Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 19, pp. 1–14.

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

NON-RADIAL NORMALIZED SOLUTIONS FOR A NONLINEAR

SCHRÖDINGER EQUATION

ZHI-JUAN TONG, JIANQING CHEN, ZHI-QIANG WANG

Abstract. This article concerns the existence of multiple non-radial positive

solutions of the L2-constrained problem

−∆u−Q(εx)|u|p−2u = λu, in RN ,∫
RN
|u|2dx = 1,

where Q(x) is a radially symmetric function, ε > 0 is a small parameter,

N ≥ 2, and p ∈ (2, 2 + 4
N

) is assumed to be mass sub-critical. We are

interested in the symmetry breaking of the normalized solutions and we prove

the existence of multiple non-radial positive solutions as local minimizers of
the energy functional.

1. Introduction

We consider nonlinear Schrödinger equations with a L2 constraint,

−∆u−Q(εx)|u|p−2u = λu, in RN ,∫
RN
|u|2dx = 1,

(1.1)

where 2 < p < 2 + 4
N , N ≥ 2, the potential function Q(x) is radially symmetric,

that is Q(|x|) = Q(x), and ε > 0 is a parameter. Many models of such type can
be seen in the literature. It is especially important in theory and application to
study the existence and properties of ground states and bound states solutions. The
existence of solutions for nonlinear Schrödinger equation, including their properties,
is of great interest in the field and has been studied extensively in the past (e.g.,
[1, 3, 8, 10, 11, 14, 15, 18]) and references therein. In addition, a growing number of
articles considering the existence of multiple bound states of nonlinear Schrödinger
equation, together with L2 constraint, have appeared in the field; see [4, 5, 6, 7, 12,
14, 17] and references therein.

Let d > 0 and µ > 0 be constants. First consider a case of constant potential
and define

c(d, µ) = inf
u∈H1(RN ),‖u‖22=µ

(1

2

∫
RN
|∇u|2dx− d

p

∫
RN
|u|pdx

)
. (1.2)

2020 Mathematics Subject Classification. 35J20, 35J60, 58E40.
Key words and phrases. Symmetry breaking; local minimizer; concentration;

nonlinear Schrödinger equations.
©2023. This work is licensed under a CC BY 4.0 license.

Submitted January 16, 2023 Published February 27, 2023.

1



2 Z.-J. TONG, J. CHEN, Z.-Q. WANG EJDE-2023/19

Here and after we use ‖ · ‖r to denote the Lr norm for r ≥ 1. Cazenave-Lions
[5] proved that c(d, µ) < 0 and c(d, µ) are attained, up to a translation, by a
radially symmetric minimizer. This minimizing feature leads to the orbital stability
of the standing waves associated with the time-dependent nonlinear Schrödinger
equation; see [5, 4]. Using the same method and being easier because of the compact
embedding from H1

r into Lp for N ≥ 2, one can prove that when Q(x) = Q(|x|) is a
radial function and replaces the constant d in the problem (1.2), the minimization
problem is also solvable in the space of radial functions; see [17]. Here H1

r denotes
the subspace of H1(RN ) of radially symmetric functions.

A natural question is that when Q is a radial potential whether there exist non-
radial solutions – symmetry breaking phenomenon. The question was studied by
Yang [17] in which the authors give conditions under which a ground state solution
is non-radial, i.e., symmetry breaking occurs. Then Yang [17] also gives conditions
which assure the existence of multiple non-radial bound state solutions. These
solutions are constructed as global minimizers of the energy functional in some
symmetric subspaces. More precisely it was proved that when N = 2 and N ≥ 4,
(1.1) has radial and non-radial solutions. Furthermore, there exist multiple non-
radial solutions [17] as ε → 0. However, the global minimization scheme may not
work for the case N = 3. The reason for such circumstances is that in the three-
dimensional case, multiple bumps may concentrate at the origin or concentrate
along the z-axis and run to infinity in two opposite directions, i.e., the north and
south poles (See Proposition 2.3 below for more details). Also for the existence of
multiple non-radial solutions the condition on Q in [17] depends on the number of
solutions, which is less desirable.

This article gives a different method to settle the issues related to the above
problem. We prove the existence of multiple non-radial positive solutions for all
dimensions N ≥ 2 with conditions on Q independent of the number of solutions.
The conclusion is much stronger than the original one. It is worth mentioning that
solutions given in this paper are found to be local minimizers, while the solutions
of [17] are global minimizers in symmetric subspaces. Under the conditions in this
paper we do not know whether the global minimizers exist and whether they are
non-radial solutions even if they do exist, yet local minimizers do exist under our
conditions.

To solve equation (1.1) we look for the critical points of the functional

Jε(u) =
1

2

∫
RN
|∇u|2dx− 1

p

∫
RN

Q(εx)|u|pdx, u ∈ H1(RN ) (1.3)

under the assumption ‖u‖22 = 1. We use the following assumption

(A1) Q ∈ C(RN ,R) is radially symmetric, i.e., Q(x) = Q(|x|), Q(x) achieves its
maximum on {x : |x| = 1}, and there exist a > 0 and σ0 > 0 such that

0 < a ≤ Q(x) ≤ max
RN

Q(x) = 1, for x ∈ RN , (1.4)

Q(|x|)−Q(1) < 0 for 0 < ‖x| − 1| ≤ 2σ0. (1.5)

For convenience, and without loss of generality, we have set the maximum value
of Q to be 1. Roughly speaking, Q is bounded from above and below by positive
constants and has an isolated global maximum point at |x| = 1. Now we state the
main theorem to be proved in this article.



EJDE-2023/19 SOLUTIONS FOR A SCHRÖDINGER EQUATION 3

Theorem 1.1. Assume N ≥ 2, p ∈ (2, 2 + 4
N ). Assume (A1) is satisfied. Let

k ≥ 1 be an integer. Then there exists εk > 0 such that for all 0 < ε < εk, (1.1)

has a non-radial positive solution uε,k satisfying limε→0 Jε(uε,k) = c(k
2−p
2 , 1). In

particularly, as ε→ 0, (1.1) has more and more non-radial positive solutions.

We remark that we can distinguish these solutions by separating their energies

using the asymptotic limits c(k
2−p
2 , 1) which is proved to be different for different

k in Lemma 2.1.
Finally, we summarize our work in terms of results and methods. We give a new

local minimization scheme which tracks down non-radial bound state solutions of
multi-bump type. This is motivated by the fact that in general the known global
minimization method cannot give these k-bump type concentrated solutions for
k ≥ 3 (this is particularly true for the dimension N = 3 as will be discussed in
Section 2). Our method allows to establish k-bump type solutions for any integer
k by taking ε small.

The article is organized as follows. In Section 2 we develop a basic formula
for the minimization problem c(d, µ) which will be used frequently in our proof
later.Then we introduce a local minimization scheme which will be used to construct
k−bump solutions. In Section 3 we first prove several lemmas in preparation of the
proof for the main theorem and we close with the proof of the main theorem. In
the Appendix we give the proof of a result in Section 2 showing that the global
minimization approach cannot give multi-bump type solutions in general.

2. Variational formulation and some technical results

Using scalings we first give a formula for the ground state energy in the constant
potential case. Here again for d > 0 and µ > 0, we define

c(d, µ) = inf
u∈H1(RN ),‖u‖22=µ

(
1

2

∫
RN
|∇u|2 − d

p

∫
RN
|u|pdx). (2.1)

We formulate the dependency of c(d, µ) in terms of d and µ, using and extending
the results of [13, 17].

Let ω > 0 be the unique positive radially symmetric solution of

−∆ω + ω = ωp−1, ω > 0, in RN . (2.2)

The ground state solution ω above is useful for the proof of the following lemma.

Lemma 2.1. Let d > 0, µ > 0, 2 < p < 2 + 4
N . Then

c(d, µ) = −c0d
4

4−N(p−2)µ
4−(N−2)(p−2)

4−N(p−2) , (2.3)

where c0 =
[4−N(p−2)]‖ω‖

−4
4−N(p−2)
2

4[4−(N−2)(p−2)] .

Proof. First by [17, Proposition 2.3] we have c(d, 1) = c(1, 1)d
4

4−N(p−2) . Similar

proof of this gives us c(d, µ) = c(1, µ)d
4

4−N(p−2) . For λ > 0, we use a scaling of w

where uλ(x) = λ
2
p−2ω(λx). From (2.2) we have

−∆uλ + λ2uλ =
(
uλ
)p−1

, (2.4)

‖uλ‖22 = λ
4−N(p−2)

p−2 ‖ω‖22. (2.5)



4 Z.-J. TONG, J. CHEN, Z.-Q. WANG EJDE-2023/19

Set µ = λ
4−N(p−2)

p−2 ‖ω‖22. Then λ2 = ( µ
‖ω‖22

)
2(p−2)

4−N(p−2) , i.e., uλ satisfies ‖uλ‖22 = µ and

−∆uλ + (
µ

‖ω‖22
)

2(p−2)
4−N(p−2)uλ =

(
uλ
)p−1

. (2.6)

By [13], we have almost everywhere in µ > 0,

∂

∂µ
c(1, µ) = −1

4

( µ

‖ω‖22

) 2(p−2)
4−N(p−2)

. (2.7)

Then integrating (2.7) with respect to µ, we can obtain

c(1, µ2)− c(1, µ1) = −1

4

∫ µ2

µ1

( µ

‖ω‖22

) 2(p−2)
4−N(p−2)

dµ.

Using c(1, µ1)→ 0 as µ1 → 0, we obtain

c(1, µ) = −1

4

‖ω‖
−4

4−N(p−2)

2 µ
4−(N−2)(p−2)

4−N(p−2)

4−(N−2)(p−2)
4−N(p−2)

= −c0µ
4−(N−2)(p−2)

4−N(p−2) .

�

Lemma 2.2. For k fixed there exists δk > 0, for every 0 < b ≤ δk, it holds

c(1, b) + c(k
2−p
2 , 1− b) > c(k

2−p
2 , 1). (2.8)

Proof. Consider a continuous function of b, f(b) = c(1, b) + c(k
2−p
2 , 1 − b), where

0 ≤ b ≤ 1. Then the result follows by a direct computation from (2.3). �

Since in some cases the global minimizers do not produce multi-bump solutions,
we introduce the method of local minimization to construct k-bump solutions. Be-
fore presenting the results, we introduce some notation. Let k ≥ 2 be an integer,
we define a subgroup of O(2)

G̃k =
{
g, g2, · · ·, gk = Id : g =

(
cos 2π

k − sin 2π
k

sin 2π
k cos 2π

k

)}
. (2.9)

Note that G̃k acts on R2 as rotation. Then to guarantee the invariance of the
group in the three-dimensional or higher dimensional cases, a group G is defined as
follows,

G = G̃k × Z2. (2.10)

Here Z2 is the reflection about the plane of x1, x2. In view of the above statement,
we require to establish a function space as follows

H1
G(RN ) = {u ∈ H1(RN ) : u is G-invariant}. (2.11)

In other words, if u ∈ H1
G, then u is G̃k-invariant with respect to (x1, x2) and

u is even in (x3, . . . , xN ). The solutions of (1.1) are constructed in G-invariant
subspaces, so that they are G-invariant.

To prove our main result, the existence of solutions in Theorem 1.1, we set up a
local minimization scheme. Choose δ > 0 such that

0 < δ < min{δk
2
,
σ0
2
}. (2.12)

Here δk is from Lemma 2.2 and σ0 is from condition (A1). We define

Oδ,ε = {u ∈ H1
G(RN ) :

∫
RN

u2 = 1, γε(u) ≥ 1− δ} (2.13)



EJDE-2023/19 SOLUTIONS FOR A SCHRÖDINGER EQUATION 5

where

γε(u) =

∫
T 1+σ0

ε

\T 1−σ0
ε

|u|2dx. (2.14)

Here for R > 0, TR = {x ∈ RN : |Px| < R} and P : RN → R2 is the linear
projection. Finally, we define a local minimization problem which will give us
desired k-bump solutions

c(Oδ,ε) = inf
u∈Oδ,ε,‖u‖22=1

Jε(u). (2.15)

We will locate k-bump solutions as minimizers in Oδ,ε of the energy functional Jε,
therefore as local minimizers in the full space for Jε on the L2 constraint. More
precisely, we will show that for ε > 0 small, c(Oδ,ε) is attained at an interior point
uε of Oδ,ε, and therefore uε is a critical point of Jε on ‖u‖22 = 1 and a solution
of (1.1). Then we establish an asymptotic energy estimate of Jε(uε) so we may
distinguish these solutions in k for ε > 0 small.

We finish this section by pointing out why a local minimization argument is
necessary for constructing these k-bumped solutions. We show in general the global
minimization cannot give k-bumped solutions.

Proposition 2.3. Assume N = 3, Q(x) = Q(|x|) is continuous with 0 < a ≤
Q(x) ≤ 1, Q attains its maximum at |x| = 1, and

lim sup
|x|→∞

Q(x) := q∞ < 1.

Then for ε > 0 small, the minimum of Jε with the constraint ‖u‖22 = 1 is achieved.

If Q(0) > k
2−p
2 , then the minimizers uε concentrate at the origin as ε→ 0.

The proof uses some results from [17] and is presented in the Appendix.

3. Asymptotic estimates and the proof of Theorem 1.1

Recall that for proving Theorem 1.1 we need to prove that the local minimization
problem (2.15) is solvable. We first give an auxiliary result on another minimization
problem. First define

TR = {x ∈ RN : |Px| < R}, (3.1)

where P : RN → R2 is the linear projection, and

T cR = RN \ TR. (3.2)

Base on the property of solutions to be constructed and to be compared, we establish
a space for R > 0,

X = XR = H1
0,G(T cR) = {u ∈ H1

0 (T cR) : gu = u,∀g ∈ G}. (3.3)

For R > 0 and b > 0, set

S(R, b) = inf
u∈X,‖u‖22=b

(1

2

∫
RN
|∇u|2 − 1

p

∫
RN
|u|pdx

)
. (3.4)

Next we state some asymptotic estimates for S(R, b).

Lemma 3.1. In the setting of (3.3) and (3.4), the following hold:



6 Z.-J. TONG, J. CHEN, Z.-Q. WANG EJDE-2023/19

(i) Let Rn → ∞, as n → ∞ and (un) ⊂ XRn be such that limn→∞ ‖un‖22 = b
and

lim
n→∞

(1

2

∫
T cRn

|∇un|2 −
1

p

∫
T cRn

|un|pdx
)

= A ≤ lim
R→∞

S(R, b). (3.5)

Then A = c(k
2−p
2 , b) and up to a subsequence, for every α > 0, there exists

r > 0 and (yn) ⊂ T cRn such that

lim inf
n→∞

∫
Br(yn)

|un|2dx ≥
b

k
− α. (3.6)

(ii) limR→∞ S(R, b) = c(k
2−p
2 , b).

Proof. By using special testing functions, we can easily obtain that for any R > 0,

S(R, b) ≤ c(k
2−p
2 , b).

Since S(R, b) is non-decreasing as R → ∞, limR→∞ S(R, b) exists. Now choose
a sequence Rn → ∞ and (un) ⊂ XRn , such that limn→∞ ‖un‖22 = b and A ≤
limR→∞ S(R, b). By the concentration-compactness principle [8, 9], we have three
possibilities.

If vanishing occurs, then for any r > 0, it holds

lim
n→∞

sup
y∈RN

∫
Br(y)

|un|2dx = 0.

It follows from [8, 9] that un → 0 in Lp(RN ) for 2 < p < 2∗ = 2N
N−2 . Then we

obtain limn→∞( 1
2

∫
RN |∇un|

2 − 1
p

∫
RN |un|

pdx) ≥ 0. This is a contradiction with

c(k
2−p
2 , 1) < 0.

Next because of the symmetry we can only expect compactness of the sequence
module the symmetry. If there exists b1 > 0 and (yn) ⊂ T cRn such that for any
α > 0, there exists r > 0,

lim inf
n→∞

∫
Br(yn)

|un|2dx ≥b1 − α. (3.7)

Since un is radial, one can get∫
Br(gyn)

|un|2dx ≥b1 − α.

Now we claim that kb1 = b. If kb1 < b, let η = η(t) be a smooth non-increasing
function on [0,+∞) such that η(t) = 1 for t ∈ [0, 1], η(t) = 0 for t ≥ 2 and
|η′(x)| ≤ 2. Write ηc(t) = 1− η(t). In (3.7) we choose αm → 0 and rm →∞. Then
we we can find unm and ynm such that

∫
Br(yn)

|unm |2dx ≥b1 − αm. We still name

this subsequence unm as un for simplicity of notations. Define

vn(x) =

k∑
i=1

η
( |x− giyn|

rn

)
un(x), (3.8)

ωn = un(x)− vn. (3.9)

Hence, ‖vn(x)‖22 → kb1 and ‖ωn(x)‖22 → b− kb1, as n→∞. Then

A+ o(1)



EJDE-2023/19 SOLUTIONS FOR A SCHRÖDINGER EQUATION 7

=
1

2

∫
RN
|∇un|2 −

1

p

∫
RN
|un|pdx

=
(1

2

∫
RN
|∇vn|2 −

1

p

∫
RN
|vn|pdx

)
+
(1

2

∫
RN
|∇ωn|2 −

1

p

∫
RN
|ωn|pdx

)
− C0

rn

=
(‖vn‖22

2b

∫
RN

∣∣√b∇vn
‖

vn‖2
∣∣2 − ‖vn‖p2

pbp/2

∫
RN

∣∣√bvn
‖vn‖ 2

∣∣pdx)
+
(‖ωn‖22

2b

∫
RN

∣∣√b∇ωn
‖ωn‖2

∣∣2 − ‖ωn‖p2
pbp/2

∫
RN

∣∣√bωn
‖ωn‖ 2

∣∣pdx)− C0

rn

≥ ‖vn‖
2
2

b
S(Rn, b) +

(‖vn‖22
pb

− ‖vn‖
p
2

pbp/2

)∫
RN

∣∣√bvn
‖vn‖2

∣∣pdx+
‖ωn‖22
b

S(Rn, b)−
C0

rn
.

Here C0 is independent of n. Sending n→∞, this implies

A ≥ kb1
b
A+

b− kb1
b

A+
1

p

[kb1
b
−
(kb1
b

)p/2]
lim
n→∞

∫
RN

∣∣√bvn
‖vn‖2

∣∣pdx.
Then we deduce that kb1 = b, since otherwise using limn→∞

∫
RN |un|

p
dx 6= 0 we

obtain a contradiction with the last term in the formula above positive.
Finally,choose αn → 0, rn → ∞,by doing a cut-off function which is similar to

(3.8),we can get ‖vn‖22 → b, as n → ∞. We can repeat the method above, using
limn→∞ ‖vn‖22 = b and Lemma 2.1 we obtain

A+ o(1) =
1

2

∫
RN
|∇un|2 −

1

p

∫
RN
|un|pdx+ o(1)

≥ 1

2

∫
RN
|∇vn|2 −

1

p

∫
RN
|vn|pdx+ o(1)

≥ kc(1, b
k

) + o(1)

= c(k
2−p
2 , b) + o(1).

Letting n→∞, o(1)→ 0, we obtain the result (i). The assertion (ii) follows from
(i) readily. �

For the minimization problem c(Oδ,ε), we will use the following asymptotic esti-
mates.

Lemma 3.2. (i) Let εn → 0 and (un) ⊂ Oδ,ε such that lim supn→∞ Jεn(un) ≤
c(k

2−p
2 , 1). Then there exists (yn) ⊂ R2×{−→0 } satisfying for any 0 < σ ≤ σ0

and
lim sup
n→∞

dist(yn, T 1+σ0
εn

\ T 1−σ0
εn

) <∞ (3.10)

and for any α > 0, there exists R > 0 such that

lim
n→∞

∫
BR(yn)

|un|2dx ≥
1

k
− α. (3.11)

(ii) limε→0 c(Oδ,ε) = c(k
2−p
2 , 1).

Proof. Let (un) ⊂ Oδ,εn be such that A = limn→∞ Jεn(un) ≤ c(k
2−p
2 , 1). Then for

any 0 < t ≤ 1− σ0, we have

lim sup
n→∞

∫
Tt/εn

|un|2dx = lim sup
n→∞

(1−
∫
T c
t/εn

|un|2dx)



8 Z.-J. TONG, J. CHEN, Z.-Q. WANG EJDE-2023/19

= 1− lim inf
n→∞

∫
Tt/εn

|un|2dx

≤ 1− (1− δ) = δ.

Next we select a sequence Rm+1 = 2Rm → ∞, as m → ∞. Up to a subsequence,
we suppose that

bm = lim sup
n→∞

∫
TRm

|un|2dx, b0 = lim
m→∞

bm.

Then using the fact that un ⊂ Oδ,εn we obtain b0 ≤ δ. Choose nm →∞, as m→∞
such that ∣∣bm − ∫

TRm

|unm |2dx
∣∣ ≤ 1

m
,
∣∣∣bm+1 −

∫
TRm+1

|unm |2dx
∣∣∣ ≤ 1

m
.

By doing a cut-off function which is similar to the proof of Lemma 3.1, we obtain
two sequences vn and ωn satisfying

lim
n→∞

∫
RN
|vn|2dx = b0, lim

n→∞

∫
RN
|ωn|2dx = 1− b0.

Note that (ωn) ⊂ H1
0,G(T cRn). Then by Lemma 3.1, we obtain

lim
n→∞

Jεn(ωn) ≥ lim
n→∞

S(Rn, 1− b0) = c(k
2−p
2 , 1− b0).

Claim 1: b0 = 0. Direct computation shows that there exists some C > 0 be such
that

Jεn(un)− Jεn(vn)− Jεn(ωn) ≥ − C

Rn
.

It follows that

A = lim
n→∞

Jεn(un)

≥ lim
n→∞

(Jεn(vn) + Jεn(ωn)− C

Rn
)

≥ c(1, b0) + c(k
2−p
2 , 1− b0) + o(1).

This implies c(k
2−p
2 , 1) ≥ c(1, b0)+c(k

2−p
2 , 1−b0). By Lemma 2.2, since 0 < b0 ≤ δk,

we have

c(1, b0) + c(k
2−p
2 , 1− b0) > c(k

2−p
2 , 1),

which is a contradiction. Thus, b0 = 0 and Claim 1 is proved.
Using Lemma 3.1, we conclude that for any α > 0, there exists R > 0 and

(yn) ⊂ RN be such that

lim
n→∞

inf
yn∈RN

∫
BR(yn)

|un|2dx ≥
1

k
− α. (3.12)

Claim 2: limn→∞ |P⊥yn| ≤ C where P⊥ = Id − P . If this claim is not the case,
then up to a subsequence, we can assume that limn→∞ |P⊥yn| = ∞. Since un is
G-invariant, then there exists g ∈ G such that |gP⊥yn − P⊥yn| → ∞ as n → ∞.

This gives a contradiction with the above estimate and
∫
RN |un|

2
dx = 1.



EJDE-2023/19 SOLUTIONS FOR A SCHRÖDINGER EQUATION 9

Claim 3: There exists constant C > 0 such that

lim sup
n→∞

dist(yn, εn
−1(Aσ0

)) ≤ C, (3.13)

where Aσ0
= {x : 1− σ0 ≤ |Px| ≤ 1 + σ0}. Assume the claim is not true, then up

to a subsequence we have dist(yn, εn
−1(Aσ0)) =∞.

We consider two cases. The first is |yn| < 1−σ0

εn
, then 1−σ0

εn
−|yn| → ∞; it follows

that ∫
T 1+σ0

εn

\T 1−σ0
εn

|un|2dx ≤ 1−
k∑
i=1

∫
BR(giyn)

|un|2dx ≤ kα.

Let α be sufficiently small, then it is a contradiction to γ(un) ≥ 1 − δ. For the
other case we have |yn| > 1+σ0

εn
. The proof is analogous to the above.

Claim 4: For every σ0 > σ > 0, there exists Cσ > 0 be such that

lim sup
n→∞

dist(yn, εn
−1(Aσ)) ≤ Cσ. (3.14)

If the claim is not true, then there exists σ ∈ (0, σ0) and up to a subsequence,
(yn) ⊂ εn

−1(Aσ) such that lim supn→∞ dist(yn, εn
−1(Aσ)) = ∞. Then we have

|yn| − 1+σ
εn
→∞ or 1−σ

εn
− |yn| → ∞.

Now we only consider the case |yn| − 1+σ
εn
→ ∞, the proof of the other case is

similar. Using condition (A1), and the continuity of Q(x), there exists 0 < a < 1
be such that

Q(|x|) ≤ a, for σ ≤
∣∣|x| − 1

∣∣ ≤ 2σ0.

i.e.,

Q(ε|y|) ≤ a, for
σ

ε
≤
∣∣|y| − 1

ε

∣∣ ≤ 2σ0
ε
.

Then 1+σ
ε ≤ |y| ≤

1+2σ0

ε .
Because of BR(yn) ⊂ (T 1+2σ0

εn

\ T 1+σ
εn

), we have Q(εnx) ≤ a, for x ∈ BR(yn).

Next, set vn(x) =
∑k
i=1 η( |x−g

iyn|
Rn

)un(x) satisfying limn→∞ ‖vn‖22 = 1 and (vn) ⊂
H1

0,G(T 1+2σ0
εn

). Then

c(k
2−p
2 , 1)

≥ lim
n→∞

Jεn(un)

= lim
n→∞

Jεn(vn) + o(1)

= lim
n→∞

(
1

2

∫
RN
|∇vn|2 −

1

p

∫
RN
|vn|pdx) + lim

n→∞

1

p

∫
RN

(1−Q(εx))|vn|pdx+ o(1)

≥ c(k
2−p
2 , 1) +

1− a
p

lim
n→∞

∫
RN
|vn|pdx

> c(k
2−p
2 , 1).

This is a contradiction. Thus, the proof of claim 4 is complete and the part (i) is
proved.

For part (ii), we note that it is easy to see using testing function, we have

lim sup
ε→0

c(Oδ,ε) ≤ c(k
2−p
2 , 1).

Then the assertion (ii) follows from the assertion (i). �



10 Z.-J. TONG, J. CHEN, Z.-Q. WANG EJDE-2023/19

We remark that for fixed varepsilon > 0, the function γε(u) is continuous in u,
and limn→∞ γεn(un) = 1 for the sequence in Lemma 3.2.

Lemma 3.3. For every α > 0, there exists R = R(α) > 0 and ε = ε(α) > 0 for
any 0 < ε < ε(α), if (un) ⊂ Oδ,ε is a minimizing sequence for c(Oδ,ε), then

lim inf
n→∞

∫
{x:ε−1(1− δ4 )≤|Px|≤ε−1(1+ δ

4 )}∩{x:|P⊥x|≤R}
|un|2dx ≥ 1− α. (3.15)

Proof. If the assertion is not true, then there exists α0 > 0 for any Rm → ∞ and
εm → 0, a minimizing sequence (um,n)∞n=1 (of fixed m) for c(Oδ,εm) such that

lim inf
n→∞

∫
{x | εm−1(1− δ4 )≤|Px|≤εm−1(1+ δ

4 )}∩{x |P⊥x|≤Rm}
|um,n|2dx < 1− α0.

Next, we choose a sequence nm →∞ such that

lim
m→∞

∫
{x|εm−1(1− δ4 )≤|Px|≤εm−1(1+ δ

4 )}∩{x||P⊥x|≤Rm}
|um,nm |

2
dx < 1− α0, (3.16)

lim
m→∞

Jεm(um,nm) = c(k
2−p
2 , 1). (3.17)

For convenience, here we denote um,nm by um. Applying Lemma 3.2 to (um), then
there exists (ym) ⊂ RN and εm|ym| → 1 for every α > 0, there exists R > 0 such
that

lim
m→∞

∫
BR(ym)

|um|2dx ≥
1

k
− α

2k
.

Take α = α0, the above statement also holds for some R0 > 0. Using the fact that
for any σ > 0, there exists Cσ > 0 such that

lim sup
m→∞

dist(ym, ε
−1
m (Aσ)) ≤ Cσ <∞.

Then we can assume that

(ym) ⊂
{
x : εm

−1(1− δ

8
) ≤ |Px| ≤ εm−1(1 +

δ

8
)
}
.

It follows that

BR0(ym) ⊂
{
x : εm

−1(1− δ

4
) ≤ |Px| ≤ εm−1(1 +

δ

4
)
}

and

lim inf
n→∞

∫
{x:ε−1

m (1− δ4 )≤|Px|≤ε
−1
m (1+ δ

4 )}∩{x:|P⊥x|≤Rm}
|um|2dx

≥
k∑
i=1

∫
BR0

(giym)

|um|2dx

≥ k(
1

k
− α0

2k
) ≥ 1− α0

2
,

which contradicts (3.16). Thus, the proof is complete. �

Lemma 3.4. There exists εk > 0 such that for any 0 < ε < εk, c(Oδ,ε) is attained
in the interior of Oδ,ε.



EJDE-2023/19 SOLUTIONS FOR A SCHRÖDINGER EQUATION 11

Proof. First we choose ε1 > 0 such that for any 0 < ε < ε1, it follows from Lemma

3.2 that c(Oδ,ε) ≤ 1
2c(k

2−p
2 , 1). Next, it follows from (2.3) that

c(k
2−p
2 , µ) = c(k

2−p
2 , 1)µ1+θ, where θ =

2(p− 2)

4−N(p− 2)
> 0.

Then there exists µ0 > 0 such that µ0
θ ≤ 1/2. Furthermore, applying Lemma 3.3

with α0 = 1
2 min{µ0, δ}, there exists R0 > 0 and ε2 > 0, for all 0 < ε < ε2, if

(un) ⊂ Oδ,ε is a minimizing sequence for c(Oδ,ε) such that

lim inf
n→∞

∫
{x:ε−1(1− δ4 )≤|Px|≤ε−1(1+ δ

4 )}∩{x:|P⊥x|≤R0}
|un|2dx ≥ 1− α0.

Now, we set εk = min {ε1, ε2} > 0, then fix 0 < ε < εk and let (un) ⊂ Oδ,ε be
a minimizing sequence of c(Oδ,ε). We will show this for a subsequence un → u in
L2(RN ).

Choose a sequence Rm+1 = 2Rm →∞, as m→∞. Then up to a subsequence,
we obtain

lim
n→∞

∫
BRm (0)

|un|2dx = bm.

It is noteworthy that bm ≥ 1− δ. Then it suffices to show limm→∞ bm = 1.
To the contrary, we assume limm→∞ bm = b < 1, it may produce a contradiction
as follows. Choose a sequence nm →∞, as m→∞ such that∣∣∣bm − ∫

BRm (0)

|unm |
2
dx
∣∣∣ ≤ 1

m
,
∣∣∣bm+1 −

∫
BRm+1

(0)

|unm |2dx
∣∣∣ ≤ 1

m
.

For simplicity of notation, we denote the sequence {unm}∞m=1 as {um}. Then we
define

vm(x) = η(
|x|
Rm

)um(x), ωm(x) = ηc(
|x|
Rm

)um(x).

It implies that

lim
m→∞

∫
RN
|vm|2dx = b,

lim
m→∞

∫
RN
|ωm|2dx = 1− b.

Next we claim that vm/‖vm‖2 ∈ Oδ,ε. In fact, it follows from (3.15) that

lim
n→∞

∫
{x:ε−1(1− δ4 )≤|Px|≤ε−1(1+ δ

4 )}∩{x:|P⊥x|≤R0}
|um|2dx ≥ 1− α0.

It follows from (un) ⊂ Oδ,ε and 1− b < δ that∫
T 1+σ0

ε

\T 1−σ0
ε

v2m
‖vm‖22

dx ≥ 1

‖vm‖22

∫
T 1+σ0

ε

\T 1−σ0
ε

|un|2dx ≥ 1− α0 > 1− δ.

Hence, the proof of the claim is complete.
Consequently,

c(Oδ,ε) + om(1)

=
1

2

∫
RN
|∇um|2 −

1

p

∫
RN

Q(εx)|um|pdx



12 Z.-J. TONG, J. CHEN, Z.-Q. WANG EJDE-2023/19

≥ (
1

2

∫
RN
|∇vm|2 −

1

p

∫
RN

Q(εx)|vm|pdx)

+
(1

2

∫
RN
|∇ωm|2 −

1

p

∫
RN

Q(εx)|ωm|pdx
)
− C

Rm

≥ ‖vm‖
2
2

2

∫
RN

|∇vm|2

‖vm‖22
− ‖vm‖

p
2

p

∫
RN

Q(εx)
vpm
‖vm‖p2

dx

+
(1

2

∫
RN
|∇ωm|2 −

1

p

∫
RN

Q(εx)|ωm|pdx
)
− C

Rm

≥ ‖vm‖22c(Oδ,ε) +
‖vm‖22 − ‖vm‖

p
2

p

∫
RN

Q(εx)
vpm
‖vm‖p2

dx+ c(k
2−p
2 , 1− b)− C

Rm
.

Sending m→∞, where C is a constant independent of R, we obtain

(1− b)c(Oδ,ε) ≥ c(k
2−p
2 , 1− b).

From the choice at the beginning, we derive that

1− b
2

c(k
2−p
2 , 1) ≥ c(k

2−p
2 , 1)(1− b)1+θ.

This gives 1/2 ≤ (1− b)θ which yields a contradiction with 1 − b ≤ α0 ≤ 1
2µ0.

Thus,1 − b = 0 or b = 1. Therefore, we proved un → u in L2(RN ). Then un → u
in Lp(RN ) by interpolation. Using the weakly lower semi-continuity, we deduce

c(Oδ,ε) ≤ Jε(u) = lim
n→∞

Jε(un) = c(Oδ,ε).

By the choice α0 ≤ δ
2 , we have u is in the interior of Oδ,ε. This completes the

proof. �

Proof of Theorem 1.1. The existence part of non-radial positive solution for each
k follows from Lemma 3.4 and the energy asymptotic as ε → 0, limε→0 Jε(uε,k) =

c(k
2−p
2 , 1), follows from Lemma 3.2. �

4. Appendix

Here we sketch the proof Proposition 2.3. We define a global minimization
problem with constraint

c(ε) = inf
u∈H1

G(R3),‖u‖22=1
Jε(u). (4.1)

First using suitable test functions we easily have lim supε→0 c(ε) ≤ c(k
2−p
2 , 1).

Under the conditions of Proposition 2.3, using [17, Lemma 4.2] for ε > 0 small
enough, c(ε) is achieved at some uε. Now we analyze the asymptotic behavior of
uε as ε → 0. Take a sequence εn → 0 and let un := uεn . Applying the concen-
tration compactness principle to (un), we can easily rule out the vanishing since

c(k
2−p
2 , 1) < 0. Now using the assumption Q(0) > k

2−p
2 and using testing func-

tions concentrating at the origin we have lim supε→0 c(ε) ≤ c(Q(0), 1) < c(k
2−p
2 , 1).

Now we claim un weakly converges to u 6= 0. Otherwise, by doing a cut-off we
obtain a sequence vn ∈ H1

0,G(T cRn) with Rn → ∞. Using Lemma 3.1 we have

lim supε→0 c(ε) ≥ c(k
2−p
2 , 1), a contradiction. Now if ‖u‖22 < 1 we may use Brezis-

Lieb Lemma to get a contradiction again. Thus compactness holds for the sequence
(un). Then it follows that lim supε→0 c(ε) = c(Q(0), 1) and for any α > 0 there



EJDE-2023/19 SOLUTIONS FOR A SCHRÖDINGER EQUATION 13

is R > 0 such that lim infn→∞
∫
RN u

2
ndx ≥ 1 − α. That is, un concentrate at the

origin as n→∞.

Acknowledgments. The authors are grateful to the referees for their useful com-
ments. This work was supported by Key Project of Natural Science Foundation of
Fujian (2020J02035), and by the NSFC (11831009).

References

[1] H. Berestycki, P. L. Lions; Nonlinear scalar field equations I: Existence of a ground state,

Arch, Rat. Mech. Anal., 82 (1983), 313–346.

[2] H. Brézis, E. Lieb; A relation between pointwise convergence of functions and convergence of
functionals, Proc. Amer. Math. Soc., 88 (1983), 486–490.

[3] J. Byeon; Standing waves for nonlinear Schrödinger equations with a radial potential, Non-

linear Anal., 50 (2002), Ser. A: Theory Methods, 1135–1151.
[4] T. Cazenave; Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics,

10. New York University, Courant Institute of Mathematical Sciences, New York; American
Mathematical Society, Providence, RI, 2003.

[5] T. Cazenave, P. L. Lions; Orbital stability of standing waves for some nonlinear Schrödinger

equations, Comm. Math. Phys., 85 (1982), 549–561.
[6] L. Jeanjean, S.-S. Lu; Nonradial normalized solutions for nonlinear scalar field equations,

Non-linearity, 32 (2019), 4942–4966.

[7] L. Jeanjean, S.-S. Lu,; On global minimizers for a mass constrained problem, Calc. Var.
Partial Differential Equations., 61 (2022), no. 6, Paper No. 214, 18 pp.

[8] P. L. Lions; The concentration-compactness principle in the calculus of variations, The locally

compact case. I, Ann. Inst. H. Poincaré Anal., Non Linéaire 1 (1984), 109–145.
[9] P. L. Lions; The concentration-compactness principle in the calculus of variations, The locally

compact case. II. Ann. Inst. H. Poincaré Anal., Non Linéaire 1 (1984), 223–283.

[10] Y. G. Oh; On positive multi-bump bound states of nonlinear Schrödinger equations under
multiple well potential, Comm. Math. Phys., 131 (1990), 223-253.

[11] P. H. Rabinowitz; On a class of nonlinear Schrödinger equations, Z. Angew, Math. Phys., 43
(1992), 270–291.

[12] M. Shibata; Stable standing waves of nonlinear Schrödinger equations with a general nonlinear

term, Manuscripta Math., 143 (2014), 221–237.
[13] A. Stefanov,; On the normalized ground states of second order PDE’s with mixed power

non-linearities, Comm. Math. Phys., 369 (2019), 929–971.

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

[15] Z.-Q. Wang; Existence and symmetry of multi-bump solutions for nonlinear Schrödinger

equations, J. Differential Equations., 159 (1999), 102–137.
[16] M. Willem; Minimax theorems, Birkhäuser Boston., 1996.

[17] J. Yang, Y. Li, Z.-Q. Wang; Radial and nonradial solutions for nonlinear schrödinger equation
with a constraint, Electron. J. Differential Equations., Special Issue 01 (2021), 225-237.

[18] C. A. Stuart; Bifurcation for Dirichlet problems without eigenvalues, Proc. Lond. Math. Soc.,
45 (1982), 169–192.

Zhi-Juan Tong

College of Mathematics and Statistics, Fujian Normal University, Fuzhou, 350117,
China

Email address: qsx20200630@student.fjnu.edu.cn

Jianqing Chen

College of Mathematics and Statistics, Fujian Normal University, Fuzhou, 350117,
China

Email address: jqchen@fjnu.edu.cn



14 Z.-J. TONG, J. CHEN, Z.-Q. WANG EJDE-2023/19

Zhi-Qiang Wang

College of Mathematics and Statistics, Fujian Normal University, Fuzhou, 350117,

China.
Department of Mathematics and Statistics, Utah State University, Logan, UT 84322,

USA

Email address: zhi-qiang.wang@usu.edu


	1. Introduction
	2. Variational formulation and some technical results
	3. Asymptotic estimates and the proof of Theorem ??
	4. Appendix
	Acknowledgments

	References