Special Issue in honor of Alan C. Lazer

Electronic Journal of Differential Equations, Special Issue 01 (2021), pp. 225–237.

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

RADIAL AND NON-RADIAL SOLUTIONS FOR A NONLINEAR

SCHRÖDINGER EQUATION WITH A CONSTRAINT

JIAXUAN YANG, YONGQING LI, ZHI-QIANG WANG

In memory of Professor Alan C. Lazer

Abstract. We study the classical nonlinear Schödinger equation with a ra-
dially symmetric potential and a constraint, for the mass subcritical case. We

obtain conditions that assure the existence of non-radial solutions. Also we

show symmetry breaking of the ground states, and the existence of multiple
non-radial solutions under additional conditions.

1. Introduction

In this article, we consider the nonlinear Schrödinger equation under an L2

constraint,

−∆u−Q(x)|u|p−2u = λu,∫
RN

|u|2dx = 1, u∈H1(RN )
(1.1)

where 2 < p < 2 + 4
N is mass subcritical. That is, we seek u and λ satisfying the

above equations. We assume that the potential function is a radial, Q(x) = Q(|x|).
We investigate conditions which assure the existence of both radial and non-radial
solutions. In particular we show that, while there always exists a radial solution,
the ground state is non-radial and there can be multiple non-radial solutions.

Before stating our results we discuss some background of the problem. Solu-
tions with a prescribed L2-norm are referred as normalized solutions. This type of
problems naturally arise from the studies of standing wave solutions of the time de-
pendent Schrödinger equations. In mathematical physics, finding solutions with a
prescribed L2-norm is particularly relevant since this quantity is preserved along the
time evolution. On this line of research we have the classical work of Cazenave-Lions
[5] and Stuart [15], and some more work later works such as [1, 6, 7, 8, 9, 10, 12, 13].
We refer the reader to these and references therein for the general discussion on the
existence and orbital stability of standing waves of the time dependent nonlinear
Schrödinger equations.

2010 Mathematics Subject Classification. 35J20, 35J60.

Key words and phrases. Ground states; symmetry breaking; k-bump solutions; concentration.
c©2021 This work is licensed under a CC BY 4.0 license.

Published October 26, 2021.

225



226 J. YANG, Y. LI, Z.-Q. WANG EJDE/SI/01

Solutions of this problem are critical points of the functional

J(u) =
1

2

∫
RN

|∇u|2dx− 1

p

∫
RN

Q(x)|u|pdx, u ∈ H1(RN ) (1.2)

under the constrained mass condition

‖u‖2 = 1. (1.3)

In particular, one can consider the minimization problem of J on the constraint

c = inf{J(v) : v ∈ H1(RN ), ‖v‖2 = 1}. (1.4)

When Q is a positive constant, this goes back to the classical work of Cazenave
and Lions [5] as applications of the concentration compactness principle [9, 10].
This gives rise to the existence of a minimizer solution which can be proved to be
orbitally stable for the corresponding initial value problem of the time dependent
nonlinear Schrödinger equation. For our purpose of notations later, we recall this
result as follow.

For each positive constant d, we define

cd = inf
u∈H1,‖u‖2=1

(1

2

∫
RN

|∇u|2dx− d

p

∫
RN

|u|pdx
)
. (1.5)

Then it was proved in [5] that cd < 0 and cd is always attained. Using the same
method and being easier using the compact embedding from H1

r into L2, one can
prove that when Q(x) is a radial function, the problem is also solvable in the space
of radial functions. Here we state this without giving a proof. Assume

(A1) Q = Q(|x|) is continuous and there exist positive constants b2 ≥ b1 > 0
such that b2 ≥ Q(x) ≥ b1.

Theorem 1.1. Assume (A1), N ≥ 2, and 2 < p < 2 + 4
N . Then

crad = inf{J(v) : v ∈ H1
r (RN ), ‖v‖2 = 1}

is achieved, where H1
r (RN ) is the radial subspace of H1(RN ).

By this theorem, the problem (1.1) has a radial solution. When we consider
the minimization problem (1.4) in the full H1 the minimization problem may or
may not be solvable. Our focus in this paper is to investigate a class of potential
function Q that assure the existence of minimizers, and to give conditions for the
minimizers to be non-radial and for multiple non-radial solutions. We make the
following assumptions.

(A2) max{q0, q∞} < qM where q0 := Q(0) and q∞ := lim sup|x|→∞Q(x).

To study the symmetry breaking phenomenon we will magnify the conditions on
the potential and study the family of problems with a small parameter ε > 0,

−∆u−Q(εx)|u|p−2u = λu,

‖u‖2 = 1, u∈H1(RN ).
(1.6)

We write the functional depending on the parameter ε as

Jε(u) =
1

2

∫
RN

|∇u|2dx− 1

p

∫
RN

Q(εx)|u|pdx, u ∈ H1(RN ), (1.7)

under the constrained mass condition

‖u‖2 = 1. (1.8)



EJDE-2021/SI/01 RADIAL AND NON-RADIAL SOLUTIONS 227

In particular, one can consider the minimization problem of Jε on the constraint

c(ε) = inf{Jε(u) : u ∈ H1(RN ), ‖u‖2 = 1}. (1.9)

We will also use crad(ε) to denote the ground state energy in the radial class (which
is always achieved by Theorem 1.1)

crad(ε) = inf{Jε(u) : u ∈ H1
r (RN ), ‖u‖2 = 1}.

Theorem 1.2. Assume (A1), (A2), N ≥ 1, and 2 < p < 2 + 4
N . Then there exists

ε0 > 0 such that for all 0 < ε < ε0, c(ε) is achieved at some uε. Moreover, let uε be
a minimizer of c(ε), and urad

ε be a minimizer of crad. Then

lim
ε→0

crad(ε) := Jε(u
rad
ε ) = cq0 ,

lim
ε→0

cε = Jε(uε) = cqM .

In particular, for ε small, uε is non-radial.

Next we seek more non-radial solutions of problem (1.6). Since Q is radial there
exists a radial solution by Theorem 1.1. In Theorem 1.2 we prove that the ground
states are non-radial. Next we show that under suitable conditions the problem has
multiple non-radial solutions. These non-radial solutions appear as ground states
in some other symmetric subspaces. Here is the main result and see Section 3 for
more detailed descriptions of asymptotic profiles of these solutions.

Theorem 1.3. Assume (A1), (A2), N = 2, and 2 < p < 4. Let k be a positive
integer. Suppose

qM
max{q0, q∞}

> k
p−2
2 .

Then there exists εk > 0 such that for all 0 < ε < εk, problem (1.6) has k non-radial
solutions.

Being minimizers of some symmetric subspaces all these solutions from the above
theorems are positive solutions. We also remark that symmetry breaking was stud-
ied in [6] for the two dimensional mass critical problem under a linear trapping
potential.

The organization of the paper is as follows. In section 2 we list some preliminaries
and state and prove a useful lemma on the ground state energy in terms of constant
potential. Section 3 is devoted to the proof of Theorem 1.2 proving the existence
and symmetry breaking of ground state solutions. In Section 4 we prove Theorem
1.3 giving multiple non-radial solutions, and discuss some possible extensions.

2. Preliminaries

We donate by H1
r (RN ) the space of radially symmetric functions u(x) = u(|x|)

which satisfy u(x), ∇u(x)∈L2(RN ). We also use notation

‖u‖q =
(∫

RN

|u(x)|q
)1/q

, for q∈[1,∞) and u∈Lq(RN ), (2.1)

‖u‖H1 = (‖∇u‖22 + ‖u‖22)1/2. (2.2)

The following is a special case of the well-known Gagliardo Nirenberg inequality
[4].



228 J. YANG, Y. LI, Z.-Q. WANG EJDE/SI/01

Lemma 2.1. Let p ≥ 2, 0 ≤ θ < 1 be such that

1

p
= θ(

1

2
− 1

N
) +

1− θ
2

. (2.3)

Then there exists constant C = C(p) independent of u such that

‖u‖p ≤ C‖∇u‖θ2‖u‖1−θ2 , ∀u∈W 1,2(RN ). (2.4)

The following lemma 2.2 is the Concentration-Compactness principle [9, 10, 17].

Lemma 2.2. Let µn be a sequence of nonnegative L1 functions on RN . For r > 0
we define a family of concentration functions,

Qn(r) := sup
y∈RN

∫
Br(y)

µn.

Suppose that µn is a sequence of L1-functions on RN ,

µn ≥ 0,

∫
RN

µndx = 1.

Then there exist a subsequence of µn (still denoted by µn) such that

α = lim
m→∞

lim
n→∞

Qn(m)

exists and one of the following three statements holds.

(i) (vanishing) α = 0 and for any R > 0 we have

lim
n→∞

sup
y∈RN

∫
BR(y)

µn = 0; (2.5)

(ii) (compactness) α = 1 and for any δ > 0 there are R > 0 and {yn} ⊂ RN
such that

lim inf
n→∞

∫
BR(yn)

µn ≥ 1− δ; (2.6)

(iii) (dichotomy) for each α∈(0, 1) and δ > 0 there are R > 0 and {yn} ⊂ RN
such that for all r ≥ R and r′ ≥ R

lim sup
n→∞

∣∣α− ∫
Br(yn)

µn
∣∣+
∣∣(1− α)−

∫
RN\Br′ (yn)

µn
∣∣ < δ. (2.7)

Using the concentration compactness principle, it was proved that for any d > 0,
the cd defined in (1.5) satisfies cd < 0 and that cd is attained. When Q(x) = Q(|x|),
the attained-ness of the radial case crad can be done in a similar and even easier
by the compact embedding from H1

r (RN ) into Lp(RN ) for N ≥ 2, 2 < p < 2N
N−2

(see [14, 4]), which gives the conclusion of Theorem 1.1. For N = 1, namely the
case of even functions, one can easily rule out dichotomy to get compactness for
the existence of a minimizer. We omit the proof here.

We finish this section with a simple but useful result relating the minimum values
of cd in terms of d.

Proposition 2.3. When the potential function in (1.5) is constant, it holds

cd = d
4

4−N(p−2) c1. (2.8)



EJDE-2021/SI/01 RADIAL AND NON-RADIAL SOLUTIONS 229

Proof. From now on for any function u and λ > 0 we use the notation uλ(x) =
λN/2u(λx). Note that ‖uλ‖2 = ‖u‖2 for any λ > 0. Choose u1 such that c1 = J(u1).
Using uλ1 (x) as a testing function we have

cd ≤ λ2
(1

2

∫
RN

|∇u1|2dx−
1

p

∫
RN

λ
N(p−2)

2

λ2
d|u1|pdx

)
.

Choosing λ0 such that

λ
N(p−2)

2
0

λ2
0

d = 1,

i.e., λ0 = d
4

4−N(p−2) , we obtain

cd ≤ λ2
0J(u1) = λ2

0c1 = d
4

4−N(p−2) c1. (2.9)

In a similar way,

c1 ≤ λ2
1

(1

2

∫
RN

|∇ud|2dx−
1

p

∫
RN

λ
N(p−2)

2
1

λ2
1

|ud|pdx
)

where cd = J(ud), and

uλ1

d = λ
N/2
1 ud(λ1x).

Therefore when we set:

d =
λ

N(p−2)
2

1

λ2
1

(2.10)

and obtain
c1 ≤ λ2

0J(u1) = λ2
0c1 = d−

4
4−N(p−2) cd. (2.11)

Then the result follows from (2.9) and (2.11). �

3. Existence and symmetry breaking of the ground states

In this section, we study the existence and symmetry property of the ground
state solutions under our conditions (A1) and (A2). We will show that for ε small,
the ground states are not radially symmetric.

Proposition 3.1.
lim
ε→0

c(ε) = cqM . (3.1)

Proof. Let uM be a minimizer of cqM . By the definition of cqM , we have

cqM = inf
‖u‖2=1

1

2

∫
RN

|∇u|2dx− 1

p
qM

∫
RN

|u|pdx

=
1

2

∫
RN

|∇uM |2dx−
1

p

∫
RN

qM |uM |pdx.

Let x0 be such thatQ(x0) = qM . We set vn(x) = uM (x−x0

εn
), where n→∞, εn → 0.

Hence we have ‖vn‖2 = ‖uM‖2 = 1. Then

c(εn) = inf
‖u‖2=1

(1

2

∫
RN

|∇u|2dx− 1

p

∫
RN

Q(εnx)|u|pdx
)

≤ 1

2

∫
RN

|∇vn|2dx−
1

p

∫
RN

Q(εnx)|vn|pdx

=
1

2

∫
RN

|∇vn|2dx−
1

p
qM

∫
RN

|vn|pdx+
1

p

∫
RN

(qM −Q(εnx))|vn|pdx



230 J. YANG, Y. LI, Z.-Q. WANG EJDE/SI/01

= cqM +
1

p

∫
RN

(qM −Q(εnx))|vn|pdx.

So we turn the complicated problem into proving the claim

1

p

∫
RN

(qM −Q(εnx))|vn|pdx→ 0.

By a change of variable we have∫
RN

(qM −Q(εnx))|vn|pdx =

∫
RN

(qM −Q(εnx+ x0))|uM |pdx.

By the dominated convergence theorem we have the desired estimate. Hence we
obtain

lim sup
ε→0

c(ε) ≤ cqM . (3.2)

Next by the definition of qM we have Q(εnx) ≤ qM , and by the energy functional
form, we have

c(ε) ≥ cqM . (3.3)

Consequently, limε→0 c(ε) = cqM . �

Lemma 3.2. For a fixed ε > 0, if c(ε) < cq∞ , then c(ε) is attained.

Proof. We fix an ε > 0 such that c(ε) < cq∞ . Let un be a minimizing sequence
of c(ε). Because c(ε) < 0, there is no vanishing for un. We claim that there is no
dichotomy for un. Otherwise, by Brezis-Lieb lemma, we have α =

∫
RN |u|2dx, 1 −

α =
∫
RN |un− u|2dx+ o(1), and un ⇀ u in H1(RN ). Since 1−α ∈ (0, 1), it follows

that

αp/2 < α, (1− α)p/2 < (1− α).

Hence we obtain

c(ε) ≥ 1

2

∫
RN

|∇un|2dx−
1

p

∫
RN

Q(εx)|un|pdx+ o(1)

≥ (1− α)c(ε) + αc(ε) + o(1)

+ [(1− α)− (1− α)p/2]

∫
RN

Q(εx)
∣∣ (un − u)

‖un − u‖2
∣∣pdx

+ [α− αp/2]

∫
RN

Q(εx)
∣∣ u

‖u‖2
∣∣pdx.

Sending n→∞ we obtain c(ε) > c(ε), a contradiction.
With vanishing and dichotomy ruled out, we have compactness for un from

Lemma 2.2 (ii), i.e., up to a subsequence there exist (xn) such that for any δ > 0
there is R > 0, it holds

lim inf
n→∞

∫
BR(xn)

u2
ndx ≥ 1− δ.

Let η(t) be a cut-off function such that η(t) = 1 for |t| ≤ 1 and η(t) = 0 for |t| ≥ 2.
Define ηR(t) = η(t/R). Define vn = ηR(|x−xn|)un(x)/‖ηR(|x−xn|)un(x)‖2. Then
we have

c(ε) = inf
‖u‖2=1

Jε(u) ≥ 1− δ
2

∫
RN

|∇vn|2dx−
1

p

∫
B2R(xn)

Q(εx)|vn|pdx+O(δ)+o(1/R).



EJDE-2021/SI/01 RADIAL AND NON-RADIAL SOLUTIONS 231

Since q∞ < qM for n large we have for γ > 0, Q(εx) ≤ q∞ + γ < qM for all
x ∈ B2R(xn). Thus for n large we have

c(ε) ≥ 1− δ
2

∫
RN

|∇vn|2dx−
1

p

∫
B2R(xn)

(q∞ + γ)|vn|pdx+O(δ) + o(1/R)

≥ cq∞+γ + o(1/R) +O(ε) +O(δ).

Sending δ → 0 and R→∞ we obtain limε→0 c(ε) ≥ cq∞+γ . Since γ > 0 is arbitrary,
we obtain limε→0 c(ε) ≥ cq∞ which is a contradiction. �

Next we prove the limiting behavior of the ground state energy in the radially
symmetric class.

Proposition 3.3. limε→0 crad(ε) = limε→0 Jε(u
rad
ε ) = cq0 .

Proof. Let u a radial minimizer of cq0 . Using this as a testing function, we easily
have limε→0 Jε(u) = cq0 which implies lim supε→0 c(ε) ≤ cq0 .

Now for any sequence εn → 0 we write un = urad
εn . By Lions Lemma [17], there

is no vanishing for this sequence.
Because uns are radial functions, it is easy to rule out dichotomy of this sequence.

Also because of the radial symmetry [14], for compactness we have for any δ > 0
there is R > 0 such that lim infn→∞

∫
BR(0)

u2
ndx ≥ 1 − δ. This implies un → u in

Ls(R2) for any 2 < s <∞ [14]. Then we have

cq0 ≤
1

2

∫
RN

|∇u|2dx− 1

p

∫
RN

q0|u|pdx ≤ lim inf
n→∞

Jεn(un).

Thus we obtain the desired estimate. �

Then Theorem 1.3 follows from Theorem 1.2, Propositions 3.1, 3.2 and 3.3.

4. Multiple non-radial solutions

To construct multiple non-radial solutions we use the group invariance property
of the problem. First, for each integer k ≥ 2 we define the group G = Gk as follows

G =
{
g, g2, . . ., gk = Id : g =

(
cos 2π

k − sin 2π
k

sin 2π
k cos 2π

k

)}
. (4.1)

We will work in the G-invariant subspace of functions H1
G(R2). We define the

working space as the following

ΓG = {u∈H1
G(R2) : ‖u‖2 = 1} (4.2)

and we have the ground state energy in the G-invariant subspace

c(ε, k) = inf
u∈ΓG

Jε(u). (4.3)

Lemma 4.1.

lim sup
ε→0

c(ε, k) ≤ c
k

2−p
2 qM

(4.4)

where as in (1.5),

c
k

2−p
2 qM

= inf
u∈H1,‖u‖2=1

(1

2

∫
R2

|∇u|2dx− k
2−p
2 qM
p

∫
R2

|u|pdx
)
. (4.5)



232 J. YANG, Y. LI, Z.-Q. WANG EJDE/SI/01

Proof. First of all, let x0 be such that Q(x0) = qM and consider

uiε(x) = u(x− gixε), i = 1, 2, . . ., k (4.6)

where u is the solution that corresponds to the ground state energy c
k

2−p
2 qM

, and

xε = x0

ε . Then we can obtain ‖
∑k
i=1 u

i
ε‖22 =

∑k
i=1 ‖uiε‖22 + o(1) = k + o(1), where

o(1)→ 0 as ε→ 0. Note that
∑k
i=1 u

i
ε/‖
∑k
i=1 u

i
ε‖2 ∈ ΓG. Thus we have

c(ε, k) ≤ Jε(
k∑
i=1

ui/‖
k∑
i=1

uiε‖2)

=
1

2(k + o(1))

∫
R2

|∇(

k∑
i=1

ui)|2dx−
1

p(k + o(1))
p
2

∫
R2

Q(εx)|
k∑
i=1

ui|pdx

=
1

2

1

k

k∑
i=1

∫
R2

|∇ui|2dx−
k−

p
2

p

k∑
i=1

∫
R2

Q(εx)|ui|pdx+ o(1)

=
1

2

∫
R2

|∇u|2dx− k
2−p
2

p

∫
R2

qM |u|pdx+ o(1).

This gives
lim sup
ε→0

c(ε, k) ≤ c
k

2−p
2 qM

. �

Lemma 4.2. For fixed k, when c(ε, k) < cq∞ , c(ε, k) is achieved.

Proof. Assume that (un) is minimizing sequence, i.e. Jε(un)→ c(ε, k) < cq∞ . Then
the minimizing sequence (un) is bounded in H1, and we may assume un⇀u in H1

and un → u a.e. in R2.
We claim that u 6=0. If not, we assume u = 0. Since lim sup|x|→∞Q(x) = q∞ <

qM for any γ > 0, there exists R > 0 such that Q(εx) ≤ q∞ + γ < qM for all
x ∈ BcR(0). Then we introduce a cut-off function

η(t) =

{
1, if |t| ≥ 2,

0, if |t| ≤ 1.
(4.7)

We let vn = un·η( |x|R ). Since u = 0, when n → ∞, we have ‖vn − un‖H1 → 0
and ‖vn‖2 → 1, n→∞. Then we normalize it

ṽn =
vn
‖vn‖2

∈ΓG. (4.8)

Then we have

c(ε, k) = Jε(un) = Jε(ṽn) + o(1) =
1

2

∫
R2

|∇ṽn|2dx−
1

p

∫
Bc

R(0)

Q(εx)|ṽn|pdx+ o(1).

Hence we can obtain

c(ε, k) ≥ 1

2

∫
R2

|∇ṽn|2dx−
1

p

∫
R2

(q∞ + γ)|ṽn|pdx+ o(1).

Sending n → ∞, we have c(ε, k) ≥ cq∞+γ . Since γ > 0 is arbitrary we obtain
c(ε, k) ≥ cq∞ , a contradiction. Similar to the proof of Lemma 3.2 we can use the
Lemma 2.2 to prove that ‖u‖2 = 1. Then we can obtain un is converges strongly
to u in Lp. Using the weak lower continuity of norm and the definition of c(ε, k),
we can obtain Jε(u) = c(ε, k), which implies that c(ε, k) can be achieved. �



EJDE-2021/SI/01 RADIAL AND NON-RADIAL SOLUTIONS 233

From the previous two lemmas we see for fixed k, there exists εk > 0 such that
for all ε < εk, c(ε, k) is attained. We will examine the asymptotic behavior of c(ε, k)
as ε→ 0. Then we will be able to distinguish between these ground state energies.

Lemma 4.3. Under the conditions of Theorem 1.3, we have

lim
ε→0

c(ε, k) = c
k

2−p
2 qM

.

Proof. We just need to consider the reverse inequality

lim inf
ε→0

c(ε, k) ≥ c
k

2−p
2 qM

. (4.9)

Let εn → 0. By the last two lemmas, we assume that un∈ΓG is such that Jεn(un) =
c(εn, k). From Lemma 4.1, we have

lim sup
ε→0

c(ε, k) ≤ c
k

2−p
2 qM

< 0. (4.10)

From this and Lemma 2.1, (un) is bounded in H1. Consequently, we may assume
that un⇀u in H1, and un → u a.e. in R2.

We claim that u = 0. If not, assume u6=0. Let vn = un − u. By the Brezis-Lieb
lemma [2, 17], we can assume ‖u‖22 = α, ‖vn‖22 = 1− α+ o(1).

However, if α = 1, we can obtain c(εn, k) → cq0 , n → ∞. It contradicts Lemma
4.1, so we have α∈(0, 1). Since α∈(0, 1), we have

(1− α) > (1− α)p/2, α > αp/2. (4.11)

Hence

c(εn, k)

= Jεn(un) = Jεn(u) + Jεn(vn) + o(1)

=
1

2

∫
R2

|∇u|2dx− 1

p

∫
R2

Q(εnx)|u|pdx+
1

2

∫
R2

|∇vn|2dx

− 1

p

∫
R2

Q(εnx)|vn|pdx+ o(1)

=
1

2

∫
R2

|∇u|2dx− 1

p

∫
R2

q0|u|pdx+
1

2

∫
R2

|∇vn|2dx

− 1

p

∫
R2

Q(εnx)|vn|pdx+ o(1)

= ‖u‖22
(1

2

∫
R2

∣∣ ∇u
‖u‖2

∣∣2dx− q0

p

∫
R2

∣∣ u

‖u‖2
∣∣pdx)+

q0

p
‖u‖22

∫
R2

∣∣ u

‖u‖2
∣∣pdx

− 1

p

∫
R2

q0

∣∣ u

‖u‖2
∣∣pdx(‖u‖22)p/2

+ ‖vn‖22
(1

2

∫
R2

∣∣ ∇vn
‖vn‖2

∣∣2dx− 1

p

∫
R2

Q(εnx)
∣∣ vn
‖vn‖2

∣∣pdx)
+

1

p
‖vn‖22

∫
R2

Q(εnx)
∣∣ vn
‖vn‖2

∣∣pdx− 1

p
(‖vn‖22)p/2

∫
R2

Q(εnx)
∣∣ vn
‖vn‖2

∣∣pdx+ o(1)

≥ αcq0 +
α− αp/2

p

∫
R2

q0

∣∣ u

‖u‖2
∣∣pdx+ (1− α)c(ε, k)

+
1

p
[(1− α)− (1− α)p/2]

∫
R2

Q(εnx)
∣∣ vn
‖vn‖2

∣∣pdx+ o(1).



234 J. YANG, Y. LI, Z.-Q. WANG EJDE/SI/01

We assume A = limn→∞ c(εn, k). Then

A ≥ αcq0 + (1− α)A+
α− αp/2

p

∫
R2

q0

∣∣ u

‖u‖2
∣∣pdx

+
1

p
[(1− α)− (1− α)p/2]

∫
R2

Q(εnx)
∣∣ vn
‖vn‖2

∣∣pdx
≥ α lim sup

n→∞
c(εn, k) + (1− α)A+

α− αp/2

p

∫
R2

q0

∣∣ u

‖u‖2
∣∣pdx

+
1

p
[(1− α)− (1− α)p/2]

∫
R2

Q(εnx)
∣∣ vn
‖vn‖2

∣∣pdx
≥ A+

α− αp/2

p

∫
R2

q0

∣∣ u

‖u‖2
∣∣pdx

+
1

p
[(1− α)− (1− α)p/2]

∫
R2

Q(εnx)
∣∣ vn
‖vn‖2

∣∣pdx
which is a contradiction; therefore u = 0.

Because lim supε→0 c(ε, k) < 0, there is no vanishing for un. By Lemma 2.2,
exists |xn| → ∞, α1 > 0, for ∀δ > 0,∃R > 0, we have for i = 1, 2, . . . , k,

lim inf
n→∞

∫
BR(gixn)

|un(x)|2dx ≥ α1 − δ. (4.12)

Then we define the sequences vn, wn via a smooth non-increasing cut-off function
ξ,

vn(x) =

k∑
i=1

ξ(
|x− gixn|

R
)un(x), wn(x) =

k∑
i=1

[1− ξ( |x− gixn|
R

)]un(x) (4.13)

where xn = x0/εn and

ξ(t) =

{
1, if |t| ≤ 1,

0, if |t| ≥ 2.
(4.14)

Obviously,

un = vn + wn. (4.15)

Then we have ‖vn‖22 → kα1, ‖wn‖22 → 1 − kα1 due to Lemma 2.2. Then we need
to prove that kα1 = 1.

If not, then 1− kα1 > 0, hence

kα1∈(0, 1), (1− kα1) > (1− kα1)p/2, kα > (kα1)p/2. (4.16)

Therefore

c(εn, k)

= Jεn(un) ≥ Jεn(vn) + Jεn(wn) +O(δ) +O(
1

R
)

=
1

2

∫
R2

|∇vn|2dx−
1

p

∫
R2

Q(εnx)|vn|pdx+
1

2

∫
R2

|∇wn|2dx

− 1

p

∫
R2

Q(εnx)|wn|pdx+O(δ) +O(
1

R
)

= ‖vn‖22
(1

2

∫
R2

∣∣∣ ∇vn‖vn‖2

∣∣∣2dx− 1

p

∫
R2

Q(εnx)
∣∣ vn
‖vn‖2

∣∣pdx)



EJDE-2021/SI/01 RADIAL AND NON-RADIAL SOLUTIONS 235

+ ‖vn‖22
1

p

∫
R2

Q(εnx)
∣∣ vn
‖vn‖2

∣∣pdx− (‖vn‖22)p/2
1

p

∫
R2

Q(εnx)
∣∣ vn
‖vn‖2

∣∣pdx
+ ‖wn‖22

(1

2

∫
R2

∣∣∣ ∇wn‖wn‖2

∣∣∣2dx− 1

p

∫
R2

Q(εnx)
∣∣ wn
‖wn‖2

∣∣pdx)
+ ‖wn‖22

1

p

∫
R2

Q(εnx)
∣∣ wn
‖wn‖2

∣∣pdx
− (‖wn‖22)p/2

1

p

∫
R2

Q(εnx)
∣∣ wn
‖wn‖2

∣∣pdx+O(δ) +O(
1

R
)

≥ kα1c(εn, k) + (1− kα1)c(εn, k)

+ [kα1 − (kα1)p/2]
1

p

∫
R2

Q(εnx)
∣∣ vn
‖vn‖2

∣∣pdx
+ [(1− kα1)− (1− kα1)p/2]

1

p

∫
R2

Q(εnx)
∣∣ wn
‖wn‖2

∣∣pdx+O(δ) +O(
1

R
).

Sending δ → 0 (and therefore R → ∞) we obtain a contradiction. Consequently,
kα1 = 1.

By Lemma 2.2, there exists a sequence (xn) satisfying |xn| → ∞ such that for
any δ > 0 there exists R > 0, we have, for i = 1, . . . , k,

lim inf
n→∞

∫
BR(gixn)

|un(x)|2dx ≥ 1

k
− δ. (4.17)

We let

vn(x) =

k∑
i=1

ξ(
|x− gixn|

R
)un(x). (4.18)

Then we have

lim
n→∞

∫
R2

|vn|2dx = 1. (4.19)

Now we have

c(εn, k)

= Jεn(un)

≥ Jεn(vn) +O(δ) +O(
1

R
)

=
k

2

∫
B2R(xn)

|∇un|2dx−
k

p

∫
B2R(xn)

Q(εnx)|un|pdx+O(δ) +O(
1

R
)

≥ k·1
k

(1

2

∫
B2R(xn)

|∇un|2
1
k

dx− 1

p
α

p−2
2

1 qM

∫
B2R(xn)

|un|p
1
k

p/2
dx
)

+O(δ) +O(
1

R
)

=
1

2

∫
B2R(xn)

∣∣∣ ∇un√
1/k

∣∣∣2dx− 1

p
k

2−p
2 qM

∫
B2R(xn)

∣∣∣ un√
1/k

∣∣∣pdx+O(δ) +O(
1

R
)

=
1

2

∫
B2R(xn)

∣∣∣ ∇un‖un‖2

∣∣∣2dx− 1

p
k

2−p
2 qM

∫
B2R(xn)

∣∣∣ un
‖un‖2

∣∣∣pdx+O(δ) +O(
1

R
)

≥ c
k

2−p
2 qM

+O(δ) +O(
1

R
).

Sending n→∞ we have

lim inf
ε→0

c(ε, k) ≥ c
k

2−p
2 qM

+O(δ) +O(
1

R
).



236 J. YANG, Y. LI, Z.-Q. WANG EJDE/SI/01

Then sending δ → 0 (and therefore R→∞) we obtain the result. �

Proof of Theorem 1.3. By using the Proposition 2.3, we obtain

cd = d
− 2

N(p−2)
2

−2 c1. (4.20)

Hence by Lemma 4.1, we have

lim sup
ε→0

c(ε, k) ≤ c
k

2−p
2 qM

= (k
2−p
2 qM )

− 2
N(p−2)

2
−2 c1

= c1(qM )
4

2−N(p−2) k−
2(p−2)

2−N(p−2)

< min
{
cq0 = c1(q0)

4
4−N(p−2) , cq∞ = c1(q∞)

4
4−N(p−2)

}
.

Then, for fixed k, exists εk, when ε < εk, we have c(ε, k) < min{cq0 , cq∞}. Therefore,
by Lemma 4.2, for ε < εk, c(ε, i) is achieved at some ui for i = 1, . . . , k. Using
Lemma 4.3 we have for i = 1, . . . , k

lim
ε→0

c(ε, i) = c
i
2−p
2 qM

, (4.21)

which implies by Lemma 2.3 that these ui for i = 1, . . . , k are mutually different
non-radial solutions. �

Remark 4.4. From the proof of Lemma 4.3 we can see that these solutions are
multi-bump type solutions. In particular, for i = 1, 2, . . . , k, solution ui behaves
like a normalized sum of translations of a minimizer of c

i
2−p
2 qM

at a symmetric

orbit of the group action.

Remark 4.5. The result is still true for N ≥ 4. We may use the group G = Gk ×
O(N − 2), and consider the subspace of G-invariant functions. These functions are
radially symmetric with respect to the last N − 2 variables. Since N − 2 ≥ 2, when
we do concentration compactness analysis the concentration points stay around the
two dimensional subspace of the first two variables. Therefore our arguments go
through with little modifications. We omit the proof here. It would be interesting
to see whether the same phenomena is still valid for the case of N = 3, though our
minimization arguments seem to break down here.

For problems similar to (1.1) but without a constraint, we refer [3, 11, 16] for
references on results of similar natures in particular [3, 16] where as a small param-
eter tends to zero there are more and more non-radial solutions. However for our
problem (1.6) we do not know whether this would be the case, i.e., as ε → 0 the
number of non-radial solutions tends to infinity.

Acknowledgement. The work was supported by the NSFC (11671085, 11771324,
11831009). The authors are grateful for the anonymous referee’s helpful suggestions.

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Jiaxuan Yang

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

Email address: qsx20190557@student.fjnu.edu.cn

Yongqing Li

College of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117, China
Email address: yqli@fjnu.edu.cn

Zhi-Qiang Wang

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

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


	1. Introduction
	2. Preliminaries
	3. Existence and symmetry breaking of the ground states
	4. Multiple non-radial solutions
	Acknowledgement

	References