Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 110, pp. 1–11.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, DOI: 10.58997/ejde.2025.110

SOLUTIONS TO MAGNETIC SCHRÖDINGER EQUATIONS WITH

ARBITRARY GROWTH AT INFINITY

WENDY F. ALMEIDA, GIOVANY M. FIGUEIREDO

Abstract. This work addresses the existence of at least one radial solution to the nonlinear
magnetic Schrödinger equation( ε

i
∇−A(x)

)2
u+ V (x)u = f(|u|2)u in RN ,

where both the magnetic potential A and the electric potential V are continuous, radial func-
tions. Our main tool is the penalization method developed by del Pino and Felmer [17], which

we adapt to the complex-valued setting under magnetic effects. By using the small parameter

ε and radial symmetry, we handle nonlinearities with arbitrary growth.

1. Introduction

The purpose of this article is to investigate the magnetic Schrödinger equation(ε
i
∇−A(x)

)2

u+ V (x)u = f(|u|2)u in RN , (1.1) LE1

where u : RN → C is an unknown function, N ≥ 3, i represents the imaginary unit, A =
(A1, . . . , AN ) : RN → RN denotes the magnetic (or vector) potential, V : RN → R+ a potential
continuous and the nonlinear term f :R+ → R is a regular function that satisfies appropriate
conditions with arbitrary growth.

A standing wave solution for (1.1) is a solution of the form

ψ(x, t) = eiωtu(x), (1.2)

where ω ∈ R is a real frequency, and u : RN → C is a spatial function to be determined.
Rewriting (1.1) in terms of ψ(x, t), we have(ε

i
∇−A(x)

)2

ψ + V (x)ψ = f(|ψ|2)ψ − i
∂ψ

∂t
in RN . (1.3)

Thus, the function ψ satisfies the time-dependent Schrödinger equation with a frequency ω.
The parameter ε in (1.1) represents the Planck constant, typically denoted by ℏ in quantum

mechanics. This constant plays a fundamental role in distinguishing quantum mechanics from
classical mechanics. In the limit as ε → 0, the quantum effects vanish, and the equation transi-
tions to a classical regime, where the motion of particles follows Newtonian mechanics. Conversely,
for nonzero values of ε, wave-particle duality emerges, and solutions of the equation exhibit inter-
ference and localization effects characteristic of quantum systems. The presence of the magnetic
potential A(x) further highlights the influence of the electromagnetic field on the quantum behav-
ior of particles, modifying their trajectories according to the principles of gauge theory. For more
details on the physical motivations, see [21, 22, 24].

Recently, the study of magnetic Schrödinger equations has been approached from various per-
spectives, however only a limited number of works have addressed this topic. For instance, in
[15], the authors investigate the existence of standing waves for a class of nonlinear Schrödinger
equations in RN , incorporating both electric and magnetic fields. Under suitable non-degeneracy

2020 Mathematics Subject Classification. 35B33, 35J20, 35Q55.
Key words and phrases. Magnetic Schrödinger equations; arbitrary growth at infinity.
©2025. This work is licensed under a CC BY 4.0 license.

Submitted September 27, 2025. Published November 24, 2025.
1



2 W. F. ALMEIDA, G. M. FIGUEIREDO EJDE-2025/110

assumptions on the critical points of an auxiliary function associated with the electric field, they
establish the existence and multiplicity of complex-valued solutions in the semiclassical limit.
Moreover, they demonstrate that in this limit, while the presence of a magnetic field induces a
phase shift in the complex wave, it does not affect the spatial localization of the wave’s modulus
peaks.

In [16], a magnetic Schrödinger equation was analyzed in the context of competition between
electric potentials. In [7], the authors investigate the existence of infinitely many geometrically
distinct solutions to problem (1.1), considering nonlinearities f with either subcritical or critical
growth. Using a finite-dimensional reduction approach, [8] and [10] establish a multiplicity result
for (1.1). Another multiplicity results can be seen in [4, 5, 6, 18, 23]. For further significant
contributions to the study of the magnetic Schrödinger equation, we refer the reader to [3, 9, 11,
12, 14, 25, 28] and the references therein.

Throughout this article, we use the following assumptions:

(A1) A is a radial continuous function, that is A ∈ C(RN ,RN ) and if |x| = |y| then A(x) = A(y);
(A2) The potential V is a radial continuous function and there are positive constants R1 < r1 <

r2 < R2 such that:
(i) A(x) = (0, . . . , 0) ∈ RN in the set Ω = {x ∈ RN : r1 < |x| < r2},
(ii) V (x) = 0 in the set Ω = {x ∈ RN : r1 < |x| < r2},
(iii) there exists V0 > 0 such that V (x) ≥ V0 in the set Λc = BR1

⋃
Bc

R2
;

(A3) f ∈ C(R+,R) and lims→0+ f(s) = 0 and f(s) = 0 for all s ≤ 0;
(A4) There exists θ > 2 such that f(s)s − θF (s) > 0 for every s ∈ R with s > 0 where

F (s) =
∫ s

0
f(t) dt;

(A5) The function s→ f(s) is non-decreasing for s > 0.

Now we give typical examples of functions A(x) and V (x) satisfying conditions (A1) and (A2).
Let

Ai(x) =


r1 − |x|, |x| < r1,

0, r1 ≤ |x| ≤ r2,

|x| − r2, |x| > r2,

for i = 1, . . . N . In this case A is radial and

A(x) = (0, . . . , 0) if r1 < |x| < r2,

satisfying condition (A2)(i). Let

V (x)−


exp( 1

|x|2−r1
), x ∈ Br1(0),

0, x ∈ Br2 ∩Bc
r1 ,

|x|2 − 2r2x+ r22, x ∈ Bc
r2 .

This function is continuous and satisfies the conditions (ii) and (iii) in (A2). Both functions A
and V are continuous and radial, ensuring symmetry with respect to the origin. Now we state the
main result in this article.

mainresult Theorem 1.1. Suppose that N ≥ 3 and (A1)–(A5) hold. Then, there exists ε0 > 0 such that for
all ε ∈ (0, ε0) problem (1.1) admits a nontrivial complex solution uε with |uε(x)| → 0 as |x| → +∞
and |uε| ∈ L∞(RN ) ∩ C1,λ

loc (RN ), for λ ∈ (0, 1).

Elliptic problems with arbitrary growth can be found as partial differential equations where the
growth term can be nonlinear and increase uncontrollably wit respect to the unknown variable or
its derivatives. For example, the authors in [2], analyzed the problem

−ε2∆u+ V (x)u = f(u) in RN ,

which exhibits superlinear growth at infinity, without imposing any constraints on the growth
of the function f . Utilizing the force of the parameter ε, the space of radial functions, and
the penalization method introduced by Del Pino and Felmer [17], they establish the existence of
solutions. The particular case f(t) = tp with p > 1 was studied in [1]. Additionally, an extension
of [2] to a Hamiltonian system was investigated in [13].



EJDE-2025/110 MAGNETIC SCHRÖDINGER EQUATIONS 3

Other approaches have been used. For instance, in [30], the existence of infinitely many so-
lutions was demonstrated for various elliptic problems under Dirichlet and Neumann boundary
conditions. this was also done for a Hamiltonian system, considering nonlinearities exhibiting sub-
linear behavior at the origin. The methodology involved modifying the nonlinearity and obtaining
solutions with small L∞ norms, ensuring that each solution of the modified problem corresponded
to a solution of the original problem. The case studied in [30], where the nonlinearity could change
sign, was explored in [20]. In [19], a Kirchhoff problem was analyzed, considering nonlinearities
that exhibit both sublinear and linear behavior at the origin, employing the strategy introduced
in [30].

The main contributions of this article are as follows:

(1) When the growth of the nonlinearity is not bounded by a polynomial function of controlled
order, classical methods such as Sobolev embeddings may not directly apply. To overcome
this difficulty we are adapting to our case the arguments that can be found in [1, 2, 13].
However, with respect to [1, 2, 13], due to the presence of the magnetic potential A and
the fact that the solutions assume complex values, a more careful analysis will be needed
and some refined estimates will be given in Lemmas 3.3, 4.1, and 4.2.

(2) Proving the existence of weak solutions can be challenging, especially if the growth of
the nonlinearity does not satisfy suitable structural conditions. For this reason, it was
necessary to put forward appropriate hypotheses on the magnetic potential A that were
compatible with the existing hypotheses about the electrical potential V . This is another
point that differs our article from the articles [1], [2] and [13] that consider solutions that
assume real values and A = 0.

(3) Even if existence is guaranteed, proving sufficient regularity for the solutions can be com-
plicated, as arbitrary growth may generate instabilities or singularities. Once again, due
to the presence of the magnetic potential A and the fact that the solutions assume complex
values, it was necessary to use Moser’s iteration method in Lemma 4.2.

This article is organized as follows. In Section 2 we introduce the Banach space where we will
look for the solution to the problem and we recall an important inequality called the diamagnetic
inequality. In Section 3 we use the Del Pino Felmer Penalization [17]. With this penalty and the
fact that we are in the space of radial functions, we get around the difficulty of having an arbitrary
growth. In fact, it is possible to obtain a functional associated to the penalized problem. In this
section we show that this functional has the Geometry of the Mountain Pass Theorem and that
Palais-Smale sequences have strongly convergent subsequences. In Section 4, using the strength of
the parameter ε and the fact that we obtain a radial solution to the penalized problem, we show
that the solution to the penalized problem is a solution to the original problem.

2. Notation and variational tools
tools

To introduce the variational structure of the problem, we define the Hilbert space H1
A,ε(RN ,C)

obtained by the closure of C∞
0 (RN ,C) under the scalar product

(u, v)A,ε = Re

∫
RN

(
∇A,εu · ∇A,εv + V (x)uv

)
dx for all u, v ∈ H1

A,ε(RN ,C)

where ∇A,εu = (Dε
1u, . . . ,D

ε
Nu), D

ε
j = ε

i ∂j − Aj(x), and Re and the bar denote the real part
of a complex number and the complex conjugation respectively. The norm induced by this inner
product is

∥u∥A,ε =
(∫

RN

(|∇A,εu|2 + V (x)|u|2) dx
)1/2

for u ∈ H1
A,ε(RN ,C).

Since there is no relationship between H1
A,ε(RN ,C) and H1(RN ,R); that is, H1

A,ε(RN ,C) ̸⊂
H1(RN ,R) and H1(RN ,C) ̸⊂ H1

A,ε(RN ,C), we will frequently use in this paper the following

diamagnetic inequality (see [25, Theorem 7.21])

ε|∇|u|(x)| ≤ |∇A,εu(x)| for almost every x ∈ RN . (2.1) diam



4 W. F. ALMEIDA, G. M. FIGUEIREDO EJDE-2025/110

This implies that, if u ∈ H1
A,ε(RN ,C) then |u| ∈ H1(RN ,R). Therefore, u ∈ Lp(RN ,C) for any

p ∈ [2, 2∗].
We denote by H1

A,ε,rad(RN ,C) the subspace of H1
A,ε(RN ,C) formed by the radial functions,

that is

H1
A,ε,rad(RN ,C) = {u ∈ H1

A,ε(RN ,C) : u(x) = u(|x|) for x ∈ RN}.

3. Auxiliary problems
Auxiliary

Note that, since the functional has arbitrary growth, it is not possible to obtain a functional
directly associated to this problem. To overcome this difficulty, we will use the Del Pino and
Felmer prenalization method [17] and make strong use of the radiality of the problem to obtain a
functional that satisfies the Mountain Pass Geometry and that has compactness.

We choose k > θ/(θ−2), where θ is given by (A4), and using (A5), take a > 0 to be the unique
number such that f(a) = V0/k, with V0 given by (A2). We set

f̂(s) :=

{
f(s), if s ≤ a,

V0/k, if s > a.

Let χΛ denotes the characteristic function of the set Λ. We introduce the penalized nonlinearity
g : RN × R → R by setting

g(x, s) := χΛ(x)f(s) + (1− χΛ(x))f̂(s). (3.1) def_g

Now, we shall consider the modified problem

(
ε

i
∇−A(x))2u+ V (x)u = g(x, |u|2)u, x ∈ RN ,

u ∈ H1
A,ε(RN ,C).

(3.2) tDea

Notice that if uε is a solution of the above problem such that |uε(x)| < a1/2 in Λc ,then, in view
of the definition of g, there holds g(x, |u|2)u = f(|u|2)u for each x ∈ RN . Thus, the function u is
also a solution of the original problem (1.1).

In view of the above comment, in the sequel we study the modified problem (3.2). In the next
result we prove that some properties on the continuous function g(x, s).

lem3.1 Lemma 3.1. The continuous function g(x, s) satisfies the following properties uniformly in x ∈
RN :

(1) g(x, s) = 0 for each s ≤ 0;
(2) lims→0+ g(x, s) = 0, g and G(s) =

∫ s

0
g(t)dt are radials in x;

(3) (i) 0 ≤ θ
2G(x, s) < g(x, s)s, for each x ∈ Λ, s > 0,

(ii) 0 ≤ G(x, s) ≤ V (x)
k s and 0 ≤ g(x, s) ≤ V (x)

k , for each x ∈ Λc, s > 0;
(4) the function s 7→ g(x, s) is non-decreasing for s > 0;
(5) for all u ∈ H1

A,ε,rad(RN ,C), we have
∫
RN G(x, |u|2)dx <∞ and

∫
RN g(x, |u|2)|u|2dx <∞.

Proof. Note that from (1)–(4) follow by the definition of g. Now, for u ∈ H1
A,ε,rad(RN ,C), from

[27], there exists CN > 0, which depends just the dimension N such that

|u(x)| ≤ CN∥u∥A,ε

|x|(N−1)/2
. (3.3) Decaimento

Then ∫
RN

G(x, |u|2)dx ≤
∫
Λ

G
(
x,
C2

N∥u∥2A,ε

R1

)
dx+

∫
Λc

G(x, |u|2)dx

≤ max
Λ

G
(
·,
C2

N∥u∥2A,ε

R1

)
|Λ|+ V0

k

∫
Λc

|u|2dx <∞.

(3.4) calhou

Using the same argument we can conclude that
∫
RN g(x, |u|2)|u|2dx <∞. □



EJDE-2025/110 MAGNETIC SCHRÖDINGER EQUATIONS 5

By Lemma 3.1(5), the functional associated with (3.2), namely

Jε(u) :=
1

2

∫
RN

|∇A,εu|2dx+
1

2

∫
RN

V (x)|u|2dx− 1

2

∫
RN

G(x, |u|2)dx, u ∈ H1
A,ε,rad(RN ,C)

belongs to C1(H1
A,ε,rad(RN ,C),R) with Gâteaux differential given by

J ′
ε(u)v := Re

(∫
RN

∇A,εu∇A,εvdx+

∫
RN

V (x)uvdx−
∫
RN

g(x, |u|2)uvdx
)
, u, v ∈ H1

A,ε,rad(RN ,C).

Moreover, its critical points are the weak solutions of the modified problem (3.2).
Now we prove that the associated functional Jε satisfies the Mountain Pass Geometry.

mountpassIA Lemma 3.2. Suppose (1)–(5) of Lemma 3.1 hold. Then, the functional Jε has a Mountain Pass
Geometry, that is

(i) Jε(0) = 0;
(ii) there exist ρ, δ > 0 such that Jε(u) ≥ δ for all u ∈ H1

A,ε,(RN ,C) with ∥u∥A,ε = ρ;

(iii) there exists u0 ∈ H1
A,ε,rad(RN ,C) such that ∥u∥A,ε > ρ0 and Jε(u0) ≤ 0.

Proof. Note that by (2) of Lemma 3.1, given Υ > 0, there exists δ > 0 such that for all 0 ≤ s < δ.
Then we have

G(s) = F (s) ≤ Υ

2
|s|2.

Considering (3.3) and ∥u∥A,ε = ρ with ρ enough small such that |u(x)| ≤ CN∥u∥A,ε

R
1/2
1

≤ δ, there

exists C > 0 such that

Jε(u) =
1

2

∫
RN

|∇A,εu|2dx+
1

2

∫
RN

V (x)|u|2dx− 1

2

∫
Λ

G(x, |u|2)dx− 1

2

∫
Λc

G(x, |u|2)dx

≥ 1

2
∥u∥2A,ε −

Υ

4

∫
RN

|u|2dx− 1

2k

∫
RN

V (x)|u|2dx ≥ C∥u∥2A,ε,

and the proof of (ii) is complete, decreasing ρ if necessary.
(iii) Note that, for all x ∈ Λ , from (G3), there exist C > 0 and s0 > 0 such that G(x, s) ≤ Csθ

for all s ≥ s0. Now consider v0 ∈ C∞
0 (Λ) and note that

Jε(sv0) ≤
s2

2
∥v0∥2A,ε −

sθ

2

∫
Λ

|v0|θ dx.

Since θ > 2, we obtain Jε(sv0) → −∞ as s → +∞ thus, taken u0 = sv0 for s sufficiently large
(iii) is proved. □

By [29, Theorem 1.15] we assert the existence of a Palais-Smale sequence (un) inH
1
A,ε,rad(RN ,C)

at level dε, that is, a sequence with the property

Jε(un) → dε and J ′
ε(un) → 0,

where dε is the minimax level of the mountain pass theorem related to Jε, namely

dε = inf
ς∈J

max
t∈[0,1]

Jε(ς(t)), (3.5) beta0

where

J =
{
ς ∈ C([0, 1], H1

A,ε,rad(RN ,C) : ς(0) = 0 and Jε(ς(1)) < 0
}
.

By a reasoning similar to the one in [29, Theorem 4.2], we have

dε = inf
u∈H1

A,ε,rad(RN ,C),u̸=0
sup
t≥0

Φε(tu). (3.6) beta-e

The main feature of the modified functional is that it satisfies the Palais-Smale condition, as
we can see from the next result.

vit5 Lemma 3.3. The functional Jε satisfies the (PS)dε
condition for any level dε ∈ R.



6 W. F. ALMEIDA, G. M. FIGUEIREDO EJDE-2025/110

Proof. Suppose that (un) ⊂ H1
A,ε,rad(RN ,C) is a (PS)dε

sequence for Jε, that is, Jε(un) → dε and

J ′
ε(un) → 0. We first prove that (un) is bounded in H1

A,ε(RN ,C). Indeed, by using (3) of lemma
3.1 we obtain

dε + on(1)∥un∥A,ε ≥ Jε(un)−
1

θ
J ′
ε(un)un

≥
(1
2
− 1

θ

)
∥un∥2A,ε +

1

θ

∫
Λc

ε

g(x, |un|2)|un|2dx− 1

2

∫
Λc

ε

G(x, |un|2)dx

≥ 1

2

(θ − 2

θ
− 1

k

)
∥un∥2A,ε,

where on(1) denotes a quantity approaching zero as n → ∞. Since k > θ/(θ − 2) we conclude
from the above inequality that (un) is bounded in H1

A,ε(RN ,C).
Claim. For any given ζ > 0, there exists R = R(ζ) > 0 such that R > 4R2 and

lim sup
n→∞

∫
BR(0)c

(|∇A,εun|2 + V (x)|un|2)dx ≤ ζ. (3.7) Crucial

To prove the claim we consider ηR ∈ C∞(RN ,R) such that 0 ≤ ηR ≤ 1, ηR ≡ 0 in BR/2(0),
ηR ≡ 1 in BR(0)

c and |∇ηR| ≤ C/R, where C > 0 is a constant independent of R. Since the
sequence (ηRun) is bounded in H1

A,ε(RN ,C), we have that J ′
ε(un)(ηRun) = on(1), that is,

Re
(∫

RN

∇A,εun∇A,ε(unηR)dx
)
+

∫
RN

V (x)|un|2ηRdx =

∫
RN

g(x, |un|2)|un|2ηRdx+ on(1).

Since ηR take values in R, a direct calculation shows that

∇A,ε(unηR) = iun∇ηR + ηR∇A,εun.

The two above equalities and Lemma 3.1(3)(ii) imply that∫
RN

(
|∇A,εun|2 + V (x)|un|2

)
ηRdx

≤ 1

k

∫
RN

V (x)|un|2ηRdx+Re
(∫

RN

−iun∇A,εun∇ηR
)
dx+ on(1).

By using the definition of ηR, Hölder’s inequality and the boundedness of (un) we obtain(
1− 1

k

)∫
BR(0)c

(
|∇A,εun|2 + V (x)|un|2

)
dx ≤ C

R
∥un∥L2∥∇A,εun∥L2 + on(1)

≤ C1

R
+ on(1).

So, for any fixed ζ > 0, we can choose R > 0 large enough, such that

lim sup
n→∞

∫
BR(0)c

(|∇A,εun|2 + V (x)|un|2) ≤ ζ.

This completes the proof of the claim.
Now note that∫
RN

(
g
(
x, |un|2

)
|un|2 − g(x, |u|2)|u|2

)
dx =

∫
BR1

(
g
(
x, |un|2

)
|un|2 − g(x, |u|2)|u|2

)
dx

+

∫
BR\BR1

(
g
(
x, |un|2

)
|un|2 − g(x, |u|2)|u|2

)
dx

+

∫
Bc

R

(
g
(
x, |un|2

)
|un|2 − g(x, |u|2)|u|2

)
dx.

We shall prove that each of these terms approaches zero as n → ∞. From the boundedness of
BR1

⊂ Λc, we have un → u in L2 (BR1
). By Lemma 3.1(3) it follows that∫

BR1

(
g
(
x, |un|2

)
|un|2 − g(x, |u|2)|u|2

)
dx = on(1).



EJDE-2025/110 MAGNETIC SCHRÖDINGER EQUATIONS 7

Using the proof of lemma 3.1(5) and Lebesgue’s Dominated Convergence Theorem, we conclude
that ∫

BR\BR1

(
g
(
x, |un|2

)
|un|2 − g(x, |u|2)|u|2

)
dx = on(1).

Now note that ∣∣ ∫
Bc

R

g
(
x, |un|2

)
|un|2 − g(x, |u|2)|u|2dx

∣∣
≤

∫
Bc

R

g
(
x, |un|2

)
|un|2dx+

∫
Bc

R

g(x, |u|2)|u|2dx

≤
∫
Bc

R

(|∇A,εun|2 + V (x)|un|2) +
∫
Bc

R

g
(
x, |un|2

)
|un|2dx.

Since
∫
Bc

R
g
(
x, |un|2

)
|un|2dx is integrable and using (3.7), we obtain∫

Bc
R

(
g
(
x, |un|2

)
|un|2 − g(x, |u|2)|u|2

)
dx = on(1).

Observe that J ′
ε(un)u = on(1), which implies that

∥u∥2A,ε =

∫
RN

g(x, |u|2)|u|2dx.

Then

lim
n→∞

∥un∥2A,ε = lim
n→∞

∫
RN

g(x, |un|2)|un|2dx =

∫
RN

g(x, |u|2)|u|2dx = ∥u∥2A,ε. □

4. Proof of Theorem 1.1

From Lemmas 3.2 and 3.3, for each ε > 0, there exists uε ∈ H1
A,ε,rad(RN ,C) weak solution of

problem (3.2). That is,

Jε (uε) = dε and J ′
ε (uϵ) v = 0,∀v ∈ H1

A,ε,rad(RN ,C).
Note that, by the Principle of Symmetric Criticality [29, Theorem 1.28], we have that uε is in fact
a critical point of Jε in the space H1

A,ε(RN ,C). The next result is crucial for this section.

vit55 Lemma 4.1. ∥uε∥2A,ε → 0 as ε→ 0.

Proof. Taking ψ ∈ C∞
0, rad (Ω,R), a nonnegative function with suppψ ⊂ Ω, there is a unique

tϵ ∈ R+ such that
Jϵ (tϵψ) = max

t≥0
Jϵ(tψ).

Then, from (A2),

ϵ2
∫
Ω

|∇ψ|2dx =

∫
Ω

f
(
|tϵψ|2

)
|ψ|2dx,

and choosing Ω1 ⊂ Ω such that ψ(x) ≥ ψ0 > 0 for all x ∈ Ω1, it follows that

ϵ2
∫
Ω

|∇ψ|2dx ≥
∫
Ω1

f
(
|tϵψ|2

)
|ψ|2 ≥ ψ2

0

∫
Ω1

f
(
|tϵψ|2

)
dx.

Thus, from (A5), we conclude that tε → 0 as ε→ 0. Furthermore,

0 < dε ≤ Jϵ (tεψ) ≤
t2ε
2

∫
Ω

ϵ2|∇ψ|2dx,

which implies that dε → 0 as ε→ 0. On the other hand, there exists C > 0 such that

dε = Jϵ (uϵ) = Jϵ (uϵ)−
1

θ
J ′
ϵ (uϵ)uϵ

≥
(1
2
− 1

θ

)( ∫
RN

(
|∇A,εuε|2 +

(
1− 1

k

)
V (x)|uϵ|2

)
dx

)
≥ C∥uε∥2A,ε.



8 W. F. ALMEIDA, G. M. FIGUEIREDO EJDE-2025/110

The proof is complete. □

Now we prove a regularity result. Its proof follows the argument found in [5, Lemma 4.1.]. We
will give the proof here for completeness of this work.

leminha1 Lemma 4.2. Let (εn) be a sequence of positive numbers with εn → 0+ and un ∈ H1
A,εn,rad

(RN ,C)
be a solution of (3.2). Then |un| ∈ L∞(RN ) ∩ C1,λ

loc (RN ), for λ ∈ (0, 1). Moreover, there exists a
constant C > 0, independent on n, such that |un| ≤ C∥un∥A,εn .

Proof. We define uL,n ∈ H1
A,εn,rad

(RN ,C) and zL,n ∈ H1
A,εn,rad

(RN ,C) by setting

uL,n(x) := min{|un(x)|, L}, zL,n := u
2(β−1)
L,n un,

with β > 1 to be determined later. By using the calculation performed in [14, equation (2.2)] and
the diamagnetic inequality we obtain

Re
(
∇A,εnun∇A,εnzL,n

)
≥ u

2(β−1)
L,n |∇A,εnun|2 ≥ u

2(β−1)
L,n ε2n

∣∣∇|un|
∣∣2.

This inequality, the definition of zL,n and J ′
εn(un)zL,n = 0 imply that∫

RN

u
2(β−1)
L,n |∇|un||2dx ≤

∫
RN

(
g(x, |un|2)− V (x)

)
|un|2u2(β−1)

L,n dx. (4.1) moser1

Arguing as in (3.4), using Lemma 4.1 and Lemma 3.1(1), we obtain C1 > 0 such that

g(x, s) ≤ V0
2

+ C1|s|(2
∗−2)/2, for all (x, s) ∈ RN × R.

This, (4.1), and V (x) ≥ V0 provide∫
RN

u
2(β−1)
L,n ε2n

∣∣∇|un|
∣∣2dx ≤

∫
RN

(V0
2

+ C1|un|2
∗−2 − V (x)

)
|un|2u2(β−1)

L,n dx

≤ C1

∫
RN

|un|2
∗
u
2(β−1)
L,n dx.

(4.2) 2moser

Let S be the best constant of the embedding D1,2(RN ,R) ↪→ L2∗(RN ,R) and define ûL,n :=

|un|uβ−1
L,n . We have that

S−1∥ûL,n∥2L2∗ ≤
∫
RN

∣∣∇(
|un|uβ−1

L,n

)∣∣2dx.
But ∫

RN

∣∣∇(
|un|uβ−1

L,n

)∣∣2dx =

∫
{|un|≤L}

∣∣∇(
|un|uβ−1

L,n

)∣∣2dx+

∫
{|un|>L}

∣∣∇(
|un|uβ−1

L,n

)∣∣2dx
=

∫
{|un|≤L}

∣∣∇|un|β
∣∣2 dx+

∫
{|un|>L}

L2(β−1) |∇|un||2 dx

≤ β2

∫
RN

u
2(β−1)
L,n ε2n

∣∣∇|un|
∣∣2dx,

and therefore

∥ûL,n∥2L2∗ ≤ C3β
2

∫
RN

u
2(β−1)
L,n ε2n

∣∣∇|un|
∣∣2dx.

This and (4.2) yield

∥ûL,n∥2L2∗ ≤ C4β
2

∫
RN

|un|2
∗
u
2(β−1)
L,n dx, (4.3) 4moser

for all β > 1. The above expression and uL,n ≤ |un|, imply that

∥ûL,n∥2L2∗ ≤ C4β
2

∫
RN

|un|2
∗−2|un|2βdx. (4.4) 5moser

Now, setting

t :=
2∗2∗

2(2∗ − 2)
> 1, α :=

2t

t− 1
< 2∗, (4.5) def_t



EJDE-2025/110 MAGNETIC SCHRÖDINGER EQUATIONS 9

we can apply Hölder’s inequality with exponents t/(t− 1) and t in (4.4), to obtain

∥ûL,n∥2L2∗ ≤ C4β
2∥un∥2βLβα

(∫
RN

|un|2
∗(2∗/2)dx

)1/t

. (4.6) 6moser

Claim. There exist n0 ∈ N and K > 0 such that, for any n ≥ n0, it holds∫
RN

|un|2
∗(2∗/2)dx ≤ K.

Assuming the claim is true, we can use (4.6) to conclude that

∥ûL,n∥2L2∗ ≤ C6β
2∥un∥2βLβα .

Since

∥uL,n∥2βLβ2∗ =
(∫

RN

uβ2
∗

L,ndx
)2/2∗

≤
(∫

RN

|un|2
∗
u
2∗(β−1)
L,n dx

)2/2∗

= ∥ûL,n∥2L2∗ ≤ C6β
2∥un∥2βLβα ,

we can apply Fatou’s lemma in the variable L to obtain

∥|un|∥Lβ2∗ ≤ C
1/β
7 β1/β∥|un|∥Lβα .

We now set β := 2∗/α > 1 and note that, since |un| ∈ L2∗(RN ), the above inequality holds
for this choice of β. Moreover, since β2α = β2∗, it follows that the inequality also holds with β
replaced by β2. Hence,

∥|un|∥Lβ22∗ ≤ C
1/β2

7 β2/β2

∥|un|∥Lβ2α .

By iterating this process and recalling that βα = 2∗ we obtain, for k ∈ N,

∥|un|∥Lβk2∗ ≤ C
∑k

i=1 β−i

7 β
∑m

i=1 iβ−i

∥|un|∥L2∗ .

Since β > 1 we can take the limit as k → ∞ to obtain

∥|un|∥L∞ ≤ C8∥|un|∥L2∗ .

From the Sobolev imbedding, there exists a constant positive C, independent on n such that

∥|un|∥L∞ ≤ C∥|un|∥A,εn . (4.7) Wendy

By the elliptic regularity |un| ∈ L∞(RN ) ∩ C1,λ
loc (RN ), for λ ∈ (0, 1).

It remains to prove the claim. In fact, let β = 2∗/2. From (4.3), we have

|ûL,n|22∗ ≤ Cβ2

∫
RN

u2
∗

n u
(2∗−2)
L,n dx,

or equivalently

|ûL,n|22∗ ≤ Cβ2

∫
RN

u2nu
(2∗−2)
L,n u(2

∗−2)
n dx.

Using the Hölder inequality with exponents 2∗

2 and 2∗

2∗−2

|ûL,n|22∗ ≤ Cβ2
(∫

RN

[
unu

(2∗−2)
2

L,n

]2∗
dx

)2/2∗(∫
RN

u2
∗

n dx
) 2∗−2

2∗
.

From definition of ûL,n, we have(∫
RN

[
unu

(2∗−2)
2

L,n

]2∗
dx

)2/2∗

≤ Cβ2
(∫

RN

[
unu

(2∗−2)
2

L,n

]2∗
dx

)2/2∗(∫
RN

u2
∗

n dx
) 2∗−2

2∗
.

From Lemma 4.1, we conclude that(∫
RN

[
unu

(2∗−2)
2

L,n

]2∗
dx

)2/2∗

≤ Cβp

∫
RN

u2nu
(2∗−2)
L,n dx,

or equivalently (∫
RN

[
unu

(2∗−2)
2

L,n

]2∗
dx

)2/2∗

≤ Cβp

∫
RN

u2
∗

n dx ≤ K <∞.



10 W. F. ALMEIDA, G. M. FIGUEIREDO EJDE-2025/110

Using the Fatou’s lemma in the variable L, we have∫
RN

u2
∗(2∗/2)

n dx ≤ K <∞,

and therefore the claim holds. □

4.1. Proof of Theorem 1.1. From (4.7) in Lemma 4.2, we have

|uε| ≤ C∥uε∥A,ϵ.

From Lemma 4.1, for ε > 0 sufficiently small, we have that

|uε| < a1/2

and this completes the proof.

Acknowledgments. W. F. Almeida was supported by the CAPES/OBMEP. G. M. Figueiredo
was supported by the CNPQ, CAPES, FAP-DF.

References

[1] C. O. Alves; Existence of positive solutions for an equation involving supercritical exponent in RN , Nonlinear

Analysis, 42 (2000), 573-581.
[2] C. O. Alves, O. H. Miygaki; Existence of positive solutions to a superlinear elliptic problem, Electron. J.

Differential Eq., Vol. 2001(2001), No. 11, pp. 1-12.

[3] C. O. Alves, G. M. Figueiredo; Multiple Solutions for a Semilinear Elliptic Equation with Critical Growth and
Magnetic Field, Milan J. Math., 82 (2) (2014), 389-405.

[4] C. O. Alves, G. M. Figueiredo, F. Furtado; On the number of solutions of NLS equations with magnetics fields

in expanding domains, J. Differential Equations, 251 (2011), no. 9, 2534-2548.
[5] C. O. Alves, G. M. Figueiredo, F. Furtado; Multiple solutions for a nonlinear Schrödinger equation with

magnetic fields, Comm. Partial Differential Equations, 36 (2011), no. 9, 1565-1586.

[6] V. Ambrosio, P. D’avenia; Nonlinear fractional magnetic Schrödinger equation: existence and multiplicity, J.
Differential Equations 264 (2018), no. 5, 3336-3368.

[7] G. Arioli, A. Szulkin; A semilinear Schrödinger equations in the presence of a magnetic field, Arch. Ration.

Mech. Anal., 170 (2003), 277–295.
[8] S. Barile; Multiple semiclassical states for singular magnetic nonlinear Schrödinger equations, Electron. J.

Differential Equations, 2008 (2008), No. 37, 18 pp.
[9] D. Barseghyan, S. Bernstein, B. Schneider, M.L. Zimmermann; Magnetic Dirichlet Laplacian in curved waveg-

uides, Opuscula Math. 45, no. 3 (2025), 293-305.

[10] S. Barile; A multiplicity result for singular NLS equations with magnetic potentials, Nonlinear Anal., 68 (2008),
no. 11, 3525-3540.

[11] S. Barile, S. Cingolani, S. Secchi; Single-peaks for a magnetic Schrödinger equation with critical growth, Adv.

Differential Equations, 11 (10) (2006), 1135-1166.
[12] S. Barile, G. M. Figueiredo; An existence result for Schrödinger equations with magnetic fields and exponential

critical growth, J. Elliptic and Parabolic Equations, 3(2017), pp 105-125.
[13] J. A. Cardoso, J. M. do O, E. S. Medeiros; Hamiltonian elliptic systems with nonlinearities of arbitrary growth,

Top. Meth. Nonlinear Analysis, 47, 2 2016, 593-612, Doi: 10.12775/TMNA.2016.018.
[14] J. Chabrowski, A. Szulkin; On the Schrödinger equation involving a critical Sobolev exponent and magnetic

field, Top. Meth. Nonlinear Anal. 25 (2005), 3–21.
[15] S. Cingolani, S. Secchi; Semiclassical limit for nonlinear Schrödinger equations with electromagnetic fields, J.

Math. Anal. Appl., 275 (2002), no. 1, 108-130.
[16] S. Cingolani, Semiclassical stationary states of nonlinear Schrödinger equations with an external magnetic

field, J. Differential Equations, 188 (2003), no. 1, 52-79.
[17] M. del Pino, P.L. Felmer; Local mountain pass for semilinear elliptic problems in unbounded domains, Calc.

Var. Partial Differential Equations, 4, no. 2, (1996) 121-137.

[18] L. Gongbao, S. Peng, C. Wang; Infinitely many solutions for nonlinear Schrödinger equations with electro-

magnetic fields, J. Differential Equations 251 (2011), no. 12, 3500-3521.
[19] M. F. Furtado, H. R. Zanata; Multiple solutions for a Kirchhoff equation with nonlinearity having arbitrary

growth, Bull. Aust. Math. Soc., 96(1), 98-109 (2017).
[20] Z. Guo; Elliptic equations with indefinite concave nonlinearities near the origin, J. Math. Anal. Appl. 367

(2010) 273-277.

[21] D. J. Griffiths, D. F. Schroeter; Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press,
2018.

[22] J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, 1999.
[23] C. Ji, V. Radulescu; Multiplicity and concentration of solutions to the nonlinear magnetic Schrödinger equa-

tion, Calc. Var. Partial Differential Equations, 59 (2020), no. 4, Paper No. 115, 28 pp.



EJDE-2025/110 MAGNETIC SCHRÖDINGER EQUATIONS 11

[24] L. D. Landau, E. M. Lifshitz; Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon Press, 1977.

[25] E. H. Lieb, M. Loss; Analysis, Graduate Studies in Mathematics, Vol. 14, Amer. Math. Soc., 1997.

[26] L. Lin, H. Tao; On the logarithmic p-Laplacian Schrödinger equation with multi-well potential: Multi-peak
positive solutions, Bulletin of Mathematical Sciences, (2025) 2550019.

[27] W. A. Strauss; Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.

[28] Y. Q. Li, T. V. Nguyen, B. L. Zhang; Existence, concentration and multiplicity of solutions for (p, N )-
Laplacian equations with convolution term, Bulletin of Mathematical Sciences, Vol. 15, No. 1 (2025) 2450009.

[29] M. Willem; Minimax Theorems, Birkhäuser, Boston 1996.

[30] Z.-Q. Wang; Nonlinear boundary value problems with concave nonlinearities near the origin, NoDEA Nonlinear
Differential Equations Appl. 8 (2001), 15-33.

Wendy F. Almeida

Universidade de Braśılia, Departamento de Matemática, Campus Darcy Ribeiro, 01, CEP 70910-900,
Braśılia, DF, Brazil

Email address: wendy fda@hotmail.com

Giovany M. Figueiredo

Universidade de Braśılia, Departamento de Matemática, Campus Darcy Ribeiro, 01, CEP 70910-900,

Braśılia, DF, Brazil
Email address: giovany@unb.br


	1. Introduction
	2. Notation and variational tools
	3. Auxiliary problems
	4. Proof of Theorem 1.1
	4.1. Proof of Theorem 1.1
	Acknowledgments

	References