Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 01, pp. 1–17.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS WITH

WEIGHTED HARDY POTENTIAL AND CRITICAL EXPONENT

YU SU, HAIBO CHEN, SENLI LIU, XIANWEN FANG

Communicated by Claudianor O. Alves

Abstract. In this article we consider the fractional Schrödinger-Poisson sys-
tem

(−∆)su− µ
Φ(x/|x|)
|x|2s

u+ λφu = |u|2
∗
s−2u, in R3,

(−∆)tφ = u2, in R3,

where s ∈ (0, 3/4), t ∈ (0, 1), 2t + 4s = 3, λ > 0 and 2∗s = 6/(3 − 2s) is the
Sobolev critical exponent. By using perturbation method, we establish the

existence of a solution for λ small enough.

1. Introduction

In this article we consider the fractional Schrödinger-Poisson system

(−∆)su− µΦ(x/|x|)
|x|2s

u+ λφu = |u|2
∗
s−2u, in R3,

(−∆)tφ = u2, in R3,

(1.1)

where s ∈ (0, 3/4), t ∈ (0, 1), 2t+ 4s = 3, λ > 0 and 2∗s = 6/(3− 2s) is the Sobolev
critical exponent. The function Φ and the parameter µ satisfy the condition

(A1) If 0 6 Φ ∈ L 3
2s (S2) then µ ∈ (0,Λ(Φ)), where Λ(Φ) is defined in Lemma 2.1

below.

In quantum mechanics a zero spin relativistic particle of charge e and mass m
in the Coulomb field of an infinitely heavy nucleus of charge Z is described by the
Hamiltonian (see e.g. [23, 28])

H(p, x) = (p2 +m2)1/2 − Ze2

|x|
.

Fall and Felli [16], extended the study of H(p, x) to

H̃(p, x) = (p2 +m2)s −
Φ( x
|x| )

|x|2s
.

2010 Mathematics Subject Classification. 35B38, 35J47.
Key words and phrases. Fractional Schrödinger-Poisson system; weighted Hardy potential;

critical exponent.
c©2020 Texas State University.

Submitted October 24, 2019. Published January 6, 2020.

1



2 Y. SU, H. CHEN, S. LIU, X. FANG EJDE-2020/01

They considered the operator

H̃(i∇, x) = (−∆ +m2)s −
Φ( x
|x| )

|x|2s
.

For Φ ≡ 0, the operator H̃(i∇, x) becomes (−∆ +m2)s, for more details, we refer

to [2, 3]. In this article, we study H̃(i∇, x), i. e. when m = 0.
In nonrelativistic molecular physics, the interaction between an electric charge

and the dipole moment D ∈ RN of a molecule is described by an inverse square
potential with an anisotropic coupling strength. In particular the Schrödinger equa-
tion for the wave function of an electron interacting with a polar molecule (supposed
to be point-like) can be written as

H = − ~
2m

∆ + e
x ·D
|x|3

− E,

where D is the dipole moment of the molecule, e and m denote the charge and the
mass of the electron (see [26]). We consider the operator

Ld := −∆− 2me|D|
~

x · d
|x|3

,

in RN , where N > 3, being |D| the magnitude of the dipole moment D, and
d = D

|D| denotes the orientation of D. The Laplace operator with dipole-type

potential (purely angular multiples of radial inverse square potentials):

LΦ := −∆− µΦ(x/|x|)
|x|2

,

in RN , where N > 3. We consider the more general class of operator H̃(i∇, x) with
zero mass,

Ls,Φ := (−∆)s − µΦ(x/|x|)
|x|2s

.

If Φ ≡ 0, then Ls,Φ becomes the fractional Laplacian operator (−∆)s. If Φ ≡ 1,
then Ls,Φ is the fractional Laplacian operator with Hardy potential (−∆)s − µ

|x|2s .

If s = 1, then Ls,Φ becomes LΦ.
From the mathematical point of view, a reason of interest in potentials of the

type Φ( x
|x| )/|x|

2s relies in their criticality with respect to the differential operator

(−∆)s; indeed, they have the same homogeneity as the fractional Laplacian (−∆)s,
hence they cannot be regarded as a lower order perturbation term. We mention
that the operator with singular potentials have been widely studied, see for example
[1, 13, 14, 15, 17, 18, 19, 20, 21, 34, 36, 38, 39] and references therein.

On the other hand, if µ = 0 and λ = 1, then system (1.1) becomes the system:

(−∆)su+ φu = |u|2
∗
s−2u, in R3,

(−∆)tφ = u2, in R3.
(1.2)

For this system there are four cases: (i) s = t = 1; (ii) s = 1 and t 6= 1; (iii)
s = t 6= 1; (iv) s 6= 1, t 6= 1, and 2t+ 4s > 3.

(i) For s = t = 1, system (1.2) reduces to the Schrödinger–Poisson system

−∆u+ φu = |u|p−2u, in R3,

−∆φ = u2, in R3.
(1.3)



EJDE-2020/01 FRACTIONAL SCHRÖDINGER-POISSON SYSTEM 3

The Schrödinger-Poisson system arises in many mathematical physical context,
such as in quantum electrodynamics, to describe the interaction between a charge
particle interacting with the electromagnetic field, and in semiconductor theory,
in nonlinear optics and in plasma physics (see [8] for more details in the physics
aspects). System (1.3) has been investigated by many researchers, see for example:
[6, 11] for |u|p−2u (4 < p < 6), [47] for |u|p−2u (3 < p < 4), [40] for |u|p−2u
(1 < p < 2).

(ii) For s = 1 and t 6= 1, Mercuri, Moroz and Van Schaftingen [32] studied the
Schrödinger–Poisson–Slater equation

−∆u+ (Iα ∗ |u|q) |u|q−2u = |u|p−2u, in R3, (1.4)

where p > 1, q > 1, α ∈ (0, N), and Iα =
Γ( 3−α

2 )

2απ
3
2 Γ(α2 )

· 1
|x|3−α is the Riesz potential.

Equation (1.4) is formally equivalent to the following system (see [32, Page 146,
Line 10]):

−∆u+ φ|u|q−2u = |u|p−2u, in R3,

(−∆)α/2φ = |u|q, in R3.
(1.5)

The existence of nonnegative ground state solution of (1.4) was established when
the parameters satisfy: either

q > 3 + α and
2(α+ 2q)

α+ 2
< p < 6,

or

q < 3 + α and 6 < p <
2(α+ 2q)

α+ 2
.

See [32, Theorem 3]. Moreover, some qualitative properties of the solutions were
estudied. Li, Gao and Zhu [27] proved the existence of sign-changing solution of
Kirchhoff type system with Hartree-type nonlinearity

−(a+ b

∫
R3

|∇u|2dx)∆u+ λV (x)u+ φ|u|q−2u = f(u), in R3,

(−∆)α/2φ = l|u|q, in R3,

(1.6)

where a > 0, b, l > 0, α ∈ (0, 3), q ∈ [2, 3 + α), λ > 0 is a parameter, and the
functions V and f satisfy suitable assumptions. For recent works, we refer to [4]
and the references therein.

(iii) For s = t 6= 1, the author in [43] applied the fountain theorem to prove the
existence of infinitely many solutions to the system

(−∆)su+ V (x)u+ φu = f(x, u), in R3,

(−∆)sφ = γsu
2, in R3.

(iv) For s 6= 1, t 6= 1 and 2t+ 4s > 3, Zhang, do Ó and Squassina [46] investigated
the system

(−∆)su+ λφu = f(u), in R3,

(−∆)tφ = λu2, in R3,
(1.7)

where f is a nonlinearity of Berestycki-Lions type. The authors proved that system
(1.7) admits a positive radial solution if λ > 0 small enough. Liu and Zhang [30]



4 Y. SU, H. CHEN, S. LIU, X. FANG EJDE-2020/01

proved the existence of positive ground state solution for ε > 0 sufficiently small to
the system

ε2s(−∆)su+ V (x)u+ φu = f(u) + |u|2
∗
s−2u, in R3,

ε2t(−∆)tφ = u2, in R3,

where V ∈ C1(R3,R+) and f is subcritical. For more details and recent works, we
refer to [5, 31, 37, 41, 42] and the references therein.

In [30, 41, 42, 46] the authors considered the existence of solution for system
(1.7) in which 2t + 4s > 3. Therefore, it is natural to ask whether system (1.7)
admits a solution for 2t+ 4s = 3. To the best of our knowledge, there is no result
on such question in the current literature. Motivated by the above facts, we study
the existence of solution for system (1.1).

1.1. Statement of results. We introduce the energy functional associated with
system (1.1) by

Iλ(u) =
1

2
‖u‖2Φ +

λ

4

∫
R3

∫
R3

|u(x)|2|u(y)|2

|x− y|3−2t
dxdy − 1

2∗s

∫
R3

|u|2
∗
sdx.

where the norm ‖ · ‖Φ is defined in Section 2. There is a difficulty in applying
variational methods to the functional Iλ.
(Difficulty) Since 2∗s < 4, we know that the functional Iλ does not satisfy Mountain
Pass Theorem, the boundedness of the Palais-Smale sequence for Iλ is hard to
obtain. To overcome this difficult, by using the perturbation method in [12, 25],
for λ small enough, we look the system (1.1) as a perturbation of the equation

(−∆)su− µΦ(x/|x|)
|x|2s

u = |u|2
∗
s−2u, in R3. (1.8)

We introduce the energy functional associated with equation (1.8) by

I0(u) =
1

2
‖u‖2Φ −

1

2∗s

∫
R3

|u|2
∗
sdx.

For λ small enough, we look at the functional Iλ as a perturbation of functional I0,

Iλ(u) := I0(u) +Kλ(u),

where

Kλ(u) =
λ

4

∫
R3

∫
R3

|u(x)|2|u(y)|2

|x− y|3−2t
dxdy.

Let S be the set of ground state critical points of I0. The perturbation method
mainly consists of two aspects:

(1) the mountain pass type critical point of I0 is a ground state in Ds,2
rad(R3);

(2) the set S is compact in Ds,2
rad(R3).

If conditions (1) and (2) are satisfied, then there exists a Palais-Smale sequence of
Iλ near the set S for sufficiently small λ.

Since the perturbation method is based on the properties of set S, we have to
study the existence of ground state solution to equation (1.8) first. The following
is our first result.

Theorem 1.1. Let s ∈ (0, 3/4) and (Φ1) hold. Then (1.8) has a ground state

solution v ∈ Ds,2
rad(R3).



EJDE-2020/01 FRACTIONAL SCHRÖDINGER-POISSON SYSTEM 5

There is a difficulty in the proof of Theorem 1.1. Because of the lack of com-
pactness of the Sobolev embedding Ds,2(R3) ↪→ L2∗

s (R3), and the functional I0
is invariant under the weighted dilation, it is hard to show that the Palais-Smale
sequence of I0 has a convergence subsequence.

For Φ ≡ 1, let us review the method in [45]. Applying their method, one has the
existence of ground state solution to the equation

(−∆)su− µ u

|x|2s
= |u|2

∗
s−2u, in R3. (1.9)

Similar to Section 3 in [45], there exists a Palais-Smale sequence {un} for the
energy functional corresponding to (1.9) such that un ⇀ u in Ds,2(R3) with u
solving (1.9). However, it may occur that u ≡ 0. The key step of [45] was to rule
out the ”vanishing” of the Palais-Smale sequence by using the limit equation of

(1.9). Taking vn(x) = λ
3−2s

2
n un(λnx+ xn) where λn > 0, xn ∈ R3 and xn

λn
→∞ as

n→ +∞, they derived that vn ⇀ v in Ds,2(R3) and∫
R3

vkφ

|x+ xk
λk
|2s
→ 0 as k → +∞ (1.10)

for any φ ∈ Ds,2(R3). Then v weakly solves

(−∆)sv = |v|2
∗
s−2v, in R3. (1.11)

Using the limit equation (1.11), they ruled out the “vanishing” of the Palais-Smale
sequence for the energy functional corresponding to (1.9).

Naturally, we would hope to overcome our difficulty by using the method in [45],
but unfortunately, for Φ satisfies condition (Φ1), the behavior of

lim
n→∞

Φ
( x+ xn

λn

|x+ xn
λn
|

)
, as

xn
λn
→∞,

is unknown, so we do not have any information of the expression

lim
n→∞

∫
R3

Φ
( x+ xn

λn

|x+ xn
λn
|

) vnϕ

|x+ xn
λn
|2s

dx. (1.12)

Clearly, their method does not apply to our case, since the expression (1.12) is not
equal to zero.

For the above reason, we use another way to prove Theorem 1.1. The cru-
cial point is the utilization of the uniform decay estimate for radial function v ∈
Ds,2

rad(R3) (see [9])

|v(x)| 6 C

|x| 3−2s
2

‖v‖s 6
C

|x| 3−2s
2

.

We consider the function I0 : Ds,2
rad(R3)→ R defined by

J0 = I0|Ds,2rad(R3).

The principle of symmetric criticality implies that the critical points of J0 are also
critical points for I0. Moreover, in Theorem 1.1 we have the following results:

(1) the mountain pass type critical point of I0 is a ground state in Ds,2
rad(R3);

(2) the set S is compact in Ds,2
rad(R3).

Combining the above results and the perturbation method, we give our second
result as follows.



6 Y. SU, H. CHEN, S. LIU, X. FANG EJDE-2020/01

Theorem 1.2. Let s ∈ (0, 3/4), t ∈ (0, 1), 2t+ 4s = 3 and (Φ1) hold. Then there
exists λ0 such that system (1.1) has a radially symmetric solution for all λ ∈ (0, λ0).

This article is organized as follows. In Section 2, we present some notations. In
Section 3, we give the proof of Theorem 1.1. In Section 4, we show the proof of
Theorem 1.2.

2. Preliminaries

For s ∈ (0, 1), the space Ds,2(R3) is the completion of C∞0 (R3) with respect to
the norm

‖u‖2s =

∫
R3

∫
R3

|u(x)− u(y)|2

|x− y|3+2s
dxdy.

We denote by Ds,2
rad(R3) the space of radial functions in Ds,2(R3). We could define

the best constants

Ss := inf
u∈Ds,2(R3)\{0}

‖u‖2s
(
∫
R3 |u|2∗

sdx)
2
2∗s

. (2.1)

We know that Ss can be attained in R3 (see [33]).
For u ∈ Ds,2(R3), we have the Hardy inequality (see [22]):

Cs,2
∫
R3

|u|2

|x|2s
dx 6

∫
R3

∫
R3

|u(x)− u(y)|2

|x− y|3+2s
dxdy, (2.2)

with

Cs,2 := 2

∫ 1

0

r2s−1|1− r
3−2s

2 |2Φs,2(r)dr.

We introduce the measure dϑ induced by Lebesgues measure on the unit sphere
S2 ⊂ R3. We denote by ‖ · ‖Lq(S2) the quantity

‖Φ‖qLq(S2) =

∫
S2
|Φ(ϑ)|qdϑ.

Lemma 2.1 ([24]). Let s ∈ (0, 1) and 0 6 Φ ∈ L 3
2s (S2). Then∫

R3

∫
R3

|u(x)− u(y)|2

|x− y|3+2s
dxdy > Λ(Φ)

∫
R3

Φ(x/|x|)|u|2

|x|2s
dx,

where u ∈ Ds,2(R3) and Λ(Φ) = Cs,2|S2|
2s
3 ‖Φ‖−1

L
3
2s (S2)

.

By using Lemma 2.1 and µ ∈ (0,Λ(Φ)),

‖u‖Φ =
(∫

R3

∫
R3

|u(x)− u(y)|2

|x− y|3+2s
dxdy − µ

∫
R3

Φ(x/|x|)|u|2

|x|2s
dx
)1/2

is an equivalent norm in Ds,2(RN ).

Lemma 2.2 ([29]). Let t, r > 1 and 0 < α < 3 with 1
t + 1

r + 3−α
3 = 2, f ∈ Lt(R3)

and h ∈ Lr(R3). There exists a sharp constant C(α, t, r) > 0, independent of f, h
such that ∫

R3

∫
R3

|f(x)||h(y)|
|x− y|3−α

dxdy 6 C(α, t, r)‖f‖t‖h‖r.

If t = r = 6
3+α , then

C(α, t, r) = C(α) = π
3−α
2

Γ(α2 )

Γ(3 + α
2 )

{Γ( 3
2 )

Γ(3)

}−α/3
.



EJDE-2020/01 FRACTIONAL SCHRÖDINGER-POISSON SYSTEM 7

A measurable function u : R3 → R belongs to the Morrey space Lq,$(R3) with
q ∈ [1,∞) and $ ∈ (0, N ] if

‖u‖qLq,$(R3) = sup
R>0,x∈R3

R$−3

∫
B(x,R)

|u(y)|qdy <∞.

Lemma 2.3 ([35]). For s ∈ (0, 1), there exists C > 0 such that for ι and ϑ satisfying
2
2∗
s
6 ι < 1, 1 6 ϑ < 2∗s, we have

(∫
R3

|u|2
∗
sdx

)1/(2∗
s)

6 C‖u‖ιs‖u‖1−ι
Lϑ,

ϑ(N−2s)
2 (R3)

,

for any u ∈ Ds,2(R3).

2.1. Formulation of system (1.1). Considering u ∈ Ds,2(R3), we define the linear
functional Fu(v) : Dt,2(R3)→ R by

Fu(v) =

∫
R3

u2vdx.

Using 2t+ 4s = 3, Holder’s inequality, and (1.9), we obtain

|Fu(v)| 6
∫
R3

|u|
6−4s
3−2s |v|dx

=

∫
R3

|u|
3+2t
3−2s |v|dx

6
(∫

R3

|u|
3+2t
3−2s ·

6
3+2t dx

) 3+2t
6
(∫

R3

|v|
6

3−2t dx
) 3−2t

6

6
( 1

Ss
‖u‖2s

) 3+2t
6
( 1

St
‖v‖2t

) 3−2t
6

,

which implies that Fu is well defined and continuous in Dt,2(R3).
From the Lax-Milgram theorem it follows that, for every u ∈ Ds,2(R3), there

exists a unique φu,t ∈ Dt,2(R3) such that∫
R3

(−∆)t/2φu,t(−∆)t/2vdx =

∫
R3

u2vdx,

which implies that φu,t is a weak solution of

(−∆)tφu,t = u2,

and the representation formula holds,

φu,t =

∫
R3

|u(y)|2

|x− y|3−2t
dy. (2.3)

It follows from (2.3) that system (1.1) can be rewritten as the equivalent form

(−∆)su− µΦ(x/|x|)
|x|2s

u+ λ
(
I2t ∗ |u|2

)
u = |u|2

∗
s−2u in R3.



8 Y. SU, H. CHEN, S. LIU, X. FANG EJDE-2020/01

3. Perturbed equation

We look at equation (1.1) as a perturbation of the equation

(−∆)su− µΦ(x/|x|)
|x|2s

u = |u|2
∗
s−2u, in R3. (3.1)

We introduce the energy functional associated with (1.8) by

I0(u) =
1

2
‖u‖2Φ −

1

2∗s

∫
R3

|u|2
∗
sdx.

Define
J0 = I0|Ds,2rad(R3).

3.1. Mountain pass geometry and Nehari manifold.

Lemma 3.1. Under the assumptions of Theorem 1.1, the functional J0 possesses
the mountain pass geometry.

Proof. We prove that J0 satisfies all the conditions in Mountain Pass Theorem.
(i) For any u ∈ Ds,2

rad(R3) \ {0}, we have

J0(u) >
1

2
‖u‖2Φ −

1

2∗s · S
2∗
s
s

‖u‖2
∗
s

Φ .

Because of 2 < 2∗s, there exists a sufficiently small positive number ρ such that

ς := inf
‖u‖s=ρ

J0(u) > 0 = J0(0).

(ii) Given u ∈ Ds,2
rad(R3) \ {0}, we have

J0(tu) =
t2

2
‖u‖2Φ −

t2
∗
s

2∗s

∫
R3

|u|2
∗
sdx < 0,

for t large enough. We choose tu > 0 corresponding to u such that J0(tu) < 0 for
all t > tu and ‖tuu‖s > ρ. �

Define
c0 = inf

Υ∈Γ0

max
t∈[0,1]

J0(Υ(t)),

where
Γ0 = {Υ ∈ C([0, 1], Ds,2

rad(R3))|Υ(0) = 0, J0(Υ(1)) < 0}.
By Lemma 3.1, we obtain that for all λ > 0, there exists {un} ⊂ Ds,2

rad(R3) such
that

J0(un)→ c0 > 0 and J ′0(un)→ 0 as n→∞.
It is also easy to see that {un} is uniformly bounded in Ds,2

rad(R3).

The Nehari manifold on Ds,2
rad(R3) is defined by

N = {u ∈ Ds,2
rad(R3)|〈J ′0(u), u〉 = 0, u 6= 0},

¯̄c0 = inf
u∈N

J0(u),

c̄0 = inf
u∈Ds,2rad(R3)

max
t>0

J0(tu).

With minor changes in the proof of [44, Theorem 4.2], we can show that

¯̄c0 = c̄0 = c0.



EJDE-2020/01 FRACTIONAL SCHRÖDINGER-POISSON SYSTEM 9

The following result implies the non-vanishing property of (PS)c0 sequence.

Lemma 3.2. Assume that the assumptions of Theorem 1.1 hold. Let {un} be a
(PS)c0 sequence of J0 with c0 > 0. Then

lim
n→∞

∫
R3

|un|2
∗
sdx > 0.

Proof. It is easy to see that {un} is uniformly bounded in Ds,2
rad(R3). Then there

exists a constant 0 < C <∞ such that ‖un‖Φ 6 C. Suppose on the contrary that

lim
n→∞

∫
R3

|un|2
∗
sdx = 0. (3.2)

According to (3.2) and the definition of (PS)c0 sequence, we obtain

c0 + o(1) =
1

2
‖un‖2Φ and ‖un‖2Φ = o(1);

these imply that c0 = 0, which contradicts with 0 < c0. �

3.2. Ground state solution.

Proof of Theorem 1.1. We divide the proof of Theorem 1.1 into two Steps.

Step 1. Since {un} is a bounded sequence in Ds,2
rad(R3), up to a subsequence, we

assume that un ⇀ u in Ds,2
rad(R3), un → u a.e. in R3, un → u in Lrloc(R3) for all

r ∈ [2, 2∗s).
According to Lemmas 2.3 and 3.2, there exists C > 0 such that for any n, we

obtain

‖un‖L2,3−2s(R3) > C > 0.

On the other hand, since {un} is bounded in Ds,2
rad(R3) and

Ds,2
rad(R3) ↪→ Ds,2(R3) ↪→ L2∗

s (R3) ↪→ L2,3−2s(R3),

we have

‖un‖L2,3−2s(R3) 6 C,

for some C > 0 independent of n. Hence, there exists a positive constant which we
denote again by C such that for any n, we obtain

C 6 ‖un‖L2,3−2s(R3) 6 C
−1.

Combining the definition of Morrey space and above inequality, we deduce that for
any n ∈ N there exist σn > 0 and xn ∈ R3 such that

1

σ2s
n

∫
B(xn,σn)

|un(y)|2dy > ‖un‖2L2,3−2s(R3) −
C

2n
> C1 > 0.

Let vn(x) = σ
3−2s

2
n un(σnx). We know that

J0(vn) = J0(un)→ c0 and J ′0(vn)→ 0 as n→∞.

Thus there exists v such that vn ⇀ v in Ds,2
rad(R3), vn → v a.e. in R3, vn → v in

Lrloc(R3) for all r ∈ [2, 2∗s). Then∫
B( xnσn ,1)

|vn(y)|2dy =
1

σ2s
n

∫
B(xn,σn)

|un(y)|2dy > C1 > 0. (3.3)



10 Y. SU, H. CHEN, S. LIU, X. FANG EJDE-2020/01

Step 2. We will show that {xnσn } is bounded. Suppose on the contrary that xn
σn
→∞

as n → ∞. By using the boundedness of {un} in Ds,2
rad(R3), we have ‖vn‖s =

‖un‖s 6 C2. Applying the uniform decay estimates of radial functions, we have

|vn(x)| 6 C

|x| 3−2s
2

‖vn‖s 6
C3

|x| 3−2s
2

,

where C3 = CC2. For any
√

C1

|B(0,1)| > ε > 0, there exists M > 0, such that for

n > M , we have

|vn(x)| 6 C3

|xnσn − 1| 3−2s
2

6 ε, for all x ∈ Bc(0, |xn
σn
− 1|).

Since B(xnσn , 1) ⊂ Bc(0, |xnσn − 1|), we obtain∫
B( xnσn ,1)

|vn(y)|2dy 6 ε2

∫
B( xnσn ,1)

dy = ε2|B(
xn
σn
, 1)| = ε2|B(0, 1)| < C1.

This contradicts (3.3). Hence, {xnσn } is bounded, and there exists R > 0 such that∫
B(0,R)

|vn(y)|2dy >
∫
B( xnσn ,1)

|vn(y)|2dy > C1 > 0.

Since the embedding Ds,2
rad(R3) ↪→ Lrloc(R3), r ∈ [2, 2∗s) is compact, we deduce that∫

B(0,R)

|v(y)|2dy > C1 > 0.

Hence, v 6≡ 0.
From vn ⇀ v weakly in Ds,2

rad(R3) and limn→∞〈J ′0(vn), ϕ〉 = o(1), we obtain

〈J ′λ(v), ϕ〉 = 0.

Since v 6≡ 0, we know that v ∈ N .
Now, we show that vn → v strongly in Ds,2

rad(R3). Applying the Brézis-Lieb
lemma [10], we obtain

c0 = lim
n→∞

J0(vn)− lim
n→∞

1

2∗s
〈J ′0(vn), vn〉

= lim
n→∞

(1

2
− 1

2∗s

)
‖vn‖2Φ

>
(1

2
− 1

2∗s

)
‖v‖2Φ

= J0(v) > ¯̄c0 = c0.

(3.4)

Thus, the inequalities above have to be equalities. We know that

lim
n→∞

‖vn‖2Φ = ‖v‖2Φ.

By using Brézis-Lieb lemma [10] again, we have

lim
n→∞

‖vn‖2Φ − lim
n→∞

‖vn − v‖2Φ = ‖v‖2Φ,

which implies that

lim
n→∞

‖vn − v‖2Φ = 0.



EJDE-2020/01 FRACTIONAL SCHRÖDINGER-POISSON SYSTEM 11

Using (3.4) again, we know that J0(v) = c0. This implies that v attain the minimum
of J0 at c0. �

4. Proof of Theorem 1.2

First, we summarize the proof of Theorem 1.1 as follows.

Lemma 4.1. The energy functional J0 satisfies the following properties:

(1) there exist ρ, ς > 0 such that if ‖u‖s = ρ, then J0(u) > ς, and e ∈ Ds,2
rad(R3)

exists such that ‖e‖s > ρ and J0(e) < 0;
(2) there exists a critical point v of J0 such that

J0(v) = c0 := min
Υ∈Γ0

max
t∈[0,1]

J0(Υ(t)),

where Γ0 = {Υ ∈ C([0, 1], Ds,2
rad(R3))|Υ(0) = 0, J0(Υ(1)) < 0};

(3) c0 := inf{J0(u)|J ′0(u) = 0, u ∈ Ds,2
rad(R3) \ {0}};

(4) there exists a path Υ0(t) ∈ Γ0 passing through v at t = t0 and satisfying

J0(v) > J0(Υ0(t)) for all t 6= t0;

(5) the set S := {u ∈ Ds,2
rad(R3)|J ′0(u) = 0, J0(u) = c0} is compact in Ds,2

rad(R3)
endowed with the strong topology up to dilations in R3.

Proof. It is easy to see that J0 is invariant by dilations. As a byproduct of the
proof of Theorem 1.1, the weak convergence of the dilated subsequence can be
upgraded into strong convergence. As a direct consequence, we show that the set
S is compact in Ds,2

rad(R3) endowed with the strong topology up to dilations in R3.
The proof is completed. �

We introduce the energy functional associated with system (1.1) by

Iλ(u) =
1

2
‖u‖2Φ +

λ

4

∫
R3

∫
R3

|u(x)|2|u(y)|2

|x− y|3−2t
dxdy − 1

2∗s

∫
R3

|u|2
∗
sdx.

Define

Jλ = Iλ|Ds,2rad(R3).

As in [12, 25], we define a modified mountain pass energy level of Jλ

cλ := min
Υ∈ΥM

max
t∈[0,1]

Jλ(Υ(t)),

where

ΥM = {Υ ∈ Γ0| sup
t∈[0,1]

‖Υ(t)‖s 6M},

M = 2{sup
u∈S
‖u‖s, sup

t∈[0,1]

‖Υ(t)‖s}.

Clearly, by the choice of M , Υ0 ∈ ΓM and thus,

c0 = min
Υ∈ΓM

max
t∈[0,1]

J0(Υ(t)).

However, since ΓM ( Γ0, the standard mountain pass theorem cannot be applied,
so other arguments are needed to prove that cλ is a critical value.

Lemma 4.2. Let λ > 0. Then limλ→0 cλ = c0.



12 Y. SU, H. CHEN, S. LIU, X. FANG EJDE-2020/01

Proof. Since λ > 0, it is easy to see that cλ > c0. On the other hand, we can take
e = Tv in property (1), where

T >
(2∗s

2

) 1
2∗s−2

.

Then Υ0(t) ∈ C([0, 1], Ds,2(R3)) is defined as Υ0(t) = te = tTv, and t0 = 1
T in

property (4). We know that

lim
λ→0

cλ = lim
λ→0

Jλ(Υ0(t))

6J0(Υ0(t)) + lim
λ→0

λ

4

∫
R3

∫
R3

|Tv(y)|2|Tv(x)|2

|x− y|3−2t
dxdy

=J0(v) = c0.

The proof is complete. �

For any positive number d, we denote

Bd(u) := {v ∈ Ds,2
rad(R3)|‖u− v‖s 6 d},

and for any subset A of Ds,2
rad(R3), we set

Ad := ∪u∈ABd(u).

Lemma 4.3. Set d1 =
√

2 · 2∗s · c0/(2∗s − 2). Let d ∈ (0, d1). For any u ⊂ Sd, we
have u 6≡ 0.

Proof. For any v ⊂ Sd, we have

‖v‖2s > ‖v‖2Φ =
2 · 2∗s
2∗s − 2

(
J0(v)− 1

2∗s
〈J ′0(v), v〉

)
=

2 · 2∗s · c0
2∗s − 2

,

which gives

‖v‖s > d1. (4.1)

By the definition of Sd and u ⊂ Sd, there exists some v ⊂ Sd such that

‖u− v‖s 6 d < d1. (4.2)

Applying the triangle inequality, we have

‖v‖s = ‖v − u+ u‖s 6 ‖v − u‖s + ‖u‖s.

which gives

‖v‖s − ‖v − u‖s 6 ‖u‖s. (4.3)

Putting (4.1) and (4.2) into (4.3), we obtain

‖u‖s > d1 − ‖v − u‖s > 0.

The proof is complete. �

Lemma 4.4. Let d > 0 be a fixed number and let {uj} ⊂ Sd. Then there exists
{σj}, such that

‖ūj‖s = ‖uj‖s,

where ūj(x) = σ
3−2s

2
j uj(σjx). And, up to a subsequence, ūj ⇀ ū ∈ S2d.



EJDE-2020/01 FRACTIONAL SCHRÖDINGER-POISSON SYSTEM 13

Proof. Let {uj} ⊂ Sd. By the definition of Sd and Lemma 4.1(5), there exists
wj ∈ S such that

‖uj − wj‖s 6 d.
By property (5) again, there exists {σj}, such that w̄j ∈ S, one can prove that

w̄j → w̄ ∈ S, where w̄j(x) = σ
3−2s

2
j wj(σjx) and ūj(x) = σ

3−2s
2

j uj(σjx). Also

‖ūj‖s = ‖uj‖s, ‖ūj − w̄j‖s = ‖uj − wj‖s 6 d.
Hence, for j large enough, we obtain

‖ūj − w̄‖s = ‖ūj − w̄j + w̄j − w̄‖s 6 ‖ūj − w̄j‖s + ‖w̄j − w̄‖s 6 2d.

Thus, {ūj} is bounded and, up to a subsequence, ūj ⇀ ū in Ds,2
rad(R3). Since

B2d(w̄) is weakly closed in Ds,2
rad(R3), we obtain ū ∈ B2d(w̄) ⊂ S2d. The proof is

complete. �

Lemma 4.5. Set d ∈ (0, d1). Suppose that there exist sequences λj > 0, λj → 0,
and {uj} ⊂ Sd satisfying

lim
j→∞

Jλj (uj) 6 c0 and lim
j→∞

J ′λj (uj) = 0.

Then there is sequence {σj} such that ‖ūj‖s = ‖uj‖s, where ūj(x) = σ
3−2s

2
j uj(σjx).

And, up to a subsequence, {ūj} converges to some ū ∈ S.

Proof. Considering that limj→∞ J ′λj (uj) = 0 and that {uj} is bounded. By Lemma

4.4, up to a subsequence, ūj ⇀ ū ∈ S2d, and by the choice of d1, ū 6= 0. We may
readily verify that

lim
j→∞

Jλj (ūj) = lim
j→∞

Jλj (uj) 6 c0 and lim
j→∞

J ′λj (ūj) = 0.

For all ϕ ∈ Ds,2
rad(R3), we obtain

|〈J ′λj (ūj), ϕ〉| = |〈J
′
λj (uj), ϕ̄〉| 6 ‖J

′
λj (uj)‖s−1‖ϕ̄‖s = o(1)‖ϕ̄‖s,

where ϕ̄ = σ
− 3−2s

2
j ϕ( xσj ). Since ‖ϕ̄‖s = ‖ϕ‖s, we obtain

J ′λj (ūj)→ 0 as j →∞.
Now, we have

〈J ′0(ū), ϕ〉 = lim
j→∞
〈J ′λj (ūj), ϕ〉 − λj

∫
R3

∫
R3

|ūj(y)|2ūj(x)ϕ(x)

|x− y|3−2t
dxdy = 0.

Hence, J ′0(ū) = 0. Furthermore, applying Lemma 2.2 and ūj ∈ S2d for all j, we
obtain

lim
j→∞
〈J ′0(ūj), ϕ〉 = lim

j→∞
〈J ′λj (ūj), ϕ〉 − λj

∫
R3

∫
R3

|ūj(y)|2ūj(x)ϕ(x)

|x− y|3−2t
dxdy

= o(1)‖ϕ‖s.
On the other hand,

c0 > lim
j→∞

Jλj (ūj)

= lim
j→∞

J0(ūj) +
λj
4

∫
R3

∫
R3

|ūj(y)|2|ūj(x)|2

|x− y|3−2t
dxdy

= lim
j→∞

J0(ūj).

(4.4)



14 Y. SU, H. CHEN, S. LIU, X. FANG EJDE-2020/01

So {ūj} is a (PS)m sequence for J0 with m := limj→∞ J0(ūj). Thus, up to a
subsequence, ūj ⇀ ū and

J0(ū) =
1

2
‖ū‖2Φ −

1

2∗s

∫
R3

|ū|2
∗
sdx

=
(1

2
− 1

2∗s

)
‖ū‖2Φ

6
(1

2
− 1

2∗s

)
lim inf
j→∞

‖ūj‖2Φ

= lim inf
j→∞

(
J0(ūj)−

1

2∗s
〈J ′0(ūj), ūj〉

)
= m.

Then, property (3) implies m > J0(ū) > c0, which, combined with (4.4), yields
m = J0(ū) = c0, which implies ū ∈ S. �

Now, we set

mλ := max
t∈[0,1]

Jλ(Υ0(t)).

Then cλ 6 mλ. It is easy to see that limλ→0mλ 6 c0, this inequality together with
Lemma 4.2, provides limλ→0 cλ = limλ→0mλ = c0. We also define

Jmλλ = {u ∈ Ds,2
rad(R3)|Jλ(u) 6 mλ}.

Lemma 4.6. For any d2, d3 > 0 satisfying d3 < d2 < d1, there are constants ι > 0
and λ̃ > 0 depending on d2, d3 such that for λ ∈ (0, λ̃), we have

‖J ′λ(u)‖s−1 > ι, for all u ∈ Jmλλ ∩ (Sd2\Sd3).

Proof. To the contrary, suppose that for some d2, d3 > 0 satisfying d3 < d2 < d1,

there exist sequences {λj} with limj→∞ λj = 0 and {uj} ∈ J
mλj
λj
∩ (Sd2 \Sd3), such

that

lim
j→∞

Jλj (uj) 6 c0 and lim
j→∞

J ′λj (uj) = 0.

By using property (5), there exists a sequence {σj} such that

{ūj} ∈ J
mλj
λj
∩ (Sd2 \ Sd3), lim

j→∞
Jλj (ūj) 6 c0, lim

j→∞
J ′λj (ūj) = 0,

where ūj(x) = σ
3−2s

2
j uj(σjx). Hence, we can apply Lemma 4.4 and assert the

existence of some ū ∈ S such that ūj → ū in Ds,2
rad(R3). As a consequence,

dist(ūj ,S)→ 0 as j →∞, contradicting the relation ūj 6∈ Sd3 . �

Lemma 4.7. For d > 0, there exists δ > 0, such that if λ > 0 with λ small enough,
t ∈ [0, 1], Jλ(Υ0(t)) > cλ − δ implies Υ0(t) ∈ Sd.

Since the proof of the above lemma is quite similar to that of [25, Propositions
4], we omit it here.

Lemma 4.8. To each 0 < d < d1 corresponds a number λ0 > 0 such that for
all λ ∈ (0, λ0), there exists a sequence {uj} ∈ Jmλλ ∩ Sd such that J ′λ(uj) → 0 as
j →∞.

Since the proof of the above lemma is quite similar to that of [12, Propositions
5.3], we omit it here.



EJDE-2020/01 FRACTIONAL SCHRÖDINGER-POISSON SYSTEM 15

Proof of Theorem 1.2. Taking d ∈ (0, d1), by Lemma 4.8, there exists λ0 > 0 such

that for all λ ∈ (0, λ0), there exists a Palais Smale sequence {uλj } ⊂ S
d
2 . By

using property (5), there exists sequence {σj} such that {ūλj } ⊂ S
d
2 where ūλj (x) =

σ
3−2s

2
j uλj (σjx). It is easy to see that {ūλj } is bounded in Ds,2

rad(R3). Then by Lemma

4.4, up to a subsequence, there exists some ūλ ∈ S d2 ·2 = Sd such that ūλj ⇀ ūλ.

Then we obtain J ′λ(ūλ) = 0, and by the choice of d, ūλ 6≡ 0. Hence ūλ is a nontrivial
critical point of Jλ. The principle of symmetric criticality implies that the critical
points of Jλ are also critical points for Iλ. �

5. Open problem

In Theorem 1.1, we obtain a ground state solution v ∈ Ds,2
rad(R3) to equation

(1.8). Since Ds,2
rad(R3) ⊂ Ds,2(R3), so v just is a ground state solution in Ds,2

rad(R3),
not a ground state solution in Ds,2(R3). It is natural to ask: Does there exist a
ground state solution to equation (1.8) in Ds,2(R3)?

Acknowledgements. This research was supported by the National Natural Sci-
ence Foundation of China 11671403, and by the University-level key projects of
Anhui University of science and technology QN2019101.

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Yu Su (corresponding author)

School of Mathematics and Big Data, Anhui University of Science and Technology,

Huainan, 232001 Anhui, China.
School of Mathematics and Statistics, Central South University, Changsha, 410083

Hunan, China
Email address: yusumath@qq.com

Haibo Chen

School of Mathematics and Statistics, Central South University, Changsha, 410083
Hunan, China

Email address: math chb@163.com

Senli Liu

School of Mathematics and Statistics, Central South University, Changsha, 410083

Hunan, China
Email address: jasonliu0615@163.com

Xianwen Fang
School of Mathematics and Big Data, Anhui University of Science and Technology,

Huainan, 232001 Anhui, China

Email address: xwfang@aust.edu.cn


	1. Introduction
	1.1. Statement of results

	2. Preliminaries
	2.1. Formulation of system (??)

	3. Perturbed equation
	3.1. Mountain pass geometry and Nehari manifold
	3.2. Ground state solution

	4. Proof of Theorem ??
	5. Open problem
	Acknowledgements

	References