Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 29, pp. 1–10.

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

DOI: 10.58997/ejde.2025.29

ASYMPTOTIC PROFILE OF LEAST ENERGY SOLUTIONS TO

THE NONLINEAR SCHRÖDINGER-BOPP-PODOLSKY SYSTEM

GUSTAVO DE PAULA RAMOS

Abstract. We consider the nonlinear Schrödinger-Bopp-Podolsky system in
R3:

−∆v + v + ϕv = v|v|p−2,

β2∆2ϕ−∆ϕ = 4πv2,

where β > 0 and 3 < p < 6; the unknowns being v and ϕ : R3 → R. We prove

that, as β → 0 and up to translations and subsequences, the least energy
solutions of the above converge to a least energy solution to the nonlinear

Schrödinger-Poisson system in R3:

−∆v + v + ϕv = v|v|p−2,

−∆ϕ = 4πv2.

1. Introduction

We are interested in the asymptotic profile of solutions to the nonlinear Schrödinger-
Bopp-Podolsky (SBP) system in R3 as β → 0+:

−∆v + v + ϕv = v|v|p−2,

β2∆2ϕ−∆ϕ = 4πv2,
(1.1)

where 3 < p < 6 and we want to solve for v, ϕ : R3 → R.
The nonlinear SBP system was introduced in the mathematical literature a few

years ago by d’Avenia & Siciliano in [5], where they established existence/non-
existence results of solutions to the following system in R3 in function of the pa-
rameters p and q ∈ R:

−∆v + ωv + q2ϕv = v|v|p−2,

β2∆2ϕ−∆ϕ = 4πv2,
(1.2)

where β, ω > 0. As for the physical meaning of this system. If v, ϕ : R3 → R solve
(1.2), then v describes the spatial profile of a standing wave

ψ(x, t) := eiωtv(x)

2020 Mathematics Subject Classification. 35J61, 35B40, 35Q55, 45K05.
Key words and phrases. Schrödinger-Bopp-Podolsky system; Schrödinger-Poisson system;

nonlocal semilinear elliptic problem; variational methods; ground state;

Nehari-Pohoz̆aev manifold; Concentration-compactness.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted January 2, 2025. Published March 17, 2025.

1



2 G. D. P. RAMOS EJDE-2025/29

that solves the system obtained by the minimal coupling of the Nonlinear Schrödinger
equation with the Bopp-Podolsky electromagnetic theory and ϕ denotes the ensu-
ing electric potential (for more details, see [5, Section 2]). Since then, there has
been an increasing number of studies about systems related to (1.2). For instance,
[3, 2, 10, 11, 15, 21, 25] addressed the existence of least energy solutions; [7, 12, 13]
considered the mass-constrained problem; [8, 9, 17, 23, 22] obtained sign-changing
solutions; and [4, 6] considered semiclassical states.

As for the asymptotic behavior as β → 0, it is already known that solutions
to a number of problems related to (1.1) converge to solutions of the respective
system obtained by formally considering β = 0. For instance, [5, Theorem 1.3]
proved such a result for radial solutions; [7, Theorem D] extended this conclusion
for least energy solutions to the mass-constrained system for 2 < p < 14/5 and
a sufficiently small mass ρ (notice that these solutions are also radial due to [7,
Theorem C]); [20, Theorem 1.3] showed that solutions to the associated eigenvalue
problem in a bounded smooth domain also have such an asymptotic profile and,
more recently, [4, Theorem 1.7] verified such a behavior for the critical nonlinear
SBP system in the semiclassical regime under the effect of an external effective
potential V : R3 → [0,∞[ which vanishes at a point x0 ∈ R3.

Before explaining our contribution, let us introduce the necessary variational
framework. The function Kβ : R3 \ {0} →]0, 1/β[ defined as

Kβ(x) :=
1

|x|
(
1− e−|x|/β)

is a fundamental solution to (4π)−1(β2∆2 −∆), so u2 ∗ Kβ solves

β2∆2ϕ−∆ϕ = 4πu2

in the sense of distributions. As such, we are lead to consider the nonlinear SBP
equation in R3:

−∆v + v + (v2 ∗ Kβ)v = v|v|p−2. (1.3)

We say that v is a least energy solution to (1.3) when it solves the minimization
problem

Iβ(u) = inf{Iβ(v) : v ∈ H1 \ {0} and I ′
β(v) = 0}; u ∈ H1,

where the energy functional Iβ : H1 → R is defined as

Iβ(v) =
1

2
∥v∥2H1 +

1

4

∫
(v2 ∗ Kβ)(x)v(x)

2dx− 1

p
∥v∥pLp .

For a proof that Iβ is a well-defined functional of class C1 and a rigorous discussion
about the relationship between (1.1) and (1.3), we refer the reader to [5, Section
3.2].

Given x ∈ R3 \ {0}, Kβ(x) → 1/|x| as β → 0, so the formal limit equation
obtained from (1.3) is the nonlinear Schrödinger-Poisson equation

−∆v + v +
(
v2 ∗ | · |−1

)
v = v|v|p−2. (1.4)

We similarly introduce a notion of least energy solution to (1.4) by considering the
energy functional I0 : H1 → R given by

I0(v) :=
1

2
∥v∥2H1 +

1

4

∫∫
v(x)2v(y)2

|x− y|
dxdy − 1

p
∥v∥pLp .



EJDE-2025/29 ASYMPTOTIC PROFILE OF LEAST ENERGY SOLUTIONS 3

In [2, Theorem 1.3], Chen, Li, Rădulescu & Tang proved that (1.3) admits
least energy solutions, while it follows from Azzollini & Pomponio’s [1] that (1.4)
also admits least energy solutions. In this context, our main result is that, up to
translations and subsequences, least energy solutions to (1.3) converge to a least
energy solution to (1.4) as β → 0 when 3 < p < 6.

Theorem 1.1. Suppose that 3 < p < 6 and given β > 0, vβ denotes a least energy
solution to (1.3). Then given a sequence {βn}n∈N ⊂]0,∞[ such that βn → 0 as
n → ∞, there exists {ξn}n∈N ⊂ R3 such that, up to subsequence, {vβn

(·+ ξn)}n∈N
converges in H1 to a least energy solution to (1.4).

The theorem is proved by arguing as in Liu & Moroz’ [16], where they character-
ized the asymptotic profile of least energy solutions to the following Schrödinger-
Poisson equation in R3 as λ→ ∞:

−∆v + v +
λ

4π

(
v2 ∗ | · |−1

)
v = v|v|p−2,

where 3 < p < 6. Let us summarize the strategy of the proof. It is already known
that when 3 < p < 6, least energy solutions to (1.3) and (1.4) are minimizers of
the respective energy functionals in the associated Nehari-Pohoz̆aev manifolds (see
[2] for the SBP system and [1, 19] for the Schrödinger-Poisson system). As such,
the core of the proof consists in comparing the least energy level achieved on these
manifolds as β → 0.

Let us finish the introduction with a comment on the organization of the paper.
In Section 2, (i) we recap relevant results present in the literature; (ii) we precisely
define the Nehari–Pohoz̆aev manifold and (iii) we recall its properties which we will
use. Finally, we prove Theorem 1.1 in Section 3.

Notation. Unless denoted otherwise, functional spaces contain real-valued func-
tions defined a.e. in R3. Likewise, we integrate over R3 whenever the domain of
integration is omitted. We define D1,2 as the Hilbert space obtained as completion
of C∞

c with respect to the inner product ⟨u, v⟩D1,2 :=
∫
∇u(x) · ∇v(x)dx. In the

following sections, we always consider a fixed p ∈]3, 6[.

2. Preliminaries

We begin by recalling the following Brézis-Lieb-type splitting property (see [24,
Lemma 2.2 (i)] or [18, Proposition 4.7]).

Lemma 2.1. If wn ⇀ v0 in H1 and wn → v0 a.e. as n→ ∞, then∫∫
wn(x)

2wn(y)
2

|x− y|
dxdy −

∫∫ (
wn(x)− v0(x)

)2(
wn(y)− v0(y)

)2
|x− y|

dxdy

−−−−→
n→∞

∫∫
v0(x)

2v0(y)
2

|x− y|
dxdy.

The Pohoz̆aev-type identities in the sequence were proved in [5, Appendix A.3]
and [19, Theorem 2.2].

Proposition 2.2. (1) If v ∈ H1 is a weak solution to (1.3), then

1

2
∥v∥2D1,2 +

3

2
∥v∥2L2 +

5

4

∫∫
Kβ(x− y)v(x)2v(y)2dxdy

+
1

4β

∫∫
e−|x−y|/βv(x)2v(y)2dxdy − 3

p
∥v∥pLp = 0.

(2.1)



4 G. D. P. RAMOS EJDE-2025/29

(2) If v ∈ H1 is a weak solution to (1.4), then

1

2
∥v∥2D1,2 +

3

2
∥v∥2L2 +

5

4

∫∫
v(x)2v(y)2

|x− y|
dxdy − 3

p
∥v∥pLp = 0.

Let Pβ : H
1 → R be defined as

Pβ(v) =
3

2
∥v∥2D1,2 +

1

2
∥v∥2L2 +

3

4

∫∫
Kβ(x− y)v(x)2v(y)2dxdy+

+
(
− 1

4β

∫∫
e−|x−y|/βu(x)2u(y)2dxdy

)
− 2p− 3

p
∥u∥pLp .

To motivate the definition of Pβ , notice that every critical point of Iβ is an element
of the Nehari–Pohoz̆aev manifold

Pβ := {v ∈ H1 \ {0} : Pβ(v) = 0}.

Indeed: if I ′
β(v) = 0, then both the Nehari identity

∥v∥2H1 +

∫
(v2 ∗ Kβ)(x)v(x)

2dx− ∥v∥pLp = 0

and the Pohoz̆aev-type identity (2.1) hold, so Pβ(v) = 0.
On the one hand, it seems to be unknown whether Pβ is a natural constraint of

Iβ in the sense that if v is a critical point of Iβ |Pβ
, then I ′

β(v) = 0. On the other

hand, under more general assumptions, Chen, Li, Rădulescu & Tang proved in [2,
Lemma 3.14] that if v solves the minimization problem

Iβ(v) = mβ := inf
u∈Pβ

Iβ(u); v ∈ Pβ ,

then v is a least energy solution to (1.3). Moreover, it follows from [2, Corollary
1.6] that mβ is actually achieved and mβ > 0. As such, we will henceforth let vβ
denote any least energy solution to (1.3).

Suppose that v ∈ H1 \{0}. There exists a unique τ > 0 such that τ2v(τ ·) ∈ Pβ ,
which is obtained as the unique critical point of the mapping

]0,∞[∋ t 7→ Iβ
(
t2v(t·)

)
=
t3

2
∥v∥2D1,2 +

t

2
∥v∥2L2 +

t3

4

∫
(v2 ∗ Ktβ)(x)v(x)

2dx− t2p−3

p
∥v∥pLp .

Furthermore, Pβ

(
t2v(t·)

)
> 0 for 0 < t < τ and Pβ

(
t2v(t·)

)
< 0 for t > τ .

We let P0 : H
1 → R be given by

P0(v) =
3

2
∥v∥2D1,2 +

1

2
∥v∥2L2 +

3

4

∫∫
v(x)2v(y)2

|x− y|
dxdy − 2p− 3

p
∥v∥pLp

and we analogously define P0, m0. As before, we can associate each v ∈ H1 \ {0}
to a unique τ > 0 such that τ2v(τ ·) ∈ P0. It follows from Azzollini & Pomponio’s
[1, Theorem 1.1] that m0 > 0 and (1.4) has a least energy solution obtained as a
minimizer of I0|P0

, so we will henceforth let v0 denote any of these solutions. Let
us recall a couple of properties of P0 that follow directly from [1, Lemma 2.3] and
which will be important for us.

Lemma 2.3. (1) The Nehari-Pohoz̆aev manifold P0 is a natural constraint of
I0.

(2) infv∈P0
∥v∥Lp > 0.



EJDE-2025/29 ASYMPTOTIC PROFILE OF LEAST ENERGY SOLUTIONS 5

We will also use the fact that minimizing sequences of I0|P0
induce a sequence

of measures which falls on the compactness case in P.–L. Lions’ [14, Lemma I.1].

Lemma 2.4 ([1, Lemma 2.6]). Suppose that {un}n∈N is a minimizing sequence of
I0|P0

and given n ∈ N, µn denotes the measure which takes each Lebesgue-mea-
surable set Ω to

µn(Ω) :=

∫
Ω

p− 3

2p− 3
|∇un(x)|2 +

p− 2

2p− 3
un(x)

2

+
p− 2

2(2p− 3)

∫
un(x)

2un(y)
2

|x− y|
dydx.

It follows that there exists {ξn}n∈N ⊂ R3 for which we can associate each δ > 0
with an rδ > 0 such that µn

(
Brδ(ξn)

)
≥ m0 − δ for every n ∈ N.

3. Asymptotic profile of least energy solutions to (1.3)

Let us develop the preliminary results needed to prove the theorem. We begin
by obtaining an upper bound for lim supβ→0mβ .

Lemma 3.1. The following inequality is satisfied: lim supβ→0mβ ≤ m0.

Proof. From v0 ∈ P0, it follows that

Pβ(v0) = −1

4

∫∫ ( 3

|x− y|
+

1

β

)
e−|x−y|/βv0(x)

2v0(y)
2dxdy < 0.

As such, there exists a unique tβ ∈]0, 1[ such that t
2
βv0(tβ ·) ∈ Pβ , i.e.,

3

2
t
3
β∥v0∥2D1,2 +

1

2
tβ∥v0∥2L2 +

3

4
t
3
β

∫∫
Ktββ

(x− y)v0(x)
2v0(y)

2dxdy

−
t
2
β

4β

∫∫
e−|x−y|/(tββ)v0(x)

2v0(y)
2dxdy

=
2p− 3

p
t
2p−3
β ∥v0∥pLp .

It follows from the inclusion v0 ∈ P0 that

1

2

(
1− 1

t̄2β

)
∥v0∥2L2 +

1

4

∫∫ ( 3

|x− y|
+

1

tββ

)
e−|x−y|/(tββ)v0(x)

2v0(y)
2dxdy

=
2p− 3

p

(
1− t

2p−6
β

)
∥v0∥pLp .

(3.1)

Let us show that∫∫ ( 3

|x− y|
+

1

tββ

)
e−|x−y|/(tββ)v0(x)

2v0(y)
2dxdy −−−→

β→0
0. (3.2)

It suffices to prove that if 0 < βn → 0 as n→ ∞, then, up to subsequence,∫∫ ( 3

|x− y|
+

1

tβn
βn

)
e−|x−y|/(tβnβn)v0(x)

2v0(y)
2dxdy −−−−→

n→∞
0.

As limn→∞ tβnβn = 0, then, up to subsequence, (t̄βnβn)n∈N is decreasing, so the
limit follows from the Monotone Convergence Theorem.



6 G. D. P. RAMOS EJDE-2025/29

We claim that limβ→0 t̄β = 1. By contradiction, suppose that 0 < βn → 0 as
n→ ∞ and α := lim infn→∞ t̄βn < 1. In view of (3.1) and (3.2), it follows that

0 >
1

2

(
1− 1

α2

)
∥v0∥2L2 =

2p− 3

p

(
1− lim sup

n→∞
t
2p−6
β

)
∥v0∥pLp ≥ 0,

which is absurd, hence the result follows.
In view of (3.2), the limit t̄β → 1 as β → 0 implies

mβ ≤ Iβ
(
t
2
βv0(tβ ·)

)
=

p− 3

2p− 3
t
3
β∥v0∥2D1,2 +

p− 2

2p− 3
tβ∥v0∥2L2

+
p− 3

2(2p− 3)
t
3
β

∫
(v20 ∗ Ktββ

)(x)v0(x)
2dx

+
t
2
β

4(2p− 3)β

∫∫
e−|x−y|/(tββ)v0(x)

2v0(y)
2dxdy −−−→

β→0
I0(v0) = m0,

and the lemma is proved. □

We can use the previous lemma to control the H1-norm of least energy solutions
to (1.3) for sufficiently small β.

Lemma 3.2. lim supβ→0 ∥vβ∥H1 <∞.

Proof. As vβ ∈ Pβ , we obtain

mβ = Iβ(vβ)

=
p− 3

2p− 3
∥vβ∥2D1,2 +

p− 2

2p− 3
∥vβ∥2L2

+
p− 3

2(2p− 3)

∫∫
Kβ(x− y)vβ(x)

2vβ(y)
2dxdy

+
1

4(2p− 3)

∫∫
e−|x−y|/β

β
vβ(x)

2vβ(y)
2dxdy,

so mβ ≥ (p− 3)∥vβ∥2H1/(2p− 3) and the result follows from Lemma 3.1. □

The following estimate will also be useful for our computations.

Lemma 3.3. Given w ∈ L4, it holds that∫∫ ( 3

|x− y|
+

1

β

)
e−|x−y|/βw(x)2w(y)2dxdy ≤ 20πβ2∥w∥4L4 .

It follows that if {wβ}β>0 ⊂ H1 is such that lim supβ→0 ∥wβ∥H1 <∞, then∫∫ ( 3

|x− y|
+

1

β

)
e−|x−y|/βwβ(x)

2wβ(y)
2dxdy −−−→

β→0
0.

Proof. It follows from Hölder’s Inequality that∫∫ ( 3

|x− y|
+

1

β

)
e−|x−y|/βw(x)2w(y)2dxdy

≤
(∫ (∫ ( 3

|x− y|
+

1

β

)
e−|x−y|/βw(x)2dx

)2

dy
)1/2

∥w∥2L4 .



EJDE-2025/29 ASYMPTOTIC PROFILE OF LEAST ENERGY SOLUTIONS 7

An application of Young’s Inequality shows that(∫ (∫ ( 3

|x− y|
+

1

β

)
e−|x−y|/βw(x)2dx

)2

dy
)1/2

≤ ∥w∥2L4

∫ ( 3

|x|
+

1

β

)
e−|x|/βdx︸ ︷︷ ︸

=20πβ2

,

hence the result follows. □

Let us show that the family of Nehari–Pohoz̆aev manifolds (Pβ)β>0 is bounded
away from zero in Lp.

Lemma 3.4. infβ>0{∥v∥Lp : v ∈ Pβ} > 0.

Proof. We claim that

inf
β>0

{∥v∥H1 : v ∈ Pβ} > 0. (3.3)

Indeed, the elementary inequality re−r ≤ 1− e−r for every r ≥ 0 implies

0 = Pβ(v) ≥
1

2
∥v∥2H1 −

2p− 3

p
∥v∥pLp , (3.4)

and thus (2p−3)c∥v∥p−2
H1 /p ≥ 1/2, where c > 0 denotes the constant of the Sobolev

embedding H1 ↪→ Lp.
In this situation, the lemma follows from (3.3) and (3.4). □

The inclusion vβ ∈ Pβ implies

P0(vβ) =
1

4

∫∫ ( 3

|x− y|
+

1

β

)
e−|x−y|/βvβ(x)

2vβ(y)
2dxdy > 0,

so there exists a unique tβ > 1 such that t2βvβ(tβ ·) ∈ P0, i.e.,

3

2
t3β∥vβ∥2D1,2 +

1

2
tβ∥vβ∥2L2 +

3

4
t3β

∫∫
vβ(x)

2vβ(y)
2

|x− y|
dxdy

=
2p− 3

p
t2p−3
β ∥vβ∥pLp .

(3.5)

Our last preliminary result shows that tβ → 1 as β → 0.

Lemma 3.5. tβ → 1 and I0(t2βvβ(tβ ·)) → m0 as β → 0.

Proof. Let us prove that tβ → 1 as β → 0. We only have to show that lim supβ→0 tβ ≤
1. By contradiction, suppose that lim supβ→0 tβ > 1. In particular, we can fix
{βn}n∈N ⊂]0,∞[ such that βn → 0 as n → ∞ and α := lim infn→∞ tβn > 1. It
follows from (3.5) and from the fact that vβn

∈ Pβn
that

1

2

( 1

t2βn

− 1
)
∥vβn∥2L2

+
1

4

∫∫ ( 3

|x− y|
+

1

βn

)
e−|x−y|/βnvβn

(x)2vβn
(y)2dxdy

=
2p− 3

p
(t2p−6

βn
− 1)∥vβn

∥pLp .



8 G. D. P. RAMOS EJDE-2025/29

In view of Lemmas 3.2–3.4,

0 ≥ 1

2

( 1

α2
− 1

)(
lim sup
n→∞

∥vβn
∥2L2

)
≥ 2p− 3

p
(α2p−6 − 1)

(
lim inf
n→∞

∥vβn
∥pLp

)
> 0,

which is absurd, hence the result follows.
Now, we want to show that I0

(
t2βvβ(tβ ·)

)
→ m0 as β → 0. We have

m0 ≤ I0
(
t2βvβ(tβ ·)

)
=

p− 3

2p− 3
t3β∥vβ∥2D1,2 +

p− 2

2p− 3
tβ∥vβ∥2L2

+
p− 3

2(2p− 3)
t3β

∫
vβ(x)

2vβ(y)
2

|x− y|
dx

= t3βmβ +
p− 2

2p− 3
(tβ − t3β)∥vβ∥2L2

+
p− 3

2(2p− 3)
t3β

∫∫
e−|x−y|/β

|x− y|
vβ(x)

2vβ(y)
2dxdy

− 1

4(2p− 3)
t
3
β

∫∫
e−|x−y|/βvβ(x)

2vβ(y)
2dxdy.

Because limβ→0 tβ = 1, the result follows from Lemmas 3.1–3.3. □

Even though this limit will not be explicitly used to prove the theorem, we
remark that Lemma 3.5 implies mβ → m0 as β → 0 because Iβ(vβ) = mβ by
definition. Let us finally prove the theorem.

Proof of Theorem 1.1. From Lemma 3.5, {un := t2βn
vβn

(tβn
·)}n∈N is a minimiz-

ing sequence of I0|P0 . Let µn denote the measure defined in Lemma 2.4 and let
{ξn}n∈N ⊂ R3 be furnished by the same lemma. It follows from Lemmas 3.2 and
3.5 that {wn := un(·−ξn)}n∈N is bounded in H1, so there exists v0 ∈ H1 such that,
up to subsequence, wn ⇀ v0 in H1 as n → ∞. Due to the Kondrakov Theorem,
we can suppose further that wn → v0 a.e. as n→ ∞.

Now, we argue as in [1, Proof of Theorem 1.1] to prove that

∥wn − v0∥Lq
Lq

−−−−→
n→∞

0 for every q ∈ [2, 6[. (3.6)

By Lemma 2.4, ∥wn∥2
H1

(
R3\Brδ

(0)
) < δ for every n ∈ N. Consider a fixed δ >

0. From the Kondrakov Theorem and the fact that ∥ · ∥H1 is weakly lower-
semicontinuous, we obtain

∥wn − v0∥Lq ≤ ∥wn − v0∥Lq(Brδ
(0)) + ∥wn − v0∥Lq(R3\Brδ

(0))

≤ δ + C
(
∥wn∥

H1
(
R3\Brδ

(0)
) + ∥v0∥

H1
(
R3\Brδ

(0)
))

≤ 3δ

(3.7)

for sufficiently large n ∈ N, where C > 0 denotes the constant of the Sobolev
embedding H1 ↪→ Lq. The result then follows from the fact that given δ > 0, there
exists nδ ∈ N such that (3.7) holds for n ≥ nδ.



EJDE-2025/29 ASYMPTOTIC PROFILE OF LEAST ENERGY SOLUTIONS 9

We claim that
p− 3

2p− 3
∥wn∥2D1,2 +

p− 2

2p− 3
∥wn∥2L2 −−−−→

n→∞

p− 3

2p− 3
∥v0∥2D1,2 +

p− 2

2p− 3
∥v0∥2L2 . (3.8)

Indeed, in view of (3.6) and Lemma 2.3, we deduce that ∥v0∥Lp > 0, so v0 ̸≡ 0.
Considering (3.6), Lemmas 2.1, 3.5 and the fact that ∥ · ∥D1,2 is weakly lower-
semicontinuous, we obtain P0(v0) ≤ lim infn→∞ P0(wn) = 0. As v0 ̸≡ 0, we deduce
that there exists a unique t0 ∈ [0, 1] such that t20v0(t0·) ∈ P0. We obtain

m0 ≤ I0
(
t20v0(t0·)

)
=

p− 3

2p− 3
t30∥v0∥2D1,2 +

p− 2

2p− 3
t0∥v0∥2L2 +

p− 2

2(2p− 3)
t30

∫
v0(x)

2v0(y)
2

|x− y|
dydx

≤ p− 3

2p− 3
∥v0∥2D1,2 +

p− 2

2p− 3
∥v0∥2L2 +

p− 2

2(2p− 3)

∫
v0(x)

2v0(y)
2

|x− y|
dydx

≤ I0(wn) + on(1)

for sufficiently large n ∈ N and the result follows by taking the limit n→ ∞.
In view of (3.6) and (3.8), we obtain ∥wn − v0∥H1 → 0 as n → ∞, so v0 ∈ P0

and I0(v0) = m0. Finally, the fact that I ′
0(v0) = 0 is a corollary of Lemma 2.3. □

3.1. Notes. This article is posted at https://arxiv.org/abs/2407.19141 before its
publication.

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Gustavo de Paula Ramos
Instituto de Matemática e Estat́ıstica, Universidade de São Paulo, Rua do Matão, 1010,

05508-090 São Paulo SP, Brazil
Email address: gpramos@icmc.usp.br


	1. Introduction
	Notation

	2. Preliminaries
	3. Asymptotic profile of least energy solutions to (1.3)
	3.1. Notes

	References