Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 34, pp. 1–23.

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

DOI: 10.58997/ejde.2025.34

CONCENTRATING NORMALIZED SOLUTIONS FOR 2D NONLOCAL

SCHRÖDINGER EQUATIONS WITH CRITICAL EXPONENTIAL GROWTH

LIEJUN SHEN, MARCO SQUASSINA

Abstract. We study the existence of solutions to nonlocal Schrödinger problems with different

types of potentials

−∆u+W (x)u = σu+ κ[|x|−µ ∗ F (u)]f(u) in R2,∫
R2

|u|2dx = a2,

where a ̸= 0, σ ∈ R is known as the Lagrange multiplier, κ > 0 is a parameter, W ∈ C(R2) is

the nonnegative external potential, µ ∈ (0, 2), and F denotes the primitive function of f ∈ C(R)
which has critical exponential growth in the Trudinger-Moser sense at infinity. We prove that
the problems admit at least a positive solution, and we analyze the concentrating behavior.

1. Introduction

In this article, we aim to prove existence of positive solutions to the nonlocal Schrödinger
equation with different types of potentials

−∆u+W (x)u = σu+ κ[|x|−µ ∗ F (u)]f(u) in R2, (1.1)

under the constraint ∫
R2

|u|2dx = a2, (1.2)

where a ̸= 0, σ ∈ R is known as the Lagrange multiplier, κ > 0 is a parameter, W ∈ C(R2) is the
nonnegative external potential, µ ∈ (0, 2) and F denotes the primitive function of f ∈ C(R) which
has critical exponential growth in the Trudinger-Moser sense at infinity.

Inspired by the well-known Trudinger-Moser type inequality, we recall that a function f has
the critical exponential growth at infinity if there exists a constant α0 > 0 such that

lim
|s|→+∞

|f(s)|
eαs2

=

{
0, ∀α > α0,

+∞, ∀α < α0.
(1.3)

This definition was introduced by Adimurthi and Yadava [2], see also de Figueiredo, Miyagaki and
Ruf [30] for example.

Hereafter, we shall assume that the nonlinearity f satisfies (1.3) and the following assumptions:

(A1) f ∈ C(R,R) and f(s) ≡ 0 for all s ∈ (−∞, 0];
(A2) There is a q ∈

(
2, 6−µ

2

)
such that f(s)/sq−1 is an increasing function of s on (0,+∞).

(A3) There is a c0 > 0 such that f(s) ≥ c0s
q−1 for all s ∈ [0,+∞).

2020 Mathematics Subject Classification. 35A15, 35J10, 35B09, 35B33.
Key words and phrases. Positive normalized solution; Choquard equation, critical exponential growth;

Rabinowitz’s type potential; steep potential well; variational method.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted January 12, 2025. Published April 4, 2025.

1



2 L. J. SHEN, M. SQUASSINA EJDE-2025/34

We would like to highlight here that many functions f satisfy the above assumptions, with
α0 = 4π and c0 = 1, for example

f(s) =

{
0, s ≤ 0,

sq−1e4πs
2

, s > 0,

where q ∈ (2, 6−µ
2 ). Similar assumptions for a nonlinearity f satisfying (1.3) and (A1)–(A3) can

be found in [13, 58].
Over the past few decades, a lot of attentions have been paid to the standing wave solutions to

the time-dependent nonlinear Choquard equation

i
∂ψ

∂t
= ∆ψ −W (x)ψ + [|x|−µ ∗ F (ψ)]f(ψ) in R+ × RN , (1.4)

where ψ : RN × R → C acts as the time-dependent wave function, W : RN → R stands for the
real external potential and nonlinear term f(ψ) describes the interaction effect among particles.
Inserting the standing wave ansatz ψ(x, t) = exp(−iωt)u(x) with ω ∈ R and x ∈ RN into (1.4), it
follows that u : RN → R satisfies the Choquard equation

−∆u+ W̄ (x)u = [|x|−µ ∗ F (u)]f(u) in RN ; (1.5)

here and in the sequel W̄ (x) =W (x) + ω for all x ∈ RN .
There exist two directions in the studies of standing waves of the Choquard equation (1.5).

On the one hand, one can choose the frequency ω ∈ R to be fixed and investigate the existence
of nontrivial solutions for (1.5) obtained as the critical points of the variational functional I :
H1(RN ) → R given by

I(u) =
1

2

∫
RN

[
|∇u|2 + (W (x) + ω)u2

]
dx− 1

2

∫
RN

[|x|−µ ∗ F (u)]F (u)dx.

Actually, (1.5) is closely related to the Choquard equation arising from the studies of Bose-Einstein
condensation and can be used to describe the finite-range many-body interactions between particles
since |x|−µ can be reviewed as the classic Riesz potential. Letting N ≥ 3 and f(s) = |s|p−2s for
all s ∈ R, (1.5) is of the form

−∆u+ u = (|x|−µ ∗ |u|p)|u|p−2u, x ∈ RN . (1.6)

To describe a polaron at rest in the quantum field theory, Pekar [51] introduced the Choquard-
Pekar equation which is N = 3, µ = 1 and p = 2 in (1.6). Choquard adopted this equation
to characterization an electron trapped in its own hole as an approximation to the Hartree-Fock
theory for the one component plasma [41]. Subsequently, Lieb [40] and Lions [43] studied the
existence and uniqueness of positive solutions to (1.6) by variational methods. The authors in
[45, 47] concluded the regularity, positivity and radial symmetry of the ground state solutions
and investigated the decay properties at infinity. It should be pointed out that (1.6) was also
proposed by Morozet al. in [46] as a model for self-gravitating particles in the context as it
can be regarded as the classic Schrödinger-Newton equation, see e.g. [52, 62]. Actually, (1.6)
and its variants have received more and more attentions by many mathematicians because of
the appearance of the convolution type nonlinearities in these years. We refer the reader to
[1, 4, 5, 8, 9, 38, 11, 12, 15, 47, 48, 55] and the references therein, particularly to [49], for some
meaningful review of the Choquard equations.

On the other hand, one can consider the ω ∈ R to be unknown. In such a situation, ω is
supposed to act as a Lagrange multiplier and the L2-norm of the obtained solutions would be
prescribed since there is a conservation of mass which is said that the wave function ψ(x, t) with
its corresponding Cauchy initial function ψ(0, x) which preserves L2-mass in the following sense∫

RN

|ψ(t, x)|2dx =

∫
RN

|ψ(0, x)|2dx, ∀t ∈ (0,∞).

From the physical point of view, this spirit of research holds particular significance as it accounts
for the conservation of mass. Moreover, it provides valuable insights into the dynamic properties



EJDE-2025/34 NORMALIZED SOLUTIONS FOR CHOQUARD EQUATIONS 3

of the standing waves of (1.5), for instance stability or instability in [24, 28]. In this article, we
shall focus primarily on this direction.

Jenajean [34] used a minimax approach and compactness argument to conclude the existence
of solutions for the Schrödinger problem

−∆u+ ωu = g(u) in RN ,∫
RN

|u|2dx = a2.
(1.7)

There exist some further complements and generalizations in [36]. In [59], letting g(t) = τ |t|q−2t+
|t|p−2t with 2 < q ≤ 2 + 4

N ≤ p < 2∗, Soave obtained the existence of solutions for problem (1.7),

where 2∗ = 2N
N−2 if N ≥ 3 and 2∗ = ∞ if N = 2. For this type of combined nonlinearities, Soave

[60] also proved the existence of ground state and excited solutions when p = 2∗. For more results
for problem (1.7), we refer the reader to [7, 20, 35, 37, 63] and the references therein.

In the spirit of [34], when W̄ (x) ≡ 0 for all x ∈ RN in (1.5), the authors in [39] deduced the
existence of nontrivial solutions solutions to the nonlocal problem of Choquard type

−∆u+ ωu = [|x|−µ ∗ F (u)]f(u) in RN ,∫
RN

|u|2dx = a2,
(1.8)

provided a > 0 is sufficiently small, where f possesses the Sobolev subcritical growth at infinity.
Afterwards, Bartsch et al. [18] investigated the existence of solutions for problem (1.8) which is
simpler and more transparent than that of [39]. As to the case that the nonlinearity f admits
the critical growth, Ye, Shen and Yang [65] dealt with the existence of normalized ground state
solutions for the Hartree problem with a perturbation. There are some other interesting results
with respect to problem (1.8), see [15, 18, 29, 39] for example.

The reader may observe that the spatial dimension of problem (1.1) is two, the case therefore is
very special because 2∗ = ∞ in this situation. Explaining it more specifically, the fact H1(R2) ̸↪→
L∞(R2) shall make the problems special and quite delicate. Thus, it is not so direct to dispose
of the nonlinearity involving a critical exponential growth trivially. Letting W (x) ≡ 0 for all
x ∈ R2 in (1.1), Deng and Yu [29] supposed that the nonlinearity satisfies (1.3) and the following
assumptions

(A4) f : R → R is continuous;
(A5) f(t) = o(|t|τ ) as |t| → 0 for some τ > 3;
(A6) there exists a positive constant θ > 6−µ

2 such that 0 < θF (t) ≤ f(t)t for t ̸= 0;

(A7) there exist constants σ > 6−µ
2 and ξ > 0 such that F (t) ≥ ξ |t|σ for all t ∈ R.

Then, for some sufficiently small mass a > 0 and σ > 0 large enough, the authors used the
arguments in [34] to investigate the existence of normalized solution. Moreover, the ground state
solution was considered when f in addition has some monotone type assumptions. Afterwards,
Alves and Shen [15] handled the existence of nontrivial solutions to the problem

−∆u+ ωu = [|x|−µ ∗ (|x|−βF (u))]|x|−βf(u) in R2,∫
R2

|u|2dx = a2,
(1.9)

where β > 0, 0 < µ < 2 with 0 < 2β + µ < 2 and f admits the supercritical exponential growth
(see [10, 12]). It is worth pointing out here that either assumption (A6) or

(A7’) lim infs→+∞
F (s)

eα0s2
> 0, where α0 > 0 comes from (1.3),

plays a pivotal role in [15]. Actually, either the assumption (A7) or (A7’) is used for restoring the
compactness caused by the critical exponential growth and the whole space R2. As a consequence,
these assumptions seem indispensable to some extent in the mentioned works.

Motivated by the quoted papers above, particularly by [13, 14, 57, 58], we are going to consider
the existence of normalized solutions to nonlocal Schrödinger equations with different potentials



4 L. J. SHEN, M. SQUASSINA EJDE-2025/34

and critical exponential growth. Speaking it clearly, let us suppose that W (x) := V (εx) for all
ε > 0 and x ∈ R2 in (1.1) with the assumption

(A8) V ∈ C(R2,R) and 0 < V0 := infx∈R2 V (x) < V∞ := lim inf |x|→∞ V (x) < +∞, where
V (0) = V0.

Now, we can state the first main result in this article.

Theorem 1.1. Assume (A1)–(A3), (A8), (1.3) hold and µ ∈ (0, 2). Then there exist constants
κ∗ > 0, a∗ > 0 and ε∗ > 0 such that, for every κ ∈ (0, κ∗), a > a∗ and ε ∈ (0, ε∗), the problem

−∆u+ V (εx)u = σu+ κ[|x|−µ ∗ F (u)]f(u) in R2,∫
R2

|u(x)|2dx = a2,
(1.10)

has a pair of weak solutions (ū, σ̄) ∈ H1(R2) × R such that ū(x) > 0 for all x ∈ R2 and σ̄ < 0.
Moreover, if zε denotes the global maximum of ū, then, up to a subsequence if necessary,

lim
ε→0+

V (εzε) = V0.

We shall assume that W (x) = λV (x) for all λ > 0 and x ∈ R2 and the function V : R2 → R
satisfies the following conditions

(A9) V ∈ C(R2,R) with V (x) ≥ 0 on R2;
(A10) Ω := intV −1(0) is nonempty and bounded with smooth boundary, and Ω = V −1(0);
(A11) there exists a b > 0 such that the set Ξ := {x ∈ R2 : V (x) < b} is nonempty and admits

finite measure.

The main result in this directionreads as follows.

Theorem 1.2. Suppose (A)–(A3), (A9)–(A11), (1.3) hold and µ ∈ (0, 2). then there are κ∗ > 0,
a∗ > 0 and λ∗ > 1 such that, for all κ ∈ (0, κ∗), a > a∗ and λ > λ∗, the problem

−∆u+ λV (x)u = σu+ κ[|x|−µ ∗ F (u)]f(u) in R2,∫
R2

|u(x)|2dx = a2,
(1.11)

has a couple of weak solution (u, σ) ∈ H1(R2) × R such that u(x) > 0 for all x ∈ R2 and σ < 0.
If we denote (uλ, σλ) by the couple of weak solutions established above for all λ > λ∗, then for all
fixed a > a∗, passing to a subsequence if necessary, uλ → u0 in H1(R2) and σλ → σ0 in R as
λ→ +∞, where σ0 < 0 and (u0, σ0) is a couple of weak solution to the problem

−∆u = σu+ κ
(∫

Ω

F (u(y))

|x− y|µ
dy

)
f(u), x ∈ Ω,

u(x) = 0, x ∈ ∂Ω,∫
Ω

|u|2dx = a2.

(1.12)

The two types of potentials V appearing in Theorems 1.1 and 1.2 have been considered by
many mathematicians over the past decades, see [17, 23, 54, 33, 31, 32] and [19, 21, 22, 27, 44],
respectively. As a matter of fact, the former one is known as the Rabinowitz’s potential, while the
latter one is called by the steep potential well.

Remark 1.3. Concerning the existence of normalized solutions to some classes of local equations
with Rabinowitz’s potential, we prefer to refer the reader to [3, 6, 14, 16, 57]. The reader can
find the latest paper [58] focuses on the normalized solutions to Schrödinger-Newton system with
steep potential well.

It seems the first attempts to study the existence of normalized solutions to Choquard equations
with the above two types of potentials in a unified way.

It should be pointed out that we could not conclude the proofs of Theorems 1.1 and 1.2 simply
by repeating the approaches adopted in the previous papers mentioned in Remark 1.3. On the



EJDE-2025/34 NORMALIZED SOLUTIONS FOR CHOQUARD EQUATIONS 5

one hand, we successfully generalize the local case in [3, 6, 14, 16, 57] to the nonlocal one and so
there are some additional difficulties. On the other hand, thanks to the special structure of the
work space in [58], the key compact imbedding holds true in advance and it mainly deals with the
boundedness of minimizing sequence, while we easily obtain the boundedness and there are some
subtle efforts to recover the compactness in the proof of Theorem 1.2. As a consequence, we tend
to believe that this article may prompt some further studies on normalized solutions to a class of
nonlocal Schrödinger equations.

To conclude this section, we simply sketch the main ideas to arrive at the proofs of Theorems
1.1 and 1.2. Owing to the arguments adopted in [13, 14, 57, 58], for each fixed constant R > 0,
we introduce the following continuous function fR : R → R defined by

fR(s) =


0, if s ≤ 0,

f(s), if 0 ≤ s ≤ R,
f(R)
Rq−1 s

q−1, if R ≤ s < +∞,

(1.13)

where the constant q ∈
(
2, 6−µ

2

)
comes from (A2). For the rest of this article, we define FR(s) =∫ s

0
fR(t)dt for each s ∈ R to be the primitive function of fR. It follows from a direct computation

with (A2) that

qFR(s) ≤ fR(s)s, ∀s ≥ 0. (1.14)

We can use the monotone assumption in (A2) again to see that

fR(s) ≤
f(R)

Rq−1
sq−1, ∀s ≥ 0. (1.15)

With such a nonlinearity fR defied in (1.13), we turn to study the auxiliary problem

−∆u+W (x)u = σu+ κ[|x|−µ ∗ FR(u)]fR(u) in R2,∫
R2

|u|2dx = a2.
(1.16)

By (1.15), we can see that Problem (1.16) involves L2-subcritical growth since q < 6−µ
2 . So, the

solvability of Problem (1.16) becomes available. At this stage, we invite the reader to observe that
if the couple (uR, σR) is a solution of Problem (1.16), then it is indeed the solution to the original
Problems (1.1)-(1.2) as long as |uR|∞ ≤ R due to the definition of fR in (1.13). Having this in
mind, we shall derive the proofs of Theorems 1.1 and 1.2 combining the solvability of Problem
(1.16) and the L∞-estimate.

This article is organized as follows. In Section 1.2, we will introduce some preliminary results
handling the convolution parts. Sections 2 and 3 we obtain existence results for the auxiliary
Problem (1.16) with two different types of potentials. Finally, the detailed proofs of Theorems 1.1
and 1.2 shall be exhibited in Section 4.

1.1. Notation. From now on, we use the following notation:

• Br(x) ⊂ R2 is an open ball centered at x ∈ R2 with radius r > 0 and Br = Br(0).
• C,C1, C2, · · · denote any positive constant, whose value is not relevant.
• For all x ∈ R2, we define

u+(x) := max{u(x), 0} ≥ 0 and u−(x) := min{u(x), 0} ≤ 0.

• | · |p denotes the usual norm of the Lebesgue space Lp(R2), for every p ∈ [1,+∞]. ∥ · ∥Hi

denotes the usual norm of the Hilbert space for i ∈ {1, 2}.
• on(1) denotes a real sequence with on(1) → 0 as n→ +∞.
• “ → ” and “⇀ ” stand for the strong and weak convergence in the related function spaces,
respectively.

• We recall the celebrated Gagliardo-Nirenberg inequality, given an l ∈ [2,+∞),

|u|ll ≤ C|u|(1−γl)l
2 |∇u|γll

2 in H1(R2), γl = 2
(1
2
− 1

l

)
, (1.17)

where the constant C > 0 is just dependent of l.



6 L. J. SHEN, M. SQUASSINA EJDE-2025/34

1.2. Two basic facts. In this section, we exhibit some preliminary results adopted to prove the
main results. From now on, we shall always suppose that 0 < µ < 2 just for simplicity. Let us
first introduce the well-known Hardy-Littlewood-Sobolev inequality.

Lemma 1.4 ([42, Theorem 4.3]). Suppose that s, r > 1 and 0 < µ < N with 1
s + µ

N + 1
r = 2,

φ ∈ Ls(RN ) and ψ ∈ Lr(RN ). Then, there exists a sharp constant C = C(s,N, µ, r) > 0,
independent of φ and ψ, such that∫

RN

[|x|−µ ∗ φ(x)]ψ(x)dx ≤ C∥φ∥s∥ψ∥r. (1.18)

Since it mainly concerns the whole space R2 in this paper, we will assume that N = 2 in (1.18).
Let us conclude this section by introducing the celebrated Brézis-Lieb lemma for the nonlocal

term of Choquard type.

Lemma 1.5 ([47, Lemma 2.4]). . Let p ∈ [ 4−µ
2 ,+∞) and (un)n∈N be a bounded sequence in

L
4p

4−µ (R2). If un → u almost everywhere on R2 as n→ ∞, then

lim
n→∞

∫
R2

[(
|x|−µ ∗ |un|p

)
|un|p −

(
|x|−µ ∗ |un − u|p

)
|un − u|p

]
dx

=

∫
R2

(
|x|−µ ∗ |u|p

)
|u|pdx.

(1.19)

Moreover, for all φ ∈ C∞
0 (R2), it holds that

lim
n→∞

∫
R2

(
|x|−µ ∗ |un|p

)
|un|p−2unφdx =

∫
R2

(
|x|−µ ∗ |u|p

)
|u|p−2φdx. (1.20)

2. Truncated problem: Rabinowitz’s type potential

In this section, we are going to prove the existence of positive solutions for the nonlocal
Schrödinger equation

−∆u+ V (εx)u = σu+ κ[|x|−µ ∗ FR(u)]fR(u) in R2, (2.1)

under the constraint ∫
R2

|u|2dx = a2, (2.2)

where the potential V : R2 → R satisfies (A8), ε, κ > 0 are parameters, a > 0, σ ∈ R is known as
the Lagrange multiplier and the nonlinearity fR is defined in (1.13).

In general, to solve Problems (2.1)-(2.2), we look for critical points of the variational functional

Jε,R(u) =
1

2

∫
R2

[
|∇u|2 + V (εx)|u|2

]
dx− κ

2

∫
R2

[|x|−µ ∗ FR(u)]FR(u)dx (2.3)

restricted to the sphere S(a) defined by

S(a) =
{
u ∈ H1(R2) :

∫
R2

|u|2dx = a2
}
. (2.4)

Taking advantage of (A8) and (1.15) together with (1.18), it is simple to verify that the functional
Jε,R is of class C1(H1(R2),R) and it derivative is given by

J ′
ε,R(u)v =

∫
R2

[∇u∇v + V (εx)uv] dx− κ

∫
R2

[|x|−µ ∗ FR(u)]fR(u)vdx, ∀u, v ∈ H1(R2).

We note that since V is a positive and bounded function by (A8), then the work space H1(R2)
with its usual norm ∥ · ∥H1 will be adopted for simplicity in the present section.

The existence result concerning the Problems (2.1)-(2.2) is the following.



EJDE-2025/34 NORMALIZED SOLUTIONS FOR CHOQUARD EQUATIONS 7

Theorem 2.1. Suppose (A1)–(A3), (A8), (1.3) holda and µ ∈ (0, 2). then there exists an R∗ > 0
such that for all R > R∗, there exist a∗ = a∗(R) > 0 and ε∗ = ε∗(R) > 0 such that, for each fixed
κ ∈ (0, 1), a > a∗ and ε ∈ (0, ε∗), the minimization problem

Υε,R(a) := min
u∈S(a)

Jε,R(u) (2.5)

can be attained by some function in H1(R2). Hence, there is (uR, σR) ∈ H1(R2)× R such that it
is a couple solution of Problems (2.1)-(2.2), where uR(x) > 0 for all x ∈ R2 and σR < 0.

The proof of the above theorem will be divided into several lemmas. Before exhibiting them,
we will always suppose that the potential V and the nonlinearity fR satisfy (A8) and (1.3) with
(A1)–(A3) in this section.

Lemma 2.2. For all fixed R > 0, the variational functional Jε,R is coercive and bounded from
below on S(a) for each κ ∈ (0, 1), a > 0 and ε > 0, where Jε,R and S(a) are appearing in (2.3)
and (2.4), respectively.

Proof. By (1.14)-(1.15) and (1.18), for all u ∈ S(a), we use (1.17) with l = 4q
4−µ > 2 to obtain

Jε,R(u) ≥
1

2

∫
R2

|∇u|2 dx− C
4−µ
2 Cµf

4−µ(R)a4−µ

2R(4−µ)(q−1)q4−µ

(∫
R2

|∇u|2 dx
)q− 4−µ

2

.

As q ∈
(
2, 6−µ

2

)
, clearly q − 4−µ

2 < 1, then the statement follows. □

As a direct consequence of Lemma 2.2, for every fixed R > 0, κ ∈ (0, 1), a > 0 and ε > 0, the
real number Υε,R(a) in (2.5) is well-defined and it shall be used to look for nontrivial solutions
for Problems (2.1)-(2.2). Alternatively, we need to conclude that Υε,R(a) is uniformly bounded
above with respect to κ ∈ (0, 1) and ε > 0 and so there is the result below.

Lemma 2.3. There exists an R∗ > 0 such that for all fixed R > R∗, there is an a∗ = a∗(R) > 0
satisfying for all a > a∗, there exists a constant ΘR = Θ(R) < 0, independent of ε, such that
Υε,R(a) ≤ ΘR for all κ ∈ (0, 1) and ε > 0.

Proof. According to the definition of fR in (1.13), it holds that

fR(s)

sq−1
=

{ f(s)
sq−1 , 0 ≤ s ≤ R,

f(R)
Rq−1 , R ≤ s < +∞.

(2.6)

Since f satisfies (1.3), we apply (A2) to see that limR→+∞
f(R)
Rq−1 = +∞ which indicates that there

exists an R∗ > 0 such that, for all R > R∗, it holds that f(R)
Rq−1 ≥ c0. As a consequence, owing to

(2.6) and (A3), we arrive at

fR(s) ≥ c0s
q−1, ∀s ≥ 0 and R > R∗. (2.7)

We now fix a positive function ψ ∈ C∞
0 (R2) ∩ S(1); combining (2.7) and (A8), it follows that

Jε,R(tψ) ≤
t2

2

∫
R2

|∇ψ|2dx+
|V |∞
2

t2 − κc20t
2q

2q2

∫
R2

(|x|−µ ∗ |ψ|q)|ψ|qdx→ −∞

as t → +∞, where we have used that V (εx) ≤ |V |∞ for all ε > 0 and x ∈ R2 by (A8). Choosing
a sufficiently large t∗ = t∗(R) > 0 and letting a∗ = t∗|ψ|2, it permits us to look for a constant
ΘR = Θ(R) < 0, dependent of R, such that

Jε,R(u) ≤ ΘR, ∀R > R∗, κ ∈ (0, 1), a > a∗ and ε > 0,

provided u ∈ S(a), as asserted. The proof is complete. □

Similar to [3, 6, 14, 16, 57], we have the following result in the nonlocal case of Choquard type.

Lemma 2.4. Let a2 > a1 > a∗. Then
Υε,R(a2)

a2
2

<
Υε,R(a1)

a2
1

for all fixed R > R∗, κ ∈ (0, 1) and

ε > 0.



8 L. J. SHEN, M. SQUASSINA EJDE-2025/34

Proof. Let ξ > 1 such that a2 = ξa1 and (un) ⊂ S(a1) be a minimizing sequence with respect to
the number Υε,R(a1), that is,

Jε,R(un) → Υε,R(a1) as n→ +∞.

Setting vn = ξun, obviously vn ∈ S(a2). According to (A2), the function t 7→ FR(t)
tq is increasing

on (0,+∞), we obtain the inequality

FR(ts) ≥ tqFR(s), ∀s > 0 and t ≥ 1,

and so, by using Υε,R(a2) ≤ Jε,R(vn) = Jε,R(ξun), we have that

Υε,R(a2) ≤ ξ2Jε,R(un) +
κ

2

∫
R2

{
ξ2[|x|−µ ∗ FR(un)]FR(un)− [|x|−µ ∗ FR(ξun)]FR(ξun)

}
dx

≤ ξ2Jε,R(un) +
κ(ξ2 − ξ2q)

2

∫
R2

[|x|−µ ∗ FR(un)]FR(un)dx.

To continue the proof, we have a claim.

Clame 2.5. There exist a positive constant C > 0, independent of n ∈ N, and a positive integer
n0 ∈ N such that

∫
R2 [|x|−µ ∗ FR(un)]FR(un)dx ≥ C for all n ≥ n0.

Otherwise, there exists a subsequence of (un) ⊂ S(a1), still denoted by itself, such that∫
R2

[|x|−µ ∗ FR(un)]FR(un)dx→ 0 as n→ +∞.

Now, we apply Lemma 2.3 to obtain

ΘR ≥ Υε,R(a1) + on(1) = Jε,R(un) ≥ −κ
2

∫
R2

[|x|−µ ∗ FR(un)]FR(un)dx, n ∈ N,

which is absurd and Claim 2.5 is proved. Thanks to Claim 2.5 and the fact that ξ2 − ξ2q < 0, we
therefore reach

Υε,R(a2) ≤ ξ2Jε,R(un) + κ(ξ2 − ξ2q)C,

for n ∈ N large. Letting n→ +∞, it follows that

Υε,R(a2) ≤ ξ2Υε,R(a1) + κ(ξ2 − ξ2q)C < ξ2Υε,R(a1),

that is,
Υε,R(a2)

a22
<

Υε,R(a1)

a21
,

proving the lemma. □

Lemma 2.6. Let R > R∗, κ ∈ (0, 1) and ε > 0 be fixed, assume (un) ⊂ H1(R2) is a minimizing
sequence associated with Υε,R(a) for a > a∗. Then, there exist bounded sequence (σn) ⊂ R and
σR < 0 such that for some subsequence, still denoted by itself, one has limn→+∞ σn = σR and

∥J ′
ε,R(un)− σnΨ

′(un)∥(H1(R2))−1 → 0 as n→ +∞,

where Ψ : H1(R2) → R is given by

Ψ(u) =
1

2

∫
R2

|u|2dx.

Proof. Setting the functional Ψ : H1(R2) → R given by

Ψ(u) =
1

2

∫
R2

|u|2dx,

we see that S(a) = Ψ−1({a2/2}). Then, by Willem [64, Proposition 5.12], there exists (σn) ⊂ R
such that

∥J ′
ε,R(un)− σnΨ

′(un)∥(H1(R2))−1 → 0 as n→ +∞, (2.8)

Since (un) is bounded in H1(R2), it concludes that (σn) is also a bounded sequence, then we can
assume that σn → σR as n→ +∞ along a subsequence. This together with (2.8) leads to

J ′
ε,R(un)− σRΨ

′(un) = on(1) in (H1(R2))−1.



EJDE-2025/34 NORMALIZED SOLUTIONS FOR CHOQUARD EQUATIONS 9

Now, we prove that σR < 0. First of all, let us recall that∫
R2

[
|∇un|2 + V (εnx)|un|2

]
dx− κ

∫
R2

[|x|−µ ∗ FR(un)]fR(un)undx = σRa
2 + on(1).

Since Jε,R(un) = Υε,R(a) + on(1), one gets

2Υε,R(a) + κ

∫
R2

[|x|−µ ∗ FR(un)][FR(un)− fR(un)un]dx = σRa
2 + on(1).

By (A2), it holds that

2Υε,R(a) +
(1
q
− 1

)
κ

∫
R2

[|x|−µ ∗ FR(un)]fR(un)undx ≥ σRa
2 + on(1).

As f(s)s ≥ 0 for all s ∈ R, q > 2, we obtain

2Υε,R(a) ≥ σRa
2.

Now, according to Υε,R(a) ≤ ΘR < 0 for every R > R∗, κ ∈ (0, 1), a > a∗ and ε > 0 by Lemma
2.3, it follows that σR < 0. The proof is complete. □

Our next result is a compactness theorem on S(a) and then it is possible to find a minimizer
for Υε,R(a).

Theorem 2.7. Let R > R∗, κ ∈ (0, 1) and ε > 0 be fixed as above. Suppose that (un) ⊂ S(a) is
a minimizing sequence of Υε,R(a) for each fixed a > a∗, then un ⇀ u in H1(R2) as n → ∞. If
u ̸= 0, then un → u in H1(R2) along a subsequence as n→ ∞.

Proof. Since Jε,R is coercive on S(a), the sequence (un) is bounded, and so, un ⇀ u in H1(R2)
for some subsequence. If u ̸= 0 and |u|2 = â ̸= a, we must have â ∈ (0, a). By the Brézis-Lieb
Lemma (see [64]),

|un|22 = |un − u|22 + |u|22 + on(1).

Furthermore, arguing as (1.19) to see that

lim
n→∞

∫
R2

{[|x|−µ∗FR(un)]FR(un)−[|x|−µ∗FR(un−u)]FR(un−u)}dx =

∫
R2

[|x|−µ∗FR(u)]FR(u)dx.

Setting vn = un − u, dn = |vn|2 and supposing that |vn|2 → d, we reach a2 = â2 + d2. From
dn ∈ (0, a) for n large enough,

Υε,R(a) + on(1) = Jε,R(un) = Jε,R(vn) + Jε,R(u) + on(1) ≥ Υε,R(dn) + Υε,R(â) + on(1).

thereby, by Lemma 2.4,

Υε,R(a) + on(1) ≥
d2n
a2

Υε,R(a) + Υε,R(â) + on(1).

Letting n→ +∞, one finds

Υε,R(a) ≥
d2

a2
Υε,R(a) + Υε,R(â). (2.9)

Since â ∈ (0, a), employing Lemma 2.4 in (2.9) again, we arrive at the following inequality

Υε,R(a) >
d2

a2
Υε,R(a) +

â2

a2
Υε,R(a) =

(d2
a2

+
â2

a2

)
Υε,R(a) = Υε,R(a),

which is absurd. This asserts that |u|2 = a, or equivalently, u ∈ S(a).
As |un|2 = |u|2 = a, un ⇀ u in L2(R2) and L2(R2) is reflexive, it is well-known that

un → u in L2(R2). (2.10)

This combined with interpolation theorem in the Lebesgue space and (1.14)-(1.15) gives∫
R2

[|x|−µ ∗ FR(un)]FR(un)dx→
∫
R2

[|x|−µ ∗ FR(u)]FR(un)dx. (2.11)

These limits together with Υε,R(a) = limn→+∞ Jε,R(un) provide

Υε,R(a) ≥ Jε,R(u).



10 L. J. SHEN, M. SQUASSINA EJDE-2025/34

As u ∈ S(a), we infer that Jε,R(u) = Υε,R(a), then

lim
n→+∞

Jε,R(un) = Jε,R(u),

that combines with (2.10) and (2.11) to give

∥un∥2H1 → ∥u∥2H1 ,

The last limit permits to conclude that un → u in H1(R2). The proof is complete. □

As we can observe that it is crucial to verify that the weak limit u ̸= 0 before exploiting the
compact result established in Theorem 2.7. To arrive at it, we need to introduce the following
variational functionals J0,R and J∞,R defined by

J0,R(u) =
1

2

∫
R2

[
|∇u|2 + V0|u|2

]
dx− κ

2

∫
R2

[|x|−µ ∗ FR(u)]FR(u)dx,

J∞,R(u) =
1

2

∫
R2

[
|∇u|2 + V∞|u|2

]
dx− κ

2

∫
R2

[|x|−µ ∗ FR(u)]FR(u)dx,

(2.12)

restricted to the sphere S(a) defined in (2.4). One easily sees that J0,R, J∞,R ∈ C1(H1(R2),R)
and

J ′
0,R(u)v =

∫
R2

(∇u∇v + V0uv) dx− κ

∫
R2

[|x|−µ ∗ FR(u)]fR(u)vdx,

J ′
∞,R(u)v =

∫
R2

(∇u∇v + V∞uv) dx− κ

∫
R2

[|x|−µ ∗ FR(u)]fR(u)vdx,

for all u, v ∈ S(a). We also need to consider the minimization problems

Υ0,R(a) = min
u∈S(a)

J0,R(u)

Υ∞,R(a) = min
u∈S(a)

J0,∞(u).
(2.13)

Owing to the definitions of Υ0,R(a) and Υ∞,R(a), by (A8), it is clear to check that

Υ0,R(a) < Υ∞,R(a), ∀R > R∗, κ ∈ (0, 1) and a > a∗. (2.14)

Lemma 2.8. If R > R∗, κ ∈ (0, 1) and ε > 0 are fixed, then it holds that limε→0+ Υε,R(a) =
Υ0,R(a) for all a > a∗. Particularly, there is a small ε∗ = ε∗(R) > 0 such that Υε,R(a) < Υ∞,R(a)
for all ε ∈ (0, ε∗).

Proof. To begin wit a claim.

Clame 2.9. If R > R∗, κ ∈ (0, 1) and ε > 0 are fixed, there is a U0 ∈ H1(R2) such that

U0 ∈ S(a) and J0,R(U0) = Υ0,R(a), ∀a > a∗.

Indeed, we suppose that (Un) ⊂ S(a) is a minimizing sequence of Υ0,R(a). Similar to Lemma
2.2, (Un) is bounded and there is a Ū0 such that Un ⇀ Ū0 along a subsequence. It follows from
the Vanishing lemma, c.f. [64, Lemma 1,21], that there are ρ > 0 and (zn) ⊂ R2 such that

lim inf
n→∞

∫
Bρ(yn)

|Un|2dx > 0.

Otherwise, Un → 0 in Lp(R2) for every p ∈ (2,+∞) which together with (1.18) and (1.14)-(1.15)
implies that limn→∞

∫
R2 [|x|−µ ∗ FR(Un)]FR(Un)dx = 0. So, Υ0,R(a) = limn→∞

∫
R2 |∇Un|2dx ≥ 0

but it cannot occur using a similar arguments in Lemma 2.3. Now, we define Ūn := qUn(·+zn) and
it is still a minimizing sequence of Υ0,R(a). Hence, Ūn ⇀ U0 ̸= 0 in H1(R2) along a subsequence,
and then the Claim is proved by Theorem 2.7.

Since U0 ∈ S(a) in Claim 2.9, we arrive at

Υε,R(a) ≤ Jε,R(U0) =
1

2

∫
RN

(|∇U0|2 + V (εx)|U0|2)dx− κ

2

∫
R2

[|x|−µ ∗ FR(U0)]FR(U0)dx.



EJDE-2025/34 NORMALIZED SOLUTIONS FOR CHOQUARD EQUATIONS 11

Taking the limit as ε → 0+ and recalling V (0) = infz∈R2 V (z) = V0, the we use the Lebesgue’s
Dominated Convergence theorem as well as Claim 2.9 to obtain

lim sup
ϵ→0+

Υε,R(a) ≤ J0,R(U0) = Υ0,R(a). (2.15)

On the other hand, by (A8), one finds that

J0,R(u) ≤ Jε,R(u), ∀u ∈ H1(R2),

implying that

Υ0,R(a) ≤ Υε,R(a), ∀ε > 0.

Therefore,

Υ0,R(a) ≤ lim inf
ε→0+

Υε,R(a). (2.16)

From (2.15) and (2.16), it holds that

lim
ε→0+

Υε,R(a) = Υ0,R(a).

The limit above combined with (2.14) yields that there is a ε∗ > 0 such that Υ0,R(a) < Υ∞,R(a)
for all ε ∈ (0, ε∗). The proof is complete. □

Lemma 2.10. If R > R∗, κ ∈ (0, 1) and ε > 0 are fixed. Assume (un) ⊂ S(a) is a minimizing
sequence with respect to Υε,R(a) for all a > a∗, then there is a function u ∈ H1(R2) such that
un ⇀ u along a subsequence in H1(R2). Moreover, we have that u ̸= 0 provided that ε ∈ (0, ε∗).

Proof. The first part is a direct consequence of Lemma 2.2 and hence we omit it here. Suppose
by the contradiction that un ⇀ 0 in H1(R2). Then

Υε,R(a) + on(1) = Jε,R(un) = J∞,R(un) +
1

2

∫
R2

[V (εx)− V∞]|un|2dx.

From (A8), for each η > 0, there is a sufficiently large ρ > 0 such that

V (z) ≥ V∞ − η for |z| ≥ ρ.

Thereby, in view of (un) ⊂ S(a),

Υε,R(a) + on(1) ≥ Υ∞,R(a)− ηa2 +

∫
Oρ

[V (εx)− V∞]|un|2dx,

where Oρ := {z ∈ R2 : |z| < ε−1ρ}. Letting n→ ∞ and then tending η → 0+, we derive

Υε,R(a) ≥ Υ∞,R(a),

which contradicts with Lemma 2.8. The proof is complete. □

At this stage, we can show the detailed proof of Theorem 2.1.

Proof of Theorem 2.1. Using Lemma 2.2, we can choose a minimizing sequence (un) ⊂ S(a) as-
sociated with Υε,R(a) and there is a uR ∈ H1(R2) such that un ⇀ uR in H1(R2). In light of the
suitable R > R∗, κ ∈ (0, 1) and ε ∈ (0, ε∗), we can rely on Theorem 2.7 and Lemma 2.10 to see
that un → uR in H1(R2) and so uR is a minimizer of Υε,R(a) for every R > R∗, κ ∈ (0, 1) and
ε ∈ (0, ε∗) whenever a > a∗. Thanks to the Lagrange multiplier theorem, there is a σR ∈ R such
that (uR, σR) is a couple of weak solutions to (2.1), where σR < 0 follows directly by Lemma 2.6.
We clearly know that uR ≥ 0 by (A1); then some very similar arguments adopted in [48] reveal
that uR(x) > 0 for all x ∈ R2. The proof is complete. □

Let us finish this section by exhibiting the following theorem.

Theorem 2.11. Let uR be given as in Theorem 2.1, if zε denotes the global maximum of uR,
then, up to a subsequence if necessary,

lim
ε→0+

V (εzε) = V0.



12 L. J. SHEN, M. SQUASSINA EJDE-2025/34

Proof. Let εn → 0+, we relabel uR as un to be a solution of the problem

−∆u+ V (εnx)u = σu+ κ[|x|−µ ∗ F (u)]f(u) in R2,∫
R2

|u(x)|2dx = a2,
(2.17)

for some σ = σn ≤ 2
a2Υεn,R(a) by Lemma 2.6. From Lemma 2.8, we know that

lim
n→∞

Jεn,R(un) = lim
n→∞

Υεn,R(a) = Υ0,R(a). (2.18)

Clame 2.12. There exists a sequence (ȳn) ⊂ R2 such that vn = un(· + ȳn) contains a strongly
convergent subsequence in H1(R2). Moreover, up to a subsequence if necessary, yn = εnȳn → y
as n→ ∞, where V (y) = V0 = infz∈R2 V (x).

Indeed, there are ρ > 0, β > 0 and (ȳn) ⊂ R2 such that∫
Bρ(ȳn)

|un|2dx ≥ β, ∀n ∈ N. (2.19)

Otherwise, one has that un → 0 in Lp(R2) for all 2 < p < +∞ which together with (1.14)-(1.15)
and (1.18) implies that limn→∞

∫
R2 [|x|−µ ∗ FR(un)]FR(un)dx = 0. As a consequence, by means

of (2.18), we derive Υ0,R(a) = limn→∞
∫
R2 |∇un|2dx ≥ 0 violating Lemma 2.3. Thereby, (2.19)

holds and we could fix vn = un(· + ȳn). There is a v ̸= 0 such that vn ⇀ v in H1(R2) along a
subsequence. Since (vn) ⊂ S(a) and Jεn,R(un) ≥ J0,R(un) = J0,R(vn) ≥ Υ0,R(a), then one can
invoke from (2.18) that (vn) is a minimizing sequence of Υ0,R(a). It is very similar to Theorem
2.7 that vn → v in H1(R2) along a subsequence. Next, we shall verify that (yn) is bounded in
n ∈ N. Suppose, by contradiction, that |yn| → +∞ and so

Υ0,R(a) = lim
n→∞

{1

2

∫
R2

[
|∇vn|2 + V (εnx+ yn)|vn|2

]
dx− κ

2

∫
R2

[|x|−µ ∗ FR(vn)]FR(vn)dx
}

=
1

2

∫
R2

(
|∇v|2 + V∞|v|2

)
dx− κ

2

∫
R2

[|x|−µ ∗ FR(v)]FR(v)dx

≥ Υ∞,R(a)

which is absurd by (2.14), where we have used (2.18) and vn → v in H1(R2). Thus, passing to a
subsequence if necessary, we can assume that yn → y in R2. A similar argument shows that

Υ0,R(a) =
1

2

∫
R2

[
|∇v|2 + V (y)|v|2

]
dx− κ

2

∫
R2

[|x|−µ ∗ FR(v)]FR(v)dx ≥ ΥV (y),R(a).

If V (y) > V0, as the byproduct of Theorem 2.7, we could conclude that ΥV (y),R(a) > Υ0,R(a). So,
we must have that V (y) = V0 proving the Claim.

Recalling (2.17), (vn) ⊂ S(a) is a sequence of solutions to the equation

−∆u+ V (εnx+ yn)u = σnu+ κ[|x|−µ ∗ FR(u)]fR(u) in R2

with

lim
n→∞

σn ≤ 2

a2
Υεn,R(a) ≤ ΘR < 0.

Owing to Claim 2.12, the same arguments explored in [4, Lemma 4.3] become available in this
scenario to verify that

lim
n→∞

vn(x) = 0 uniformly in n ∈ N.

From which, given a τ > 0, there are some ρ0 > 0 and n0 ∈ N such that

vn(x) ≤ τ, ∀|x| ≥ ρ0 and n ≥ n0.

Clearly, it holds that |vn|∞ ̸→ 0. In fact, we can derive from (2.19) that |vn|2∞ ≥ βmeas(Bρ(0)).
At this stage, let us fix τ > 0 such that |vn|∞ ≥ 2τ and let ŷn ∈ R2 satisfy vn(ŷn) = |vn|∞
for all n ∈ N. Therefore, according to the above discussions, it holds |ŷn| ≤ ρ0 for all n ∈ N.
Furthermore, if we denote zn by un(zn) = |un|∞ for all n ∈ N, then zn = ŷn + ȳn and

lim
n→∞

V (εnzn) = lim
n→∞

V (εnŷn + εnȳn) = lim
n→∞

V (εnŷn + yn) = V (y) = V0



EJDE-2025/34 NORMALIZED SOLUTIONS FOR CHOQUARD EQUATIONS 13

completing the proof. □

3. Truncated problem: steep potential well

In this section, we shall conclude the existence of positive solutions for the nonlocal Schrödinger
equation

−∆u+ λV (x)u = σu+ κ[|x|−µ ∗ FR(u)]fR(u) in R2, (3.1)

under the constraint ∫
R2

|u|2dx = a2, (3.2)

where the potential V : R2 → R satisfies the assumptions (A9)–(A11), λ, κ > 0 are parameters,
a > 0, σ ∈ R is known as the Lagrange multiplier and the nonlinearity fR is defined in (1.13).

Before solving Problems (3.1)-(3.2), we have to determine a suitable work space. Proceeding
as [53, 56, 58], given a fixed λ > 0, by (A9), we define the space

Eλ :=
{
u ∈ L2

loc(R2) : |∇u| ∈ L2(R2) and

∫
R2

λV (x)|u|2dx < +∞
}

which is indeed a Hilbert space equipped with the inner product and norm

(u, v)Eλ
=

∫
R2

[
∇u∇v + λV (x)uv

]
dx, ∥u∥Eλ

=
√

(u, u)Eλ
, ∀u, v ∈ Eλ.

From here onwards, we shall denote E and ∥ · ∥E by Eλ and ∥ · ∥Eλ
for λ = 1, respectively. It is

simple to observe that ∥ · ∥E ≤ ∥ · ∥Eλ
for every λ ≥ 1. Therefore, owing to [56, Lemma 2.4], Eλ

could be continuously imbedded into H1(R2) for all λ ≥ 1.
Define the variational functional Jλ,R : Eλ → R by

Jλ,R(u) =
1

2

∫
R2

[
|∇u|2 + λV (x)|u|2

]
dx− κ

2

∫
R2

[|x|−µ ∗ FR(u)]FR(u)dx. (3.3)

Obviously, combining (1.15) and (1.18), one can easily show that Jλ,R belongs to C1(Eλ,R) and
it derivative is

J ′
λ,R(u)v =

∫
R2

[∇u∇v + λV (x)uv] dx− κ

∫
R2

[|x|−µ ∗ FR(u)]fR(u)vdx, ∀u, v ∈ Eλ.

To solve Problems (3.1)-(3.2), we consider the minimization problem

Ῡλ,R(a) = min
u∈S(a)

Jλ,R(u), (3.4)

where, with λ ≥ 1, the sphere in defined by

S(a) =
{
u ∈ H1(R2) :

∫
R2

|u|2dx = a2
}
. (3.5)

The existence result for Problems (3.1)-(3.2) in this section can be stated as follows.

Theorem 3.1. Suppose rm (A1)–(A3) , (A9)–(A11) and (1.3) hold, and µ ∈ (0, 2), then there is
an R∗ > 0 such that for every R > R∗, there exist some a∗ = a∗(R) > 0 and λ∗ = λ∗(R) > 1 such
that, for all κ ∈ (0, 1), a > a∗ and λ > λ∗, the minimization problem (3.4) can be achieved by
some function in Eλ. Moreover, there is (uR, σR) ∈ H1(R2)× R such that it is a couple solution
of Problems (3.1)-(3.2), where uR(x) > 0 for all x ∈ R2 and σR < 0.

Arguing as we did in Section 2, we introduce several lemmas to prove Theorem 3.1. For
simplicity, when there is no misunderstanding, we will also suppose that the potential V and the
nonlinearity fR satisfy (A9)–(A11) and (1.3) with (A1)–(A3) in this section.

Using the same calculations as in the proof of Lemma 2.2, we have the following result.

Lemma 3.2. For all fixed R > 0, the variational functional Jλ,R is coercive and bounded from
below on S(a) for each κ ∈ (0, 1), a > 0 and λ ≥ 1, where Jλ,R and S(a) are appearing in (3.3)
and (3.5), respectively.



14 L. J. SHEN, M. SQUASSINA EJDE-2025/34

The proof of the above lemms is the same as that of Lemma 2.2 and so we omit it here.
Employing some necessary modifications in the proof of Lemma 2.3, we are able to conclude the
following lemma.

Lemma 3.3. There exists an R∗ > 0 such that for all fixed R > R∗, there is an a∗ = a∗(R) > 0
satisfying for all a > a∗, there exists a constant Θ̄R = Θ̄(R) < 0, independent of λ, such that
Ῡλ,R(a) ≤ Θ̄R for all κ ∈ (0, 1) and λ ≥ 1.

Proof. The main idea originates from [58, Lemma 3.3], we show the details for the convenience
of the reader. Without loss of generality, we are assuming that 0 ∈ intV −1(0). Therefore, there
exists a sufficiently small r > 0 such that Br(0) ⊂ intV −1(0). Choose ψ ∈ C∞

0 (Br(0)) to be a
function satisfying

∫
Br(0)

|ψ|2dx = 1 and so ψ ∈ S(1). Thanks to the definition of Ω, it holds that∫
R2

V (x)|ψ|2dx =

∫
Ω

V (x)|ψ|2dx+

∫
Ωc

V (x)|ψ|2dx = 0. (3.6)

Proceeding as the proof of Lemma 2.3, we could determine a sufficiently large t∗ = t∗(R) > 0 and
then t∗ = t∗|ψ|2 to find a constant Θ̄R < 0, dependent of R, such that

Jλ,R(u) ≤ Θ̄R, ∀R > R∗, κ ∈ (0, 1), a > a∗ and λ ≥ 1,

provided u ∈ S(a). The proof is complete. □

Owing to the essential feature of steep potential well, there is no need to certify the similar
result in Lemma 2.4. In other words, we shall conclude the counterpart of Theorem 2.7 directly.

Theorem 3.4. Let R > R∗, κ ∈ (0, 1) and λ ≥ 1 be fixed. Suppose (un) ⊂ S(a) is a minimizing
sequence of Ῡλ,R(a) for all a > a∗, then un ⇀ u in Eλ as n → ∞. If in addition u ̸= 0, there is
a sufficiently large λ′∗ = λ′∗(R) > 1 such that un → u in Eλ along a subsequence as n→ ∞ for all
λ > λ′∗.

Proof. The first part is the same as its counterpart in Theorem 2.7, and we omit it here. To derive
the remaining part, we define vn := un−u ⇀ 0 in Eλ. Let us recall from (A11) that the nonempty
set Ξ := {x ∈ R2 : V (x) < b} has finite measure, then∫

R2

|vn|2dx =

∫
R2\Ξ

|vn|2dx+

∫
Ξ

|vn|2dx+ =

∫
R2\Ξ

|vn|2dx+ on(1)

≤ 1

λb

∫
R2\Ξ

λV (x)|vn|2dx+ on(1) ≤
1

λb
∥vn∥2Eλ

+ on(1)

which together with (1.17) with l = 4q
4−µ > 2 gives that∫

R2

|vn|ldx ≤ C(λb)−
(1−γl)l

2 ∥vn∥lEλ
+ on(1).

From the inequality above, combining (1.14)-(1.15) and (1.18), we see that

κ

∫
R2

[|x|−µ ∗ FR(vn)]FR(vn)dx ≤ C
4−µ
2 Cµf

4−µ(R)

q4−µR(4−µ)(q−1)
(λb)−

4−µ
2 ∥vn∥2qEλ

+ on(1). (3.7)

On the other hand, obviously |vn|2 ∈ (0, a), then Lemma 3.3 indicates that Ῡλ,R(|vn|2) ≤ 0.
Moreover, ∥vn∥Eλ

is bounded, namely there exists a ζ = ζ(R) > 0 such that ∥vn∥Eλ
≤ ζ for all

n ∈ N. Combining these facts jointly with (3.7), it holds that

0 ≥
[1
2
− C

4−µ
2 Cµf

4−µ(R)

2q4−µR(4−µ)(q−1)
(λb)−

4−µ
2 ζ2(q−1)

]
∥vn∥2Eλ

+ on(1). (3.8)

Consequently, we shall determine a sufficiently large λ′∗ = λ′∗(R) > 1 to satisfy ∥vn∥2Eλ
= on(1)

whenever λ > λ′∗. The proof is complete. □

To apply Theorem 3.4 successfully, we need the following lemma to show that the weak limit
of u is not 0.



EJDE-2025/34 NORMALIZED SOLUTIONS FOR CHOQUARD EQUATIONS 15

Lemma 3.5. Under the assumptions of Theorem 3.4, there exists a λ∗ = λ∗(R) > λ′∗ such that
u ̸= 0 for all λ > λ∗.

Proof. We collect the methods used in [53, 56, 58] to reach the proof. Firstly, we have a claim.

Clame 3.6. For some q0 ∈ (2,+∞), there exists a constant β0 > 0, independent of λ ≥ 1, such
that

lim
n→∞

sup
z∈R2

∫
Bρ(z)

|un|q0dx = β0.

To demonstrate this Claim, we can suppose that there exists a constant βλ = β(λ) > 0 such
that limn→∞ supy∈R2

∫
Bϱ(y)

|un|q0dx = βλ. Otherwise, un → 0 in Ls(R2) for every s ∈ (2,+∞)

jointly with (1.14)-(1.15) and (1.18) yields that limn→∞
∫
R2 [|x|−µ ∗FR(un)]FR(un)dx = 0. Hence,

we can conclude that Ῡλ,R(a) = limn→∞ Jλ,R(un) ≥ 0 and it is impossible because of Lemma 3.3.
With such a βλ, we are able to verify this Claim. Suppose, by contradiction, that the uniform
control from below of Lq0(R2)-norm is false. Consequently, for any k ∈ N, k ̸= 0, there are λk > 1
and a minimizing sequence (uk,n) of Ῡλk,R such that

|uk,n|q0 <
1

k
, definitely.

Then, by a diagonalization argument, for any k ≥ 1, it permits us to find an increasing sequence
(nk) ⊂ N and (unk

) ⊂ Eλnk
such that

(unk
) ⊂ S(a), Jλnk

,R(unk
) = Ῡλnk

,R(a) + ok(1) and |unk
|q0 = ok(1).

where ok(1) → 0 as k → +∞. In this situation, we could repeat the calculations above to reach a
contradiction Ῡλnk

,R(a) ≥ 0, again. So, the Claim is proved.

Thanks to Claim 3.6, there exist a sequence (zn) ⊂ R2 and a subsequence (un), still denoted
by itself, such that ∫

Bρ(zn)

|un|2dx =
1

2
β0. (3.9)

Clame 3.7. The sequence (zn) above is uniformly bounded in n ∈ N.

Otherwise, we suppose by contradiction to choose a subsequence if necessary that |zn| → ∞.
Define

Ξ1
n := {x ∈ Bρ(zn) : V (x) < b} and Ξ2

n := {x ∈ Bρ(zn) : V (x) ≥ b}.
Since the set Ξ := {x ∈ R2 : V (x) < b} is nonempty and has finite measure, one concludes that

meas(Ξ1
n) ≤ meas({x ∈ R2 : |x| ≥ |yn| − 2, V (x) < b}) → 0 as n→ ∞.

For λ ≥ 1, one sees |un|r with r > 2 is uniformly bounded in n ∈ N by Lemma 3.2 and then∫
Ξ1

n

|un|2dx ≤ [meas(Ξ1
n)]

r−2
r |un|2r = on(1)

which together with (3.9) reveals that∫
Ξ2

n

|un|2dx =

∫
Bρ(zn)

|un|2dx−
∫
Ξ1

n

|un|2dx =
1

2
β0 + on(1).

Thanks to V (x) ≥ 0 for all x ∈ R2 by (A9), using the definition of Ξ2
n, we obtain∫

R2

V (x)|un|2dx ≥
∫
Ξ2

n

V (x)|un|2dx ≥ b

∫
Ξ2

n

|un|2dx =
1

2
bβ0 + on(1). (3.10)

It follows from (1.14)-(1.15) and (1.18) that

sup
n∈N

{∫
R2

[|x|−µ ∗ FR(un)]FR(un)dx
}
≤ C, ∀λ ≥ 1, (3.11)

where C > 0 is independent of n ∈ N and λ ≥ 1. So, we deduce by (3.10) and (3.11) that

Ῡλ,R(a) ≥
1

2

∫
R2

λV (x)|un|2dx− C + on(1) ≥
λbβ0
4

− C + on(1) (3.12)



16 L. J. SHEN, M. SQUASSINA EJDE-2025/34

where the positive constants b, β0 and C are independent of λ ≥ 1. Adopting Lemma 3.3 again,
there is a sufficiently large λ∗ = λ∗(R) > λ′∗(R) such that (3.12) is impossible provided λ > λ∗.
Hence, the Claim is proved.

Owing to Claim 3.7, passing to a subsequence if necessary, we suppose that zn → z0 in R2.
Since un → u in L2

loc(R2), then we can arrive at the proof of this lemma. □

Proof of Theorem 3.1. There is a minimizing sequence (un) ⊂ S(a) associated with Ῡλ,R(a) by
Lemma 3.2 and thus un ⇀ uR in Eλ for some λ ≥ 1. As a consequence of Theorem 3.4 and
Lemma 3.5, for all R > R∗, κ ∈ (0, 1), a > a∗ and λ > λ∗, we see that un → uR in Eλ and so uR is
a minimizer of Ῡλ,R(a). By exploiting the Lagrange multiplier theorem again, a similar argument
in Lemma 2.6 makes sure a σR < 0 that (uR, σR) is a couple of weak solutions to (3.1). Finally,
the reader can derive uR > 0 as in Theorem 2.1. The proof is complete. □

As we can observe from the proof of Theorem 3.1, the couple (uR, σR) exists for all R > R∗,
κ ∈ (0, 1), a > a∗ and λ > λ∗. In other words, if R > R∗, κ ∈ (0, 1) and a ∈ (0, a∗) are fixed, the
couple (uR, σR) would also rely on λ > λ∗. It is therefore that we shall relabel it by (uλ, σλ) when
R > R∗, κ ∈ (0, 1) and a ∈ (0, a∗) are fixed.

Letting λ→ +∞, we have the following result.

Theorem 3.8. Let (uλ, σλ) ∈ Eλ × R denote by the couple of weak solutions established above
for all λ > λ∗, passing to a subsequence if necessary, uλ → u0 in H1(R2) and σλ → σ0 in R as
λ→ +∞, where σ0 < 0 and (u0, σ0) is a couple of weak solution to the problem

−∆u = σu+ κ
(∫

Ω

FR(u(y))

|x− y|µ
dy

)
fR(u), x ∈ Ω,

u(x) = 0, x ∈ ∂Ω,∫
Ω

|u|2dx = a2.

(3.13)

Proof. Let λn → +∞, we study the subsequence of (uλ, σλ) ∈ Eλ ×R, namely (uλn
, σλn

) satisfies
(uλn

) ⊂ S(a) and Jλn,R(uλn
) = Ῡλn,R. By Lemma 3.3, the sequence (uλn

) is uniformly bounded
in n ∈ N. Similar to the proof of Lemma 2.6, it holds

σλn =
1

a2

{∫
R2

[
|∇un|2 + λnV (x)|un|2

]
dx− κ

∫
R2

[|x|−µ ∗ FR(un)]fR(un)undx
}
+ on(1)

showing that (σλn
) is uniformly bounded in n ∈ N. Up to a subsequence if necessary, uλn

⇀ u0
in H1(R2) and σλn → σ0 in R as n→ +∞. In view of Lemmas 2.6 and 3.3 again, it holds that

σ0 = lim
n→∞

σλn
≤ lim

n→∞

2

a2
Ῡλn,R(a) ≤ Θ̄R < 0.

Clame 3.9. u0 ≡ 0 in Ωc := R2\Ω and so u0 ∈ SΩ(a) :=
{
u ∈ H1

0 (Ω) :
∫
Ω
|u|2dx = a2

}
.

Otherwise, there exists a compact subset Ω̂u0
⊂ Ωc with dist(Ω̂u0

, ∂Ωc) > 0 such that u0 ̸= 0

on Ω̂u0
and by Fatou’s lemma

a2 = lim inf
n→∞

∫
R2

u2λn
dx ≥

∫
Θ̂u0

u0
2dx > 0. (3.14)

Moreover, there exists ζ0 > 0 such that V (x) ≥ ζ0 for every x ∈ Ω̂u0
by the assumptions (A9) and

(A10). Combining Lemma 3.3, (1.14) and (3.14), we obtain

0 ≥ lim inf
n→∞

Ῡλn,R = lim inf
n→∞

Jλn,R(uλn)

= lim inf
n→∞

{
Jλn,R(uλn)−

1

q

[
J ′
λn,R(uλn)uλn − σλna

2
] }

≥ q − 2

2q
ζ0

(∫
Θ̂u0

u20dx
)
lim inf
n→∞

λn +
σ0
q
a2 = +∞

which is impossible. Consequently, u0 ∈ H1
0 (Ω) by the fact that ∂Ω is smooth. By taking some

similar calculations explored in (3.8) to show uλn
→ u0 in H1(R2) and so u0 ∈ SΩ(a).



EJDE-2025/34 NORMALIZED SOLUTIONS FOR CHOQUARD EQUATIONS 17

Clame 3.10. JΩ,R(u0) = ῩΩ,R(a), where ῩΩ,R := infu∈SΩ(s) JΩ,R(u) and the variational func-

tional JΩ,R : H1
0 (Ω) → R is defined by

JΩ,R(u) =
1

2

∫
Ω

|∇u|2dx− κ

2

∫
Ω

∫
Ω

FR(u(x))FR(u(y))

|x− y|µ
dxdy, ∀u ∈ H1

0 (Ω).

Actually, it is simple to see that SΩ(a) ⊂ S(a) and so ῩΩ,R(a) ≥ Ῡλn,R(a). As a consequence,
it holds ῩΩ,R(a) ≥ lim infn→∞ Ῡλn,R(a). On the other hand, we gather these facts together with
the Fatou’s lemma to obtain

ῩΩ,R(a) ≥ lim inf
n→∞

Ῡλn,R(a) = lim inf
n→∞

Jλn,R(uλn
) ≥ JΩ,R(u0) ≥ ῩΩ,R(a)

proving the Claim.
Finally, we shall prove that J ′

Ω(u0)− σ0u0 = 0 in (H1
0 (Ω))

−1. To see it, for every ψ ∈ C∞
0 (Ω),

Combining (1.20) and σλn
→ σ0, it holds that

lim
n→∞

{
J ′
λn,R(uλn

)ψ − σλn

∫
R2

uλn
ψdx

}
= 0, ∀ψ ∈ C∞

0 (Ω),

we can arrive at the desired result. The proof is complete. □

4. Proofs of main results

In this section, we are concerned with the existence and concentrating behavior of positive
solutions to the nonlocal Schrödinger equation (1.1) under the mass-constraint (1.2).

Firstly, we shall provide some growth conditions with the nonlinearity f and fR which play
foremost roles in this section. It can infer from (A1) and (A2) that

lim
s→0+

fR(s)

s
= 0, lim

s→0+

f(s)

s
= 0. (4.1)

Actually, using (A1) and (A2) with q > 2 again we obtain

0 ≤ lim
s→0+

fR(s)

s
= lim

s→0+

f(s)

s
= lim

s→0+

f(s)

sq−1
sq−2 ≤ f(1) lim

s→0+
sq−2 = 0.

Combining (1.3) and (4.1), given a fixed ε > 0, for every p̄ > 2 and ν > 1, we are able to search

for two constants such that b̃1 = b̃1(p̄, α, ε) > 0 and b̃2 = b̃2(p̄, α, ε) > 0 satisfying

|f(s)| ≤ ε|s|+ b̃1|s|p̄−1(e4πνs
2

− 1), ∀s ∈ R, (4.2)

|F (s)| ≤ ε|s|2 + b̃2|s|p̄(e4πνs
2

− 1), ∀s ∈ R. (4.3)

taking in to accoun that the nonlinearity f has the critical exponential growth at infinity, the
following Trudinger-Moser inequality found in [61, 50, 26] will play a crucial role in this section.

Lemma 4.1. If α > 0 and u ∈ H1(R2), then∫
R2

(eα|u|
2

− 1)dx < +∞.

Moreover, if |∇u|22 ≤ 1, |u|22 ≤ M < +∞ and α < 4π, then there exists Kα,M = K(M,α) such
that ∫

R2

(eα|u|
2

− 1)dx ≤ Kα,M . (4.4)

Now, we are ready to exhibit the detailed proofs of Theorems 1.1 and 1.2. In order to show
them clearly, we shall divide into two subsections.



18 L. J. SHEN, M. SQUASSINA EJDE-2025/34

4.1. Proof of Theorem 1.1. Because of Theorem 2.1, we know that the minimization constant
Υε,R(a) defined in (2.5) can be attained by some nontrivial function in H1(R2) for every fixed
R > R∗, κ ∈ (0, 1), a > a∗ and ε ∈ (0, ε∗). In other words, there is a function uR ∈ H1(R2) such
that

uR ∈ S(a) and Jε,R(uR) = Υε,R(a), ∀R > R∗, κ ∈ (0, 1), a > a∗ and ε ∈ (0, ε∗). (4.5)

Moreover, there is a σR < 0 such that the couple (uR, µR) is a solution of Problems (2.1)-(2.2) for
all R > R∗, κ ∈ (0, 1), a > a∗ and ε ∈ (0, ε∗), where uR(x) > 0 for all x ∈ R2.

According to the introduction, the reader can observe that if uR in (4.5) satisfies |uR|∞ ≤ R,
then uR is in fact a solution of the original (1.1) with σ = σR. Therefore it is posssible to arrive
at the proof of Theorem 1.1. As a consequence, the foremost objection for us is to take the
L∞-estimate on uR. To this aim, we establish the uniform estimate on |∇uR|22 below.

Lemma 4.2. Suppose that V satisfies (A8) and f meets (1.3) with (A1)–(A3). Let uR be given
by (4.5) for each R > R∗, a > a∗ and ε ∈ (0, ε∗), then there exists a κ∗ = κ∗(R) ∈ (0, 1) such that
if κ ∈ (0, κ∗), it holds that |∇uR|22 <

2−µ
2(2+µ)ν2 for every R > R∗, a > a∗ and ε ∈ (0, ε∗), where the

constant ν > 1 is appearing in (4.2) and (4.3).

Proof. Since uR ∈ S(a), we borrow the calculations in Lemma 2.2 to obtain

Jε,R(uR) ≥
1

2

∫
R2

|∇uR|2 dx− κC
4−µ
2 Cµf

4−µ(R)a4−µ

2R(4−µ)(q−1)q4−µ

(∫
R2

|∇uR|2 dx
)q− 4−µ

2

.

Since 2 < q < 6−µ
2 , by means of the Young’s inequality, there is a C1 > 0 independent of R > R∗

such that

κC
4−µ
2 Cµf

4−µ(R)a4−µ

2R(4−µ)(q−1)q4−µ

(∫
R2

|∇uR|2 dx
)q− 4−µ

2

≤ C1

[κC 4−µ
2 Cµf

4−µ(R)a4−µ

2R(4−µ)(q−1)q4−µ

] 2
6−µ−2q

+
1

4

∫
R2

|∇uR|2 dx.

Thereby, for every R > R∗, a > a∗ and ε ∈ (0, ε∗), it holds that

|∇uR|22 ≤ 4Jε,R(uR) + 4C1

[κC 4−µ
2 Cµf

4−µ(R)a4−µ

2R(4−µ)(q−1)q4−µ

] 2
6−µ−2q

.

Assuming that

4C1

[κC 4−µ
2 Cµf

4−µ(R)a4−µ

2R(4−µ)(q−1)q4−µ

] 2
6−µ−2q ≤ 2− µ

2(2 + µ)ν2
,

we arrive at

|∇uR|22 ≤ 4Jε,R(uR) +
2− µ

2(2 + µ)ν2
, ∀R > R∗, a > a∗, ε ∈ (0, ε∗).

In light of Jε,R(uR) = Υε,R(a) ≤ 0 by Lemma 2.3 and (4.5), so it permits us to choose

κ∗ = κ∗(R) := min
{[ 2R(4−µ)(q−1)q4−µ

C
4−µ
2 Cµf4−µ(R)a4−µ

][ 2− µ

8C1(2 + µ)ν2

] 6−µ−2q
2

, 1
}

and then we can complete the proof. □

With Lemma 4.2 in hand, we can derive the following result.

Lemma 4.3. Suppose that V satisfies (A8) and f meets (1.3) with (A1)–(A3). Then, for every
fixed R > R∗, κ ∈ (0, κ∗), a > a∗ and ε ∈ (0, ε∗), there exists a constant C ∈ (0,+∞) which is
independent of R > R∗ and ε ∈ (0, ε∗) such that

Γ(x) := |x|−µ ∗ F (uR) ≤ C,

where uR comes from (4.5).



EJDE-2025/34 NORMALIZED SOLUTIONS FOR CHOQUARD EQUATIONS 19

Proof. Since uR ∈ S(a), adopting Lemma 4.2 and (1.17), there is a constant T ∈ (0,+∞) which
is independent of R > R∗ and ε ∈ (0, ε∗) such that

|u|ll ≤ T, ∀l ∈ (2,+∞). (4.6)

By (4.6), we find a constant C0 ∈ (0,+∞) which is independent of R > R∗ and ε ∈ (0, ε∗) such
that ∫

R2

|uR(y)|2

|x− y|µ
dy =

∫
|x−y|<1

|uR(y)|2

|x− y|µ
dy +

∫
|x−y|≥1

|uR(y)|2

|x− y|µ
dy

≤ C̄µ
(∫

R2

|uR(y)|
2(2+µ)
2−µ dy

) 2−µ
2+µ

+ a2 ≤ C0.

(4.7)

Let us define ūR = ν
√

2(2+µ)
2−µ uR, then uR ∈ S(a) and Lemma 4.2 give us

|ūR|22 =
2ν2a2(2 + µ)

2− µ
and |∇ūR|22 ≤ 1

which together with (4.4) and ν > 1 implies that∫
R2

(e
8πν(2+µ)

2−µ |uR(y)|2 − 1)dy =

∫
R2

(e4πν
−1|ūR(y)|2 − 1)dy ≤ K(a, ν, µ). (4.8)

The above inequality shall determine a constant C1 ∈ (0,+∞) which is independent of R > R∗

and ε ∈ (0, ε∗) to reach∫
|x−y|<1

|uR(y)|p̄(e4πν|uR(y)|2 − 1)

|x− y|µ
dy

≤ Cµ
(∫

R2

|uR(y)|
p̄(2+µ)
2−µ (e

4πν(2+µ)
2−µ |uR(y)|2 − 1)dy

) 2−µ
2+µ

≤ Cµ
(∫

R2

|uR(y)|
2p̄(2+µ)

2−µ dy

) 2−µ
2(2+µ) (∫

R2

(e
8πν(2+µ)

2−µ |uR(y)|2 − 1)dy
) 2−µ

2(2+µ) ≤ C1.

Using similar calculations, we have that∫
|x−y|≥1

|uR(y)|p̄(e4πν|uR(y)|2 − 1)

|x− y|µ
dy ≤

∫
R2

|uR(y)|p̄(e4πν|uR(y)|2 − 1)dy

≤
(∫

R2

|uR(y)|2p̄dy
)1/2(∫

R2

(e8πν|uR(y)|2 − 1)dy
)1/2

≤ C2.

where C2 ∈ (0,+∞) is independent of R > R∗ and ε ∈ (0, ε∗). It follows from these two facts that∫
R2

|uR(y)|p̄(e4πν|uR(y)|2 − 1)

|x− y|µ
dy ≤ C1 + C2. (4.9)

Recalling (4.3) with (4.7) and (4.9), the proof will be done by choosing C = C0 + C1 + C2. □

With the help of the study made above, we can get the estimate for |uR|∞ as follows.

Lemma 4.4. Suppose that V satisfies (A8) and f meets (1.3) with (A1)–(A3). Then, for every
fixed R > R∗, κ ∈ (0, κ∗), a > a∗ and ε ∈ (0, ε∗), there exists a constant M ∈ (0,+∞) which is
independent of R > R∗ and ε ∈ (0, ε∗) such that |uR|∞ ≤M , where uR comes from (4.5).

Proof. In view of the definition of fR in (1.13), one has fR(s) ≤ f(s) and FR(s) ≤ F (s) for all
R > 0 and s ∈ R. Since (uR, σR) with uR(x) > 0 for all x ∈ R2 and σR < 0 is a couple of weak
solution to (2.1), we then apply (A8) and Lemma 4.3 to arrive at

−∆uR + uR ≤ f̄(uR) := uR + Cf(uR) in R2.

Proceeding with calculations similar to those in Lemma 4.3, we are able to prove that |f̄(uR)|2 ≤
K, where K ∈ (0,+∞) is a constant which is independent of R > R∗ and ε ∈ (0, ε∗). It then
follows from the Lax-Milgram theorem that there is a wR ∈ H1(R2) such that

−∆wR + wR = f̄(uR) in R2.



20 L. J. SHEN, M. SQUASSINA EJDE-2025/34

Moreover, it can choose wR to be positive in R2. At this stage, we can follow the methods used in
[10, 12, 13, 15, 56, 58] to complete the proof. For the completeness, we shall exhibit the details.
To this end, we have the claim.

Clame 4.5. For all R > R∗, κ ∈ (0, κ∗), a > a∗ and ε ∈ (0, ε∗), it holds

0 < uR(x) ≤ wR(x), ∀x ∈ R2.

Actually, we define the test function

ϕ(x) := (uR − wR)
+(x) ∈ H1(R2).

Muitiplying by ϕ on both sides of −∆(uR −wR) + (uR −wR) ≤ 0 in R2, we obtain the inequality∫
R2

[∇(uR − wR)∇ϕ+ (uR − wR)ϕ]dx ≤ 0.

An elementary computation gives us∫
R2

[|∇(uR − wR)
+|2 + |(uR − wR)

+|2]dx = 0

yielding the claim.
Owing to Claim 4.5, the proof of this lemma becomes available. From [25, Theorem 9.25] in

invokes that there is a K2 > 0 independent of R > R∗ and ε ∈ (0, ε∗) such that

∥wR∥H2 ≤ K2|fR(uR)|2, ∀R > R∗ and ε ∈ (0, ε∗).

leading to
∥wR∥H2 ≤ K3, ∀R > R∗ and ε ∈ (0, ε∗),

for some K3 > 0 independent of R > R∗ and ε ∈ (0, ε∗). In view of the continuous embedding
H2(R2) ↪→ L∞(R2), there exists K4 > 0 independent of R > R∗ and ε ∈ (0, ε∗) such that

|wR|∞ ≤ K4, ∀R > R∗ and ε ∈ (0, ε∗).

From which, we are derived from Claim 4.5 that

|uR|∞ ≤M, ∀R > R∗ and ε ∈ (0, ε∗).

Consequently, the proof is complete. □

Proof. Proof of Theorem 1.1] According to the above discussions, we can arrive at the first part
of the proof of Theorem 1.1 by fixing R > {R∗,M}, because in this case the function uR ∈ S(a)
is a positive solution of (1.1) with σ = σR < 0 for each κ ∈ (0, κ∗), a > a∗ and ε ∈ (0, ε∗). The
remaining part follows Theorem 2.11 directly. The proof is complete. □

4.2. Proof of Theorem 1.2. Recalling Theorem 3.1, there is a couple (uR, σR) ∈ Eλ×R which
is a weak solution to (3.1) with σ = σR < 0 for every R > R∗, κ ∈ (0, 1), a > a∗ and λ > λ∗,
where uR(x) > 0 for each x ∈ R2. Moreover, it holds that

uR ∈ S(a) and Jλ,R(uR) = Ῡε,R(a), ∀R > R∗, κ ∈ (0, 1), a > a∗ and λ > λ∗. (4.10)

Proceeding as in Subsection 4.1, we are able to conclude the counterparts of Lemmas 4.2, 4.3
and 4.4 as follows. Because there are no essential differences, we just present them without the
detailed proofs.

Lemma 4.6. Suppose that V satisfies (A9)–(A11) and f meets (1.3) with (A1)–(A3). Let uR be
given by (4.10) for each R > R∗, a > a∗ and λ > λ∗, then there exists an κ∗ = κ∗(R) ∈ (0, 1)
such that if κ ∈ (0, κ∗), it holds that |∇uR|22 <

2−µ
2(2+µ)ν2 for every R > R∗, a > a∗ and λ > λ∗,

where the constant ν > 1 is appears (4.2) and (4.3).

Lemma 4.7. Suppose that V satisfies (A9)–(A11) and f requires (1.3) with (A1)–(A3). Then,
for every fixed R > R∗, κ ∈ (0, κ∗), a > a∗ and λ > λ∗, there exists a constant C̄ ∈ (0,+∞) which
is independent of R > R∗ and λ > λ∗ such that

Γ̄(x) := |x|−µ ∗ F (uR) ≤ C̄,

where uR comes from (4.10).



EJDE-2025/34 NORMALIZED SOLUTIONS FOR CHOQUARD EQUATIONS 21

Lemma 4.8. Suppose that V satisfies (A9)–(A11) and f requires (1.3) with (A1)–(A3). Then,
for all fixed R > R∗, κ ∈ (0, κ∗), a > a∗ and λ > λ∗, there is M̄ ∈ (0,+∞) which is independent
of R > R∗ and λ > λ∗ such that |uR|∞ ≤ M̄ , where uR comes from (4.10).

Proof of Theorem 1.2. We are able to fix R > {R∗,M}, and Theorem 3.1 thereby indicates the
first part of the proof of Theorem 1.2 to satisfy that uR ∈ S(a) is a positive solution of (1.1) with
σ = σR < 0 for all κ ∈ (0, κ∗), a > a∗ and λ > λ∗. As for the remaining part of the proof of
Theorem 1.2, we refer to Theorem 3.8. The proof is complete. □

Acknowledgments. L. J. Shen is partially supported by the NSFC (Grant #12201565) and by
the ZJNSF (Grant LMS25A010006). M. Squassina is member of Gruppo Nazionale per l’Analisi
Matematica, la Probabilita e le loro Applicazioni (GNAMPA).

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Liejun Shen

Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China

Email address: ljshen@zjnu.edu.cn

Marco Squassina

Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via della Garzetta 48,
25133, Brescia, Italy

Email address: marco.squassina@unicatt.it


	1. Introduction
	1.1. Notation
	1.2. Two basic facts

	2. Truncated problem: Rabinowitz's type potential
	3. Truncated problem: steep potential well
	4. Proofs of main results
	4.1. Proof of Theorem 1.1
	4.2. Proof of Theorem 1.2
	Acknowledgments

	References