽ) in Lp(RN )× Lq(RN ) for 2 < p, q < 2∗s;

(un, vn) → (ũ, ṽ) a.e. in RN ;

(λ1, λ2) → (λ̃1, λ̃2) in R2.

(4.10)

Then, according to (4.9), (4.10) and Lemma 2.4, we conclude (ũ, ṽ) is a weak solution of (1.1).
Then, by Lemma 2.9, we have P (ũ, ṽ) = 0.

Next, we claim that ũ ̸= 0, ṽ ̸= 0. Suppose by contradiction that ũ = 0. There are two cases.
If ṽ = 0, then from P (un, vn) = on(1) and Lemma 2.1, we obtain that

∥(−∆)s/2un∥22 + ∥(−∆)s/2vn∥22 = β

∫
RN

|un|r1 |vn|r2 dx+ on(1)

≤ βS
−2∗s/2
r1,r2

(
∥(−∆)s/2un∥22 + ∥(−∆)s/2vn∥22

)2∗s/2

+ on(1).

Assuming that ∥(−∆)s/2un∥22 + ∥(−∆)s/2vn∥22 → ν ≥ 0, we obtain ν = 0 or ν ≥ β−N−2s
2s S

N
2s
r1,r2 .

Then, we have

cβ(a, b) = lim
n→∞

E(un, vn) = 0

or

cβ(a, b) = lim
n→∞

E(un, vn) =
s

N
ν ≥ s

N
β−N−2s

2s S
N
2s
r1,r2 ,

and both of them are contradicted with Lemma 4.8. Now if ṽ ̸= 0, by the maximum principle we
have

(−∆)sṽ + λ̃2ṽ = (Iα ∗ |ṽ|p)|ṽ|p−2ṽ, in RN ,

ṽ > 0.

If λ̃2 ≤ 0, then by [33, Proposition 2.17] we obtain ṽ ≡ 0, which contradicts with ṽ > 0. Hence,

we obtain λ̃2 > 0. Let ûn = un − ũ, v̂n = vn − ṽ. Then by (4.10), Lemma 2.5 and Lemma 2.6 we
have

∥(−∆)s/2vn∥22 = ∥(−∆)s/2v̂n∥22 + ∥(−∆)s/2ṽ∥22 + on(1),



20 Z. CHEN, Y. YANG EJDE-2025/49∫
RN

(
Iα ∗ |vn|q

)
|vn|q dx =

∫
RN

(
Iα ∗ |ṽ|q

)
|ṽ|q dx+ on(1),∫

RN

|un|r1 |vn|r2 dx =

∫
RN

|un|r1 |v̂n|r2 dx+ on(1).

It follows from P (un, vn)− P (ũ, ṽ) = on(1) that

∥(−∆)s/2un∥22 + ∥(−∆)s/2v̂n∥22 = β

∫
RN

|un|r1 |v̂n|r2 dx+ on(1).

As before, it holds ∥(−∆)s/2un∥22 + ∥(−∆)s/2v̂n∥22 → 0 or

lim inf
n→∞

(
∥(−∆)s/2un∥22 + ∥(−∆)s/2v̂n∥22

)
≥ β−N−2s

2s S
N
2s
r1,r2 .

If ∥(−∆)s/2un∥22 + ∥(−∆)s/2v̂n∥22 → 0, then we have

cβ(a, b) = lim
n→∞

E(un, vn)

= lim
n→∞

s

N

(
∥(−∆)s/2un∥22 + ∥(−∆)s/2v̂n∥22

)
+ Jq(ṽ)

≥ cβ(0, b),

which is a contradiction. On the other hand, if lim infn→∞

(
∥(−∆)s/2un∥22 + ∥(−∆)s/2v̂n∥22

)
≥

β−N−2s
2s S

N
2s
r1,r2 , we obtain

cβ(a, b) = lim
n→∞

s

N

(
∥(−∆)s/2un∥22 + ∥(−∆)s/2v̂n∥22

)
+ Jq(ṽ)

≥ s

N
β−N−2s

2s S
N
2s
r1,r2 + cβ(0, b),

which is a contradiction. Hence, we can conclude that ũ ̸= 0. Similarly, ṽ ̸= 0.
Finally, we show the strong convergence. As before, we obtain that

∥(−∆)s/2ûn∥22 + ∥(−∆)s/2v̂n∥22 = β

∫
RN

|ûn|r1 |v̂n|r2 dx+ on(1).

Then there are two cases: ∥(−∆)s/2ûn∥22 + ∥(−∆)s/2v̂n∥22 → 0 or

lim inf
n→∞

(
∥(−∆)s/2ûn∥22 + ∥(−∆)s/2v̂n∥22

)
≥ β−N−2s

2s S
N
2s
r1,r2 .

If the second case occur, then

cβ(a, b) = lim
n→∞

s

N

(
∥(−∆)s/2ûn∥22 + ∥(−∆)s/2v̂n∥22

)
+ Jp(ũ) + Jq(ṽ)

≥ s

N
β−N−2s

2s S
N
2s
r1,r2 + cβ(∥ũ∥2, ∥ṽ∥2)

≥ s

N
β−N−2s

2s S
N
2s
r1,r2 + cβ(a, b),

where we used that ∥ũ∥2 ≤ a, ∥ṽ∥2 ≤ b and Lemma 4.7(ii). This is a contradiction. Consequently,
∥(−∆)s/2ûn∥22 + ∥(−∆)s/2v̂n∥22 → 0. In addition, by the maximum principle, (ũ, ṽ) is a positive

solution of (1.1). Then, if λ̃1 ≤ 0, λ̃2 ≤ 0, then by [33, Proposition 2.17] we obtain ũ ≡ 0, ṽ ≡ 0,

which contradicts ũ > 0, ṽ > 0. We obtain λ̃1 > 0 and λ̃2 > 0. Noting that

∥(−∆)s/2ûn∥22 + λ̃1∥ûn∥22 + ∥(−∆)s/2v̂n∥22 + λ̃2∥v̂n∥22
=

(
E′(un, vn) + λ1,nun + λ2,nvn

)
[(ûn, v̂n)]−

(
E′(ũ, ṽ) + λ̃1ũ+ λ̃2ṽ

)
[(ûn, v̂n)] + on(1)

= on(1),

we conclude that (un, vn) → (ũ, ṽ) in Hr. This completes the proof. □



EJDE-2025/49 FRACTIONAL SCHRÖDINGER-CHOQUARD SYSTEMS 21

Proof of Theorem 1.4. By Lemma 4.3, there exists a (PSP ) sequence {(un, vn)} ⊂ Sr(a, b) for
E|Sr(a,b) at level cβ(a, b) > 0. In view of Corollary 4.5, we know that there exists β∗ > 0 such
that as β > β∗, cβ(a, b) < min{cβ(a, 0), cβ(0, b)}. As a consequence, by Lemma 4.9, when N ≥ 4s,

0 < α < N , N+2s+α
N < p, q < 2∗α,s or 2s < N < 4s, 0 < α < N , max{N+2s+α

N , 2∗α,s − 2s
4s−N } <

p, q < 2∗α,s, and let β̃ = max{β∗, β∗}, if β > β̃, we can deduce that (ũ, ṽ) is a mountain pass
type normalized solution for the problem (1.1)-(1.2) with the corresponding Lagrange multipliers

λ̃1, λ̃2 > 0, which is positive and radially symmetric. □

Acknowledgement. The authors would like to express sincere thanks to the anonymous referees
for their valuable comments that helped to improve the manuscript. This research was supported
by the NSFC (No. 11501252, No. 11571176).

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Zilin Chen

School of Science, Jiangnan University, Wuxi, Jiangsu 214122, China
Email address: 6231204001@stu.jiangnan.edu.cn

Yang Yang (corresponding author)

School of Science, Jiangnan University, Wuxi, Jiangsu 214122, China
Email address: yangyangli@jiangnan.edu.cn


	1. Introduction and main results
	2. Preliminaries
	3. The case 2< p, q < N + 2s + N
	4. The case N + 2s + N < p, q < 2,s
	Acknowledgement

	References