Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 114, pp. 1–14.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, DOI: 10.58997/ejde.2025.114

NORMALIZED GROUND STATE SOLUTIONS FOR KIRCHHOFF

EQUATION WITH SUBCRITICAL OR CRITICAL PERTURBATION

ZIAN FAN

Abstract. This article concerns the Kirchhoff equation with subcritical or critical perturbation.
We establish the existence of normalized ground state solutions by using Pohozaev manifold and

subcritical approximation methods.

1. Introduction and main results

In this article, we study the Kirchhoff equation with subcritical or critical perturbation

−(a+ b

∫
R3

|∇u|2dx)∆u+ λu = η|u|p−2u+ |u|q−2u in R3,∫
R3

|u|2dx = c, u ∈ H1(R3),

(1.1)

where a, b > 0 are two constants, 2 < p < q ≤ 2∗ = 6, c, η > 0, λ ∈ R. In the previous
two decades, a large number of scholars have studied the existence of nontrivial solutions of the
following Kirchhoff type equation

−(a+ b

∫
RN

|∇u|2dx)∆u+ V (x)u = f(x, u),

see [1, 16, 6, 7, 2, 24, 4]. In these references, Alves, Perera and He proved the existence results of
the nontrivial solutions by the variational methods. Zhang [24] considered one Kirchhoff equation
with critical Sobolev exponent. By Nehari manifold, Chen [4] established the existence of positive
solutions to (1.1) for 1 < p < 2 < q < 2∗ in a bounded domain.

Recently, more attention is paid to the existence of solutions to nonlinear Schrödinger (NLS)
equation with prescribed mass, ∫

RN

|u|2dx = c.

Such solution is usually called a normalized solution. Soave [17, 18] proved the existence of
normalized ground states and properties of ground states for NLS equation

−∆u− λu = µ|u|q−2u+ |u|p−2u in RN ,∫
RN

|u|2dx = a2, u ∈ H1(RN ),

where 2 < q < p ≤ 2∗. Yao [21] studied normalized solutions for the Choquard equations with
lower critical exponent

−∆u+ λu = γ(Iα ∗ |u| α
N +1)|u| α

N −1u+ µ|u|q−2u in RN ,∫
RN

|u|2dx = c, u ∈ H1(RN ),

2020 Mathematics Subject Classification. 35J15, 35J20, 35J60, 35B33.

Key words and phrases. Normalized solutions; Kirchhoff equation; critical growth; Pohozaev manifold.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted June 4, 2025. Published December 9, 2025.

1



2 Z. FAN EJDE-2025/114

where α ∈ (0, N), 2 < q ≤ 2N/(N − 2). By employing the Sobolev subcritical approximation
method, Li, Nie and Zhang [10] studied the problem (1.1) for η sufficiently large and obtained the
existence of normalized ground states. Kong and Chen [8] studied (1.1) in for the case N = 4.
By decomposing Pohozaev manifold and constructing fiber map, they proved the existence of a
positive normalized ground state. For more references on normalized solution of Kirchhoff equation
and Schrödinger equation, we refer to [5, 9, 11, 12, 22, 23, 25, 26].

Motivated by the above papers, we examine the existence of normalized solutions for Kirchhoff
equation in the subcritical case 2 < q < 2∗ and the critical case q = 2∗. For subcritical case,
by using the constrained minimization method on a suitable submanifold and using Schwartz
symmetrization rearrangements we obtain the existence of normalized ground state solutions. For
critical case, the main difficulty in solving the problem (1.1) is to study the estimate of Mountain
pass Level and prove inf Iq(u) is attained in a suitable submanifold. Firstly we compare the zeros
of some functions and study the Mountain pass Level, then we construct an increasing sequence
to get the properties of inf Iq(u). Lately in the aid of subcritical approximation method, we prove
the existence of normalized ground state solutions for q = 2∗ for any η > 0 .

Let E = H1(R3) be the usual Sobolev space equipped with the norm

∥u∥ =
(∫

R3

(|∇u|2 + u2)dx
)1/2

.

The norm of the space Lj(R3)(1 < j ≤ 2∗) is defined as

∥u∥j =
(∫

R3

|u|jdx
)1/j

.

Moreover, Er = H1
rad(R3) denotes the subspace of radial functions

Er = H1
rad(R3) = {u ∈ H1(R3) : u(x) = u(|x|)}.

From [20] we know that the embedding Er → Lj(R3)(2 < j ≤ 2∗) is continuous and compact for
2 < j < 2∗.

For 0 ̸= u ∈ D1,2(R3), we define

S0 = inf

∫
R3 |∇u|2dx(∫

R3 |u|2∗dx
)2/2∗ .

We now introduce the main results in this article.

Theorem 1.1. Assume that c > 0, 14
3 ≤ p < q < 2∗, then problem (1.1) admits at least a couple

of ground state solution (u, λ) ∈ H1(R3)× R for any η > 0. Moreover, the solution is real-valued
positive and radially symmetric non-increasing function with λ > 0.

Theorem 1.2. Assume that c > 0, 14
3 ≤ p < q, q = 2∗, then problem (1.1) admits at least a

couple of ground state solution (u, λ) ∈ H1(R3) × R for any η > 0. Moreover, the solution is
real-valued positive and radially symmetric non-increasing function with λ > 0.

We set

γp =
3(p− 2)

2p
, γq =

3(q − 2)

2q
.

Remark 1.3. Direct calculations show that under the assumption of Theorem 1.1 or Theorem
1.2, it results that p > 2, q > 2, qγq > pγp ≥ 4.

2. Variational framework and technical lemmas

In this section, we assume p, q satisfy the conditions of Theorem 1.1 or Theorem 1.2. By Remark
1.3 we know that q > p > 2 and qγq > pγp ≥ 4.

The functional associated to (1.1) is

Iq(u) =
a

2

∫
R3

|∇u|2dx+
b

4

(∫
R3

|∇u|2dx
)2

− η
1

p

∫
R3

|u|pdx− 1

q

∫
R3

|u|qdx



EJDE-2025/114 KIRCHHOFF EQUATION WITH SUBCRITICAL OR CRITICAL PERTURBATION 3

=
1

2
A(u) +

1

4
D(u)− 1

p
L(u)− 1

q
K(u)

with the constrain

Sc = {u ∈ E :

∫
R3

|u|2dx = c}.

Where

A(u) = a

∫
R3

|∇u|2dx, D(u) = b
(∫

R3

|∇u|2dx
)2

,

L(u) = η

∫
R3

|u|pdx, K(u) =

∫
R3

|u|qdx.

The Gagliardo-Nirenberg inequality can be founded in [19]:

∥u∥q ≤ C(q)∥∇u∥γq

2 ∥u∥1−γq

2 , 2 < q < 2∗. (2.1)

If q = 2∗, by using the definition of S0 we obtain∫
R3

|u|2
∗
dx ≤ S

−2∗/2
0 ∥∇u∥2

∗

2 . (2.2)

Since the embedding H1(R3) → Lj(R3)(2 < j ≤ 2∗) is continuous, then we deduce that Iq ∈
C1(E,R). For u ∈ E, we define

Gq(u) = A(u) +D(u)− γpL(u)− γqK(u).

If u ∈ E is a solution to the first equation of (1.1),

−(a+ b

∫
R3

|∇u|2dx)∆u+ λu = η|u|p−2u+ |u|q−2u in R3, (2.3)

then

A(u) + λB(u) +D(u) = L(u) +K(u),

where

B(u) =

∫
R3

u2dx.

Moreover, by using an argument similar to [15, 14] we have the identity

1

2
A(u) + λ

3

2
B(u) +

1

2
D(u)− 3

p
L(u)− 3

q
K(u) = 0.

Hence if u ∈ E is a solution to (2.3), then u satisfies the Pohozaev equality

Gq(u) = A(u) +D(u)− γpL(u)− γqK(u) = 0. (2.4)

Suppose that u ∈ E, by computations we obtain

⟨I ′q(u), ϕ⟩ = (a+ b

∫
R3

|∇u|2dx)
∫
R3

∇u∇ϕdx− η

∫
R3

|u|p−2uϕdx−
∫
R3

|u|q−2uϕdx

for every ϕ ∈ E. Hence the normalized solutions of problem (1.1) are the critical points of the
energy functional Iq under the constrain Sc.

We say that uc is a ground state of (1.1) on Sc if (uc, λc) ∈ E × R is a normalized solution to
(1.1) and uc has the minimal energy among all nontrivial solutions belonging to Sc, that is,

Iq(u) = inf{Iq(v) : v ∈ Sc, (Iq |Sc)
′(v) = 0}.

Let u ∈ E, and ut = t3/2u(tx), t > 0. By computations we obtain

h(t) = Iq(u
t) =

at2

2

∫
R3

|∇u|2dx+
bt4

4

(∫
R3

|∇u|2dx
)2

− η
tpγp

p

∫
R3

|u|pdx− tqγq

q

∫
R3

|u|qdx.

Then Iq(u
t) → −∞ as t → +∞, Iq is not bounded from below.

Consider the Pohozaev set

Mc,q =
{
u ∈ Sc : Gq(u) = 0

}
.



4 Z. FAN EJDE-2025/114

From (2.4) we deduce that if u ∈ E is a solution to (1.1), then u ∈ Mc,q, that is, any critical points
of Iq |Sc stay in Mc,q. By computations we obtain

h′(t) = tA(u) + t3D(u)− γpt
pγp−1L(u)− γqt

qγq−1K(u).

It is clear that Gq(u) = 0 if and only if h′(1) = 0. Next, we claim that Mc,q is a natural constraint.

Lemma 2.1. The following results hold:

(i) Mc,q is a smooth manifold of codimension 1 in Sc;
(ii) If u ∈ Mc,q is a critical point of Iq |Mc,q

, then u is a critical point of Iq |Sc
.

Proof. (1) Note that

Mc,q = {u ∈ E : Gq(u) = 0, G1(u) = 0} ,
where

G1(u) =

∫
R3

|u|2dx− c.

Then it is easy to see that Gq(u), G1(u) are of class C1 in E. If G′
q(u) = 0, then by the Lagrange

multipliers rule there exists some λ1 ∈ R and u ∈ Mc,q such that

G′
q(u) + λ1G1

′(u) = 0,

which implies that

−2a∆u− 4b
(∫

R3

|∇u|2dx
)
∆u+ 2λ1u = ηpγp|u|p−2u+ qγq|u|q−2u.

By Pohozaev equality (2.4) we deduce that

2A(u) + 4D(u) = pγ2
pL(u) + qγ2

qK(u),

then

−2A(u) + γp(4− pγp)L(u) + γq(4− qγq)K(u) = 0.

This contradicts Remark 1.3. Hence we obtain that G′
q(u) ̸= 0 and Mc,q is a smooth manifold of

codimension 1 in Sc.
(2) If u ∈ Mc,q is a critical point of Iq |Mc,q

, by the Lagrange multipliers rule there exists some
λ, λ2 ∈ R such that for any φ ∈ E,

⟨I ′q(u), φ⟩+ λ

∫
R3

uφdx+ λ2⟨G′
q(u), φ⟩ = 0,

then u satisfies the equation

− (2λ2 + 1)a∆u− (4λ2 + 1)b

(∫
R3

|∇u|2dx
)
∆u+ λu

= η(λ2pγp + 1)|u|p−2u+ (λ2qγq + 1)|u|q−2u.

From Pohozaev equality (2.4) we deduce that

(2λ2 + 1)A(u) + (4λ2 + 1)D(u) = γp(λ2pγp + 1)L(u) + γq(λ2qγq + 1)K(u). (2.5)

Since Gq(u) = 0, then from (2.4) and (2.5) we obtain

λ2(−2A(u)− (pγp − 4)γpL(u)− (qγq − 4)γqK(u)) = 0.

Then we obtain that λ2 = 0, the proof is complete. □

Lemma 2.2. For any u ∈ Sc, there exists a unique t0 such that ut0 ∈ Mc,q and

Iq(u
t0) = max

t>0
Iq(u

t).

Moreover, if Gq(u) ≤ 0, then 0 < t0 ≤ 1.



EJDE-2025/114 KIRCHHOFF EQUATION WITH SUBCRITICAL OR CRITICAL PERTURBATION 5

Proof. Suppose that u ∈ Sc, then

h(t) = Iq(u
t) =

t2

2
A(u) +

t4

4
D(u)− tpγp

p
L(u)− tqγq

q
K(u),

h′(t) = tA(u) + t3D(u)− γpt
pγp−1L(u)− γqt

qγq−1K(u),

= t3(t−2A(u) +D(u)− γpt
pγp−4L(u)− γqt

qγq−4K(u)) = t3g(t).

Then it is clear that h(0) = 0 and h(t) → −∞ as t → +∞. There exists t0 such that h′(t) > 0 for
t ∈ (0, t0) and h′(t) < 0 for t ∈ [t0, t1), t1 > t0 > 0, it follows that h(t) achieves its local maximum
at t = t0 and h′(t0) = 0. Since pγp ≥ 4 and qγq > 4, then g(t) is strictly decreasing. On the other
hand, we have

{t > 0 : h′(t) = 0} = {t > 0 : g(t) = 0},

which implies that h(t) has a positive unique critical point which corresponds to its maximum.
By computations we have

Gq(u
t0) = t0[at0

∫
R3

|∇u|2dx+ bt30

(∫
R3

|∇u|2dx
)2

− γpt
pγp−1
0 L(u)− γqt

qγq−1
0 K(u)]

= t0h
′(t0).

From h′(t0) = 0, we obtain that Gq(u
t0) = 0 and ut0 ∈ Mc,q.

Suppose Gq(u) ≤ 0, we claim that 0 < t0 ≤ 1. Assume by contradiction that t0 > 1, using that
Gq(u

t0) = 0 we obtain

0 = Gq(u
t0) = t40[t

−2
0 A(u) +D(u)− γpt

pγp−4
0 L(u)− γqt

qγq−4
0 K(u)]

< t40[A(u) +D(u)− γpL(u)− γqK(u)]

= t40Gq(u) ≤ 0,

which is a contradiction, then 0 < t0 ≤ 1. This completes the proof. □

Let u ∈ Mc,q, we obtain

Iq(u) = Iq(u)−
1

4
Gq(u) =

1

4
A(u) + (

γp
4

− 1

p
)L(u) + (

γq
4

− 1

q
)K(u). (2.6)

In view of Remark 1.3, we deduce that Iq(u) > 0 and Iq(u) is bounded below on Mc,q.
Since that Iq(u) is bounded below on Mc,q, we may define

m(c, q) = inf
u∈Mc,q

Iq(u).

Lemma 2.3. There exists C1 = C1(a, p, q, η) > 0 such that m(c, q) > C1.

Proof. Let u ∈ Mc,q. Then

A(u) +D(u) = γpL(u) + γqK(u) ≥ a0∥∇u∥22, (2.7)

where a0 = min{1, a}. Using (2.1)-(2.2) and (2.7) we can deduce that

a0 ≤ ηC(p)γp∥∇u∥pγp−2
2 c

p(1−γp)

2 + C(q)γq∥∇u∥qγq−2
2 c

q(1−γq)

2 , 2 < q < 2∗,

a0 ≤ ηC(p)γp∥∇u∥pγp−2
2 c

p(1−γp)

2 + γqS
−2∗
2

0 ∥∇u∥2
∗−2

2 , q = 2∗.

Then there exists C > 0 such that ∥∇u∥22 > C > 0. It follows from (2.6)-(2.7) that

Iq(u) ≥
1

4
A(u) =

a

4
∥∇u∥22. (2.8)

Then there exists C1 = C1(a, p, q, η) > 0 such that m(c, q) > C1. □



6 Z. FAN EJDE-2025/114

3. Proof of Theorem 1.1

In this section, we study the subcritical case 14
3 ≤ p < q < 2∗. From Remark 1.3 we have

q > p > 2, qγq > pγp ≥ 4. Let {un} ⊂ Mc,q be a minimizing sequence for Iq(u), that is

lim
n→∞

Iq(un) = inf
u∈Mc,q

Iq(u) = m(c, q).

Then we have the following Lemma.

Lemma 3.1. Under the assumption of Theorem 1.1, m(c, q) is attained by a real- valued positive
and radially symmetric non-increasing function.

Proof. Let vn = |un|∗ be the Schwartz symmetrization rearrangement of |un|. Then by [13] we
obtain

A(vn) ≤ A(un), D(vn) ≤ D(un), B(vn) = B(un) = c, (3.1)

K(vn) = K(un), L(vn) = L(un). (3.2)

Therefore,

Iq(vn) ≤ Iq(un), Gq(vn) ≤ Gq(un) = 0.

Since vn ∈ Sc, then by Lemma 2.2 there exists a unique 0 < tn ≤ 1 such that {vtnn } ⊂ Mc,q, where

vtnn = t
3/2
n vn(tnx).

From (3.1) and (3.2) we deduce that

m(c, q) ≤ Iq(v
tn
n ) = Iq(v

tn
n )− 1

pγp
Gq(v

tn
n )

= (
1

2
− 1

pγp
)t2nA(vn) + (

1

4
− 1

pγp
)t4nD(vn) + (

γq
pγp

− 1

q
)tqγq

n K(vn)

≤ (
1

2
− 1

pγp
)A(un) + (

1

4
− 1

pγp
)D(un) + (

γq
pγp

− 1

q
)K(un)

= Iq(un)−
1

pγp
Gq(un) = Iq(un) → m(c, q)

as n → ∞. Then we obtain that

Iq(v
tn
n ) → m(c, q), n → ∞.

Hence {vtnn } is a minimizing sequence for Iq(u).
We claim {vtnn } is bounded in E. Otherwise, if ∥vtnn ∥ → ∞ as n → ∞, then by (2.8) we have

that Iq(v
tn
n ) → ∞, that is a contradiction. Therefore {vtnn } is bounded in E. We can extract a

subsequence(still denoted {vtnn }) and v ∈ E such that

vtnn ⇀ v in Er;

vtnn → v in Li(R3)(2 < i < 2∗),

vtnn → v a.e. in R3

as n → ∞. Therefore we obtain K(vtnn ) → K(v), L(vtnn ) → L(v). On the other hand, by Lemma
2.3, we have

0 < C1

< m(c, q)

≤ Iq(v
tn
n )

= Iq(v
tn
n )− 1

2
Gq(v

tn
n )

= −1

4
t4nD(vtnn ) + (

γp
2

− 1

p
)tpγp

n L(vtnn ) + (
γq
2

− 1

q
)tqγq

n K(vtnn )

≤ (
γp
2

− 1

p
)L(vtnn ) + (

γq
2

− 1

q
)K(vtnn )



EJDE-2025/114 KIRCHHOFF EQUATION WITH SUBCRITICAL OR CRITICAL PERTURBATION 7

→ (
γp
2

− 1

p
)L(v) + (

γq
2

− 1

q
)K(v),

which implies that v ̸= 0. Moreover, by Fatou’s Lemma,

Gq(v) = A(v) +D(v)− γpL(v)− γqK(v)

≤ lim inf
n→∞

[A(vtnn ) +D(vtnn )]− γp lim
n→∞

L(vtnn )− γq lim
n→∞

K(vtnn )

= lim inf
n→∞

Gq(v
tn
n ) = 0,

which implies that

Gq(v) ≤ 0. (3.3)

Next we shall prove that ∫
R3

|v|2dx = c.

Set

0 <

∫
R3

|v|2dx = c1 ≤ c, ξ =
c1
c
, v1(x) = ξ

1
q−2 v(ξ

q
3q−6x).

Note that 2 < p < q < 6. Then

A(v1) = ξ
6−q
3q−6A(v) ≤ A(v), B(v1) = c, D(v1) ≤ D(v), (3.4)

L(v1) = ξ
p−q
q−2L(v) ≥ L(v), K(v1) = K(v). (3.5)

It follows from (3.3)-(3.5) that

Gq(v1) ≤ Gq(v) ≤ 0.

From Lemma 2.2, there exists a unique 0 < τ ≤ 1 such that Gq(v
τ
1 ) = 0 and vτ1 ∈ Mc,q. Then by

using (3.4) and (3.5) we deduce that

m(c, q) ≤ Iq(v
τ
1 ) = Iq(v

τ
1 )−

1

pγp
G(vτ1 )

= (
1

2
− 1

pγp
)τ2A(v1) + (

1

4
− 1

pγp
)τ4D(v1) + (

γq
pγp

− 1

q
)τ qγqK(v1)

≤ (
1

2
− 1

pγp
)A(v) + (

1

4
− 1

pγp
)D(v) + (

γq
pγp

− 1

q
)K(v)

≤ (
1

2
− 1

pγp
) lim inf

n→∞
A(vtnn ) + (

1

4
− 1

pγp
) lim inf

n→∞
D(vtnn ) + (

γq
pγp

− 1

q
) lim
n→∞

K(vtnn )

= lim inf
n→∞

[Iq(v
tn
n )− 1

pγp
Gq(v

tn
n )] = m(c, q).

(3.6)

Which implies that Iq(v
τ
1 ) = m(c, q). We claim that τ = 1, ξ = 1. Otherwise, if 0 < τ < 1 or

0 < ξ < 1, we replace ≤ with < in (3.6) and get a contradiction. Therefore we deduce that

Iq(v
τ
1 ) = m(c, q), τ = 1, v1 = v, c = c1.

We deduce that m(c, q) is attained by a real-valued nonnegative and radially symmetric non-
increasing function. By strong maximum principle v > 0, then v is positive. □

Proof of Theorem 1.1. From Lemma 3.1 we know that v is a critical point of Iq |Mc,q and v ∈ Mc,q

is a real-valued positive and radially symmetric non-increasing function. It follows from Lemma
2.1 that v is a critical point of Iq |Sc

. By using the Lagrange multipliers rule there exists some
λ ∈ R such that for any φ ∈ E

⟨I ′q(v), φ⟩+ λ

∫
R3

vφdx = 0,

then v satisfies the equation

−(a+ b

∫
R3

|∇v|2dx)∆v + λv = η|v|p−2v + |v|q−2v.



8 Z. FAN EJDE-2025/114

Since γp < 1, γq < 1, then from Pohozaev equality (2.4) we obtain

λ

∫
R3

v2dx = (1− γp)L(v) + (1− γq)K(v) > 0,

consequently, λ > 0. The proof is complete. □

4. Proof of Theorem 1.2

Now we turn to the critical case 14
3 ≤ p < q, q = 2∗. In this case, we have

m(c, 2∗) = inf
u∈Mc,2∗

I2∗(u).

We define

qn = 2∗ − 1

n
> 2, n = 1, 2...,

then {qn} is increasing and qn → 2∗ as n → ∞. We first give the following properties of m(c, qn).

Lemma 4.1. The following results hold:

(i) lim infn→∞ m(c, qn) > 0;
(ii) lim supn→∞ m(c, qn) ≤ m(c, 2∗).

Proof. Item (i) follows from Lemma 2.3.
(ii) From the definition of m(c, 2∗) there exists u ∈ Mc,2∗ such that

I2∗(u) < m(c, 2∗) + ε, 0 < ε < 1. (4.1)

Let ut = t3/2u(tx), t > 0. It is clear that

h1(t) = Iqn(u
t) =

t2

2
A(u) +

t4

4
D(u)− tpγp

p
L(u)− tqnγqn

qn
Kn(u),

where

Kn(u) =

∫
R3

|u|qndx.

Note that |u|qn ≤ |u|2 + |u|2∗ and |u|qn → |u|2∗ as n → ∞, then from Lebesgue dominated
convergence theorem we obtain

tqnγqn

qn

∫
R3

|u|qndx =
t
3qn−6

2

qn

∫
R3

|u|qndx → t2
∗

2∗

∫
R3

|u|2
∗
dx.

Which implies that

|Iqn(ut)− I2∗(u
t)| < ε. (4.2)

On the other hand, it is easy to see that h1(0) = 0 and h1(t) → −∞ as t → +∞. By the proof of
Lemma 2.2, we deduce that h1(t) achieves its unique maximum at t = t0 for t0 ∈ [0, t1), t1 > t0 > 0.
Moreover, we have ut0 ∈ Mc,qn . Since u ∈ Mc,2∗ , it follows that

I2∗(u) = max
t>0

I2∗(u
t).

From (4.1) and (4.2), we obtain

m(c, qn) ≤ Iqn(u
t0) ≤ I2∗(u

t0) + ε ≤ I2∗(u) + ε < m(c, 2∗) + 2ε,

which implies that

lim sup
n→∞

m(c, qn) ≤ m(c, 2∗).

The proof is complete. □

Let Bδ(0) be a ball centered at the origin with radius δ > 0. Let ρ(x) ∈ C∞
0 (R3) be a radial

cut-off function such that ρ(x) = 1 for |x| ≤ δ and ρ(x) = 0 for |x| ≥ 2δ, where |∇ρ(x)| ≤
C,B2δ(0) ⊂ R3. Let 0 < ε < 1, we choose the function uε(x) as follows

uε(x) = ρ(x)Uε(x),



EJDE-2025/114 KIRCHHOFF EQUATION WITH SUBCRITICAL OR CRITICAL PERTURBATION 9

where

Uε(x) =
(3ε2)1/4

(ε2 + |x|2)1/2
.

If ε > 0 is small enough, by using [14] we have that the function uε(x) satisfies∫
R3

|∇uε|2dx = S
3/2
0 +O(ε); (4.3)∫

R3

|uε|2
∗
dx = S

3/2
0 +O(ε3); (4.4)∫

R3

|uε|2dx = C2ε+O(ε2). (4.5)

Where C2 is a positive constant independent of ε. By computations we have∫
Ω

|uε|pdx ≥


C3ε

3− p
2 , p > 3,

C3ε
3/2|ln ε|, p = 3,

C3ε
p/2, p < 3,

(4.6)

where C3 is a positive constant independent of ε. We set

0 <

∫
R3

|uε|2dx = c2, ξ1 =
c2
c
, vε(x) = ξ

1/4
1 uε(ξ

1/2
1 x).

By computations we have

A(vε) = A(uε), B(vε) = c, D(vε) = D(uε), (4.7)

L(vε) = ξ
p−6
4

1 L(uε), K(vε) = K(uε). (4.8)

Lemma 4.2. Under the assumption of Theorem 1.2, we have

0 < m(c, 2∗) < (
1

2
− 1

2∗
)ay1 + (

1

4
− 1

2∗
)by21 = c∗,

where y1 is given in the following discussion.

Proof. From Lemma 2.3, we obtain m(c, 2∗) > 0. We will show m(c, 2∗) < c∗. Since B(vε) = c,
from Lemma 2.2, there exists a unique τ0 such that vτ0ε ∈ Mc,2∗ and

I2∗(v
τ0
ε ) = max

t>0
I2∗(v

t
ε).

Consider the function

H(t) = I2∗(v
t
ε) =

at2

2

∫
R3

|∇vε|2dx+
bt4

4

(∫
R3

|∇vε|2dx
)2

− η
tpγp

p

∫
R3

|vε|pdx− t2
∗

2∗

∫
R3

|vε|2
∗
dx.

By using (4.7) and (4.8) we deduce

H(t) =
t2

2
A(uε) +

t4

4
D(uε)−

tpγp

p
ξ

p−6
4

1 L(uε)−
t2

∗

2∗
K(uε).

We set

H1(t) =
1

2
at2∥∇uε∥22 +

bt4

4
∥∇uε∥42 −

t2
∗

2∗

∫
R3

|uε|2
∗
dx.

Then from (4.3) and (4.4) we have

H1(t) ≤ H2(t) + C3(t)ε
3 + C4(t)ε, (4.9)

and

H2(t) =
a

2
t2Sβ

0 +
bt4

4
S2β
0 − t2

∗

2∗
Sβ
0 , β =

3

2
.

By the same argument as in Lemma 2.2, we deduce that H2(t) achieves its maximum at t = t2 > 0
and

aSβ
0 + bt22S

2β
0 − t2

∗−2
2 Sβ

0 = 0. (4.10)



10 Z. FAN EJDE-2025/114

We consider the function

F (t) = a+ bt− S
−2∗
2

0 t
2∗−2

2 , t > 0.

Suppose y1 is the unique zero of F (t), then by (β − 1) 2
∗

2 = β, we know that y1 and t22S
β are both

zeros of the function F (t). Since the zero of the function F (t) is unique for some t > 0, then we

deduce that y1 = t22S
β
0 . From (4.10) we obtain

H2(t) ≤ H2(t2) =
a

2
t22S

β
0 +

bt42
4
S2β
0 − t2

∗

2

2∗
Sβ
0

=
(1
2
− 1

2∗
)
at22S

β
0 +

(1
4
− 1

2∗
)
bt42S

2β
0

= (
1

2
− 1

2∗
)ay1 + (

1

4
− 1

2∗
)by21 = c∗.

(4.11)

By using (4.5), (4.6) and (4.9)-(4.11) we deduce that

H(t) ≤ H2(t2) + C4ε
3 + C4ε− C5τ

pγp

0 ξ
p−6
4

1 L(uε)

≤ c∗ + C4ε
3 + C4ε− C5ητ

pγp

0 c
6−p
4 c

p−6
4

2 L(uε)

≤ c∗ + C4ε
3 + C4ε− C5ητ

pγp

0 c
6−p
4 ε

6−p
4 ,

(4.12)

where Ci (i = 4, 5) are positive constants independent of ε.
We claim that there exists a constant M1 independent of ε such that τ0 ≤ M1. If τ0 → ∞

as ε → 0, then H(τ0) → −∞ and m(c, 2∗) ≤ 0, which contradicts m(c, 2∗) > 0. On the other
hand, since 0 is a local minimum of H(t), there exists a constant M2 independent of ε such
that H(τ0) ≥ M2 > 0. This implies that there exists a constant M3 independent of ε such that
τ0 ≥ M3 > 0.

Note that p > 2, then
6− p

4
< 1. (4.13)

By (4.12) and (4.13) we deduce that

m(c, 2∗) ≤ sup
t>0

H(t) < c∗.

The proof is complete. □

The proof of the following Lemma can be founded in [3].

Lemma 4.3. Let N ≥ 3 and 1 ≤ t < +∞. If u ∈ Lt(RN ) is a radial non-increasing function,
then one has

|u(x)| ≤ |x|−N/t(
N

|SN−1|
)∥u∥t, x ̸= 0,

where |SN−1| is the area of the unit sphere in RN .

Lemma 4.4. Under the assumption of Theorem 1.2, m(c, 2∗) is attained by a real-valued positive
and radially symmetric non-increasing function.

Proof. For qn = 2∗ − 1
n , by Lemma 3.1, m(c, qn) is attained by a sequence of real-valued positive

and radially symmetric non-increasing functions {un}, and {un} ⊂ Mc,qn , Iqn(un) = m(c, qn).
From the proof of Theorem 1.1 we know that there exists λn > 0 such that un satisfies

−(a+ b

∫
R3

|∇un|2dx)∆un + λnun = η|un|p−2un + |un|qn−2un. (4.14)

From Pohozaev equality (2.4) we obtain

λn

∫
R3

u2
ndx = (1− γp)L(un) + (1− γqn)K(un),

which implies that {λn} is bounded. Then there exists λ > 0 such that up to a subsequence,
limn→∞ λn = λ.



EJDE-2025/114 KIRCHHOFF EQUATION WITH SUBCRITICAL OR CRITICAL PERTURBATION 11

From the proof of Lemma 3.1 we deduce that {un} is bounded in Er. We can extract a
subsequence (still denoted un) and w ∈ Er such that

un ⇀ w in Er,

un → w in Li(R3)(2 < i < 2∗),

un → w a.e. in R3a

as n → ∞. Then {|un|p−2un} is bounded in L
p

p−1 (R3), which implies that∫
R3

|un|p−2unφdx →
∫
R3

|u|p−2uφdx

for φ ∈ C∞
0 (R3).

Note that For ϕ ∈ Lt(R3), t > 1, by the Young inequality, the Hölder inequality and Lemma
4.3 there exists a constant C > 0 independent of n such that

||un|qn−2unϕ| ≤ C(|un|2−1|ϕ|+ |un|2
∗−1|ϕ|)

≤ C(|x|− 1
2 |ϕ|+ |x|

−(2∗−1)
2 |ϕ|) ∈ L1(R3).

By using the Lebesgue dominated convergence theorem and passing to the limit in (4.14) we obtain
for φ ∈ C∞

0 (R3),

0 = ⟨I ′qn(un), φ⟩ → ⟨I ′2∗(w), φ⟩.
Therefore w is a solution of the equation

−(a+ b

∫
R3

|∇w|2dx)∆w + λw = η|w|p−2u+ |w|2
∗−2w.

Then G2∗(w) = 0.
We claim that w ̸= 0. Otherwise, we assume that un ⇀ w = 0, then from (3.3) we have

L(un) → 0. (4.15)

From Young inequality we obtain

|un|qn ≤ 2∗ − qn
2∗ − θ

|un|θ +
qn − θ

2∗ − θ
|un|2

∗
, 2 < θ < qn < 2∗.

On the other hand, by using the definition of S0 we obtain∫
R3

|u|2
∗
dx ≤ S

−2∗
2

0

(∫
R3

|∇un|2dx
)2∗/2

.

Hence we deduce that

Kn(un) =

∫
R3

|un|qndx ≤ qn − θ

2∗ − θ
S

−2∗
2

0

(∫
R3

|∇un|2dx
)2∗/2

+ o(1). (4.16)

Note that un ∈ Mc,qn , then

0 = Gqn(un) = A(un) +D(un)− γpL(un)− γqnKn(un). (4.17)

From the proof of Lemma 2.3, there exists a constant C > 0 such that ∥∇un∥22 ≥ C > 0, then we
may therefore assume that

lim inf
n→∞

∫
R3

|∇un|2dx = l, l > 0.

Let n → ∞, from (4.15)-(4.17) we have

a+ bl − S
−2∗
2

0 l
2∗−2

2 ≤ 0, l > 0. (4.18)

We set

F (y) = a+ by − S
−2∗
2

0 y
2∗−2

2 , y > 0,

where F (y) has been defined in the proof of Lemma 4.2. From F ′(y) = 0 we obtain y = y0 > 0
and F (0) = a > 0. it is easy to see that F (y) → −∞ as y → +∞. Moreover we have F ′(y) > 0



12 Z. FAN EJDE-2025/114

for y ∈ [0, y0) and F ′(y) < 0 for y ∈ [y0,∞). It follows that F (y) achieves its maximum at y = y0.
If y > 0, then F (y) have only one zero point y = y1 > 0. Since F (y1) = 0, we deduce that

a+ by1 − S
−2∗
2

0 y
2∗−2

2
1 = 0. (4.19)

From (4.18) and (4.19) we deduce that l ≥ y1.
By using Lemma 4.1 and (4.15) we obtain

m(c, 2∗) ≥ lim sup
n→∞

m(c, qn)

= lim sup
n→∞

Iqn(un)

= lim sup
n→∞

[Iqn(un)−
1

qnγqn
Gqn(un)]

= lim sup
n→∞

[(
1

2
− 1

qnγqn
)A(un) + (

1

4
− 1

qnγqn
)D(un)]

≥ lim inf
n→∞

[(
1

2
− 1

qnγqn
)A(un) + (

1

4
− 1

qnγqn
)D(un)]

= a(
1

2
− 1

2∗
)l + b(

1

4
− 1

2∗
)l2

≥ a(
1

2
− 1

2∗
)y1 + b(

1

4
− 1

2∗
)y21 = c∗.

Which contradicts Lemma 4.2, so we obtain w ̸= 0.
We set

0 <

∫
R3

|w|2dx = c3 ≤ c, ξ2 =
c3
c
, w1(x) = ξ

1/4
2 w(ξ

1/2
2 x).

Then by computation we have

A(w1) = A(w), B(w1) = c, D(w1) = D(w). (4.20)

L(w1) = ξ
p−6
4

2 L(w) ≥ L(w), K(w1) = K(w), (4.21)

which implies that

G2∗(w1) ≤ G2∗(w) ≤ 0.

From Lemma 2.2, there exists a unique 0 < σ ≤ 1 such that G2∗(w
σ
1 ) = 0 and wσ

1 ∈ Mc,2∗ . Then
by Lemma 4.1, (4.20), and (4.21), we deduce that

m(c, 2∗) ≤ I2∗(w
σ
1 )

= I2∗(w
σ
1 )−

1

pγp
G2∗(w

σ
1 )

= (
1

2
− 1

pγp
)σ2A(w1) + (

1

4
− 1

pγp
)σ4D(w1) + (

γ2∗

pγp
− 1

2∗
)σ2∗

∫
R3

|w1|2
∗
dx

≤ (
1

2
− 1

pγp
)A(w) + (

1

4
− 1

pγp
)D(w) + (

γ2∗

pγp
− 1

2∗
)

∫
R3

|w|2
∗
dx

≤ lim inf
n→∞

[(
1

2
− 1

pγp
)A(un) + (

1

4
− 1

pγp
)D(un) + (

γqn
pγp

− 1

qn
)

∫
R3

|un|qndx]

≤ lim inf
n→∞

[Iqn(un)−
1

pγp
Gqn(un)] = lim inf

n→∞
Iqn(un)

= lim inf
n→∞

m(c, qn) ≤ lim sup
n→∞

m(c, qn) ≤ m(c, 2∗).

By a similar method to that of Lemma 3.1, we have

I2∗w
σ
1 ) = m(c, 2∗), σ = 1, w1 = w, c = c3,

then we deduce that m(c, 2∗) is attained by a real-valued nonnegative and radially symmetric
non-increasing function. By strong maximum principle v > 0, then v is positive. □



EJDE-2025/114 KIRCHHOFF EQUATION WITH SUBCRITICAL OR CRITICAL PERTURBATION 13

Proof of Theorem 1.2. From Lemma 4.4 we know that w is a critical point of I2∗ |Mc,2∗ and
w ∈ Mc,2∗ is a real-valued positive and radially symmetric non-increasing function. It follows
from Lemma 2.1 that w is a critical point of I2∗ |Sc . By using the Lagrange multipliers rule there
exists some λ ∈ R such that for any φ ∈ E,

⟨I ′2∗(w), φ⟩+ λ

∫
R3

wφdx = 0.

Then w satisfies

−(a+ b

∫
R3

|∇u|2dx)∆u+ λu = η|u|p−2u+ |u|2
∗−2u.

Similarly to the proof of Theorem 1.1, we have λ > 0, The proof is complete. □

Acknowledgements. The authors would like to express sincere gratitude to the anonymous
referees for their valuable comments that helped to improve this article.

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Zian Fan
School of Mathematics and Statistics, Hubei Engineering University, Xiaogan, Hubei 432000, China

Email address: fza2016@163.com


	1. Introduction and main results
	2. Variational framework and technical lemmas
	3. Proof of Theorem 1.1
	4. Proof of Theorem 1.2
	Acknowledgements

	References