Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 19, pp. 1–20.

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

NODAL SOLUTIONS OF FOURTH-ORDER KIRCHHOFF

EQUATIONS WITH CRITICAL GROWTH IN RN

HONGLING PU, SHIQI LI, SIHUA LIANG, DUŠAN D. REPOVŠ

Abstract. We consider a class of fourth-order elliptic equations of Kirchhoff

type with critical growth in RN . By using constrained minimization in the

Nehari manifold, we establish sufficient conditions for the existence of nodal
(that is, sign-changing) solutions.

1. Introduction

In this article we studies the existence of nodal solutions to the fourth-order
elliptic equations of Kirchhoff type with critical growth in RN ,

∆2u−
(

1 + b

∫
RN
|∇u|2 dx

)
∆u+ V (x)u = λf(u) + |u|2

∗∗−2u, x ∈ RN , (1.1)

where ∆2u is the biharmonic operator, 2∗∗ = 2N/(N − 4) is the critical Sobolev
exponent with 5 ≤ N < 8, and b and λ are positive parameters. The continuous
functions V (x) and f(u) satisfy the following conditions:

(A1) V ∈ C(RN ,R) satisfies inf
x∈RN

V (x) ≥ V0 > 0, where V0 is a positive constant.

For each M > 0, meas{x ∈ RN : V (x) ≤ M} < ∞, where meas(·) denotes
the Lebesgue measure on RN ;

(A2) f ∈ C1(R,R) and f(u) = o(|u|), as u→ 0;
(A3) There exists p ∈ (4, 2∗∗) such that limu→∞ f(u)/up−1 = 0;
(A4) limu→∞ F (u)/u4 = +∞, where F (u) =

∫ u
0
f(t)dt;

(A5) f(u)/|u|3 is a strictly increasing function for u ∈ R \ {0}.
Problem (1.1) originates from the Kirchhoff equation

−
(
a+ b

∫
Ω

|∇u|2 dx
)

∆u = f(x, u) in Ω, (1.2)

where Ω ∈ RN is a bounded domain, a > 0, b ≥ 0, and u satisfies certain boundary
conditions. The above equation stems from a typical model proposed by Kirchhoff
[11],

utt −
(
a+ b

∫
Ω

|∇u|2 dx
)

∆u = f(x, u), (1.3)

2010 Mathematics Subject Classification. 35A15, 35J60, 47G20.
Key words and phrases. Fourth-order elliptic equation; Kirchhoff problem; critical exponent;

variational methods; nodal solution.
c©2021 Texas State University.

Submitted January 24, 2021. Published March 25, 2021.

1



2 H. PU, S. LI, S. LIANG, D. D. REPOVŠ EJDE-2021/19

which serves as a generalization of the classical D’Alembert wave equation

ρ
∂2u

∂t2
−
(ρ0

h
+

E

2L

∫ L

0

|∂u
∂x
|2dx

)∂2u

∂x2
= f(x, u),

by taking into account the effects of changes in the length of strings during vibra-
tions. The nonlocal term thus appears. See for example [6, 24] for more background
on such problems. Thanks to the pioneering work of Lions [19] on problem (1.3), a
lot of attention has been drawn to these nonlocal problems during the last decade.
That was followed by some interesting results on the existence of various solutions to
(1.2), including positive solutions, multiple solutions, bound state solutions, multi-
bump solutions, and semiclassical state solutions, both on bounded domains and
on the entire space. For more results on the Kirchhoff-type equations we refer to
[7, 15, 16, 13, 23, 27] and the references therein. Problem (1.2) with critical non-
linearity, however, is seldom covered, mainly because of the challenge - the lack of
compactness - presented by the presence of the critical Sobolev exponent. We also
refer the interested readers to [9, 18, 25, 34, 33, 35] on the fractional Kirchhoff type
problems.

Recently, various approaches have been adopted for considering the fourth-order
elliptic equations of the Kirchhoff type,

∆2u−
(
a+ b

∫
Ω

|∇u|2 dx
)

∆u = f(x, u), x ∈ Ω,

u = ∆u = 0, x ∈ ∂Ω,

where ∆2u is the biharmonic operator, with different hypotheses on the nonlinearity.
For instance, Ma [21] studied the existence and multiplicity of positive solutions to
the fourth-order equation with the fixed point theorems in cones of ordered Banach
spaces. Wang et al. [30] applied the mountain pass and the truncation methods to
get the existence of nontrivial solutions to the fourth-order elliptic equations of the
Kirchhoff type with one parameter λ. Liang and Zhang [17] used the variational
methods to obtain the existence and multiplicity of solutions to the fourth-order
elliptic equations of the Kirchhoff type with critical growth in RN .

The motivation for this paper comes from [26, 28, 29, 36]. In [26], the existence
was proved of one least energy nodal solution ub to problem (1.2), with its energy
strictly larger than the ground state energy. Meanwhile, the asymptotic behavior
of ub, as the parameter b ↘ 0, was investigated as well. Later, under some more
weak assumptions on f (especially, with the Nehari type monotonicity condition
removed), Tang and Cheng [28] improved and generalized some results obtained in
[26] with some new analytical skills and the non-Nehari manifold method. In [29],
the authors obtained the existence of least energy nodal solutions to the Kirchhoff-
type equation with critical growth in bounded domains by using the constraint
variational method and the quantitative deformation lemma. In [36], the authors
studied the fourth-order elliptic equation of the Kirchhoff-type,

∆2u−
(
a+ b

∫
RN
|∇u|2 dx

)
∆u+ V (x)u = f(u), x ∈ RN ,

u ∈ H2(RN ),

where a > 0 and b ≥ 0 are constants. By the constraint variational method and the
quantitative deformation lemma, they proved that the problem possesses one least
energy nodal solution. For more results on nodal solutions to the Kirchhoff-type



EJDE-2021/19 NODAL SOLUTIONS OF FOURTH-ORDER KIRCHHOFF EQUATIONS 3

equations, please refer to [8, 12, 20, 37] and the references therein. However, to the
best of our knowledge, there are no such results concerning the existence of nodal
solutions of the problem (1.1) involving critical nonlinearities in the whole space.

The purpose of this article is to study the existence, energy estimates and con-
vergence properties of the least energy nodal solutions to the fourth-order elliptic
equation (1.1). The novelty of this paper is that problem (1.1) concerns the crit-
ical case on the entire space. Based on these facts, the problem turns out to be
extremely complicated and more difficult than the one without critical nonlinear-
ities in bounded domains. Since problem (1.1) involves critical exponents in the
nonlinearity, it is rather difficult to show that the energy functional reaches a lower
infimum on the Nehari manifold because of the lack of compactness caused by the
critical term. As we will see, this problem prevents us from using the approach
in [2, 26, 28, 36]. So we need some new ideas to overcome the above difficulties.
Moreover, we use the constraint variational method, the topological degree theory
and the quantitative deformation lemma to prove our main results. Thus, our main
results generalize papers [2, 26, 28, 36] in several directions.

Before stating our main results, we define

H2(RN ) := {u ∈ L2(RN ) : |∇u|,∆u ∈ L2(RN )},

endowed with the norm

‖u‖H2(RN ) =
(∫

RN

(
(∆u)2 + (∇u)2 + u2

)
dx
)1/2

.

Now, we introduce the space

E :=
{
u ∈ H2(RN ) :

∫
RN

V (x)|u|2 dx <∞
}

with the inner product

〈u, v〉 =

∫
RN

(∆u∆v +∇u∇v + V (x)uv) dx

and the norm

‖u‖ =

∫
RN

(
|∆u|2 + |∇u|2 + V (x)|u|2

)
dx.

Under condition (A1), it is known that the embedding E ↪→ H2(RN ) ↪→ Lp(RN )
for p ∈ (2, 2∗∗) is compact, and continuous for p ∈ [2, 2∗∗] (see [4]), and

Sp|u|p ≤ ‖u‖, for every u ∈ E. (1.4)

In particular, the best Sobolev constant for the embedding E ↪→ L2∗∗
(RN ) is

S = inf
{∫

RN
|∆u|2dx :

∫
RN
|u|2

∗∗
dx = 1

}
.

Definition 1.1. We say that u ∈ E is a weak solution to problem (1.1), if∫
RN

(∆u∆φ+∇u∇φ+ V (x)uφ) dx+ b

∫
RN
|∇u|2dx

∫
RN
∇u∇φdx

= λ

∫
RN

f(u)φdx+

∫
RN
|u|2

∗∗−2uφ dx,

for every φ ∈ E.



4 H. PU, S. LI, S. LIANG, D. D. REPOVŠ EJDE-2021/19

For convenience, we will omit the term weak throughout this paper. The corre-
sponding energy functional Iλb : E → R to problem (1.1) is defined by

Iλb (u) =
1

2

∫
RN

(
|∆u|2 + |∇u|2 + V (x)|u|2

)
dx+

b

4

(∫
RN
|∇u|2 dx

)2

− λ
∫
RN

F (u) dx− 1

2∗∗

∫
RN
|u|2

∗∗
dx.

(1.5)

It is easy to see that Iλb belongs to C1(E,R) and the critical points of Iλb are the
solutions to (1.1). For every u ∈ E we can write

u+(x) = max{u(x), 0} and u−(x) = min{u(x), 0} .
Then every solution u ∈ E to problem (1.1) with the property that u± 6= 0 is a
nodal solution to problem (1.1).

Our objective is to find the least energy nodal solutions to problem (1.1). There
exist several interesting studies on the following typical semilinear equation, which
is related to problem (1.1) (see [3, 4]),

−∆u+ V (x)u = f(x, u) in RN . (1.6)

These methods, however, depend heavily upon the decompositions:

J(u) = J(u+) + J(u−), (1.7)

〈J ′(u), u+〉 = 〈J ′(u+), u+〉 and 〈J ′(u), u−〉 = 〈J ′(u−), u−〉, (1.8)

where J is the energy functional of (1.6), given by

J(u) =
1

2

∫
RN

(|∇u|2 + V (x)u2) dx−
∫
RN

F (x, u) dx.

However, if b > 0, the energy functional Iλb cannot be decomposed in the same way
as it is done in (1.7) and (1.8). In fact, we have

Iλb (u) = Iλb (u+) + Iλb (u−) +
b

2

∫
RN
|∇u+|2dx

∫
RN
|∇u−|2dx;

if u+ 6≡ 0, then

〈(Iλb )′(u), u+〉 = 〈(Iλb )′(u+), u+〉+ b

∫
RN
|∇u+|2dx

∫
RN
|∇u−|2dx > 〈(Iλb )′(u+), u+〉;

if u− 6≡ 0, then

〈(Iλb )′(u), u−〉 = 〈(Iλb )′(u−), u−〉+ b

∫
RN
|∇u−|2dx

∫
RN
|∇u+|2dx > 〈(Iλb )′(u−), u−〉.

Therefore, the methods used for obtaining nodal solutions to the local problem (1.6)
do not seem applicable to problem (1.1). In this paper, we follow the approach in
[5] by defining the constrained set

N λ
b = {u ∈ E : u± 6= 0, 〈(Iλb )′(u), u±〉 = 0} (1.9)

and considering a minimization problem of Iλb on N λ
b . Shuai [26] proved that N λ

b 6=
∅, in the absence of the nonlocal term, by applying the parametric method and the
implicit theorem. However, it is the nonlocal terms in problem (1.1), the biharmonic
operator and the nonlocal term involved, that add to our difficulties. Roughly
speaking, compared to the general Kirchhoff type problem (1.2), decompositions
(1.7) and (1.8) corresponding to Iλb , are much more complicated, which accounts for
some technical difficulties during the proof of the nonemptiness of N λ

b . Moreover,



EJDE-2021/19 NODAL SOLUTIONS OF FOURTH-ORDER KIRCHHOFF EQUATIONS 5

the parametric method and implicit theorem are not applicable to problem (1.1)
because the complexity of the nonlocal problem there. Hence, inspired by [1], we
follow a different path, specifically, we resort to a modified Miranda’s theorem (see
[22]). It is also feasible to prove that the minimizer of the constrained problem is
also a nodal solution via the quantitative deformation lemma and degree theory.
We can now present our first main result.

Theorem 1.2. Assume that (A1)—(A5) hold. Then there exists λ∗ > 0 such that
for all λ ≥ λ∗, problem (1.1) has a least energy nodal solution ub ∈ N λ

b such that
Iλb (ub) = infu∈Nλb I

λ
b (u).

Another goal of this paper is to establish the so-called energy doubling property
(cf. [31]), i.e., the energy of any nodal solution to problem (1.1) is strictly larger
than twice the ground state energy. The conclusion is trivial for the semilinear
equation problem (1.6). When b > 0, a similar result was obtained by Shuai [26] in
a bounded domain Ω. We are also interested in whether energy doubling property
still holds for problem (1.1). To answer this question, we prove the following result.

Theorem 1.3. Assume that (A1)–(A5) hold. Then there exists λ∗∗ > 0 such that
for all λ ≥ λ∗∗, c∗ := infu∈Mλ

b
Iλb (u) > 0 is achieved, and Iλb (u) > 2c∗, where

Mλ
b = {u ∈ E \ {0} : 〈(Iλb )′(u), u〉 = 0} and u is the least energy nodal solution

obtained in Theorem 1.2. In particular, c∗ > 0 is achieved either by a positive or a
negative function.

It is obvious that the energy of the nodal solution ub obtained in Theorem 1.2
depends on b. Next, we establish a convergence property of ub as b → 0, which
demonstrates a relationship between b > 0 and b = 0 for problem (1.1).

Theorem 1.4. Assume that (A1)–(A5) hold. Then for any sequence {bn} with
bn → 0 as n→∞, there exists a subsequence, still denoted by {bn}, such that {un}
strongly converges to u0 in E as n→∞, where u0 is a least energy nodal solution
to the problem

∆2u−∆u+ V (x) = λf(u) + |u|2
∗∗−2u in RN . (1.10)

The structure of this article is as follows: Section 2 contains the proof of the
achieving the least energy for the constraint problem (1.1). While section 3 is
devoted to the proofs of our main theorems.

Throughout this paper, we use standard notation. For simplicity, we use “→”
and “⇀” to denote the strong and weak convergence in the related function space,
respectively. By C and Ci we denote various positive constants, and by “:=”
definitions. To simplify the notation, we denote a subsequence of a sequence {un}n
also as {un}n, unless otherwise specified.

2. Some technical lemmas

To begin, fix u ∈ E with u± 6= 0. Consider the function ϕ : R+ × R+ → R and
the mapping W : R+ × R+ → R2, where

ϕ(α, β) = Iλb (αu+ + βu−), (2.1)

W (α, β) =
(
〈(Iλb )′(αu+ + βu−), αu+〉, 〈(Iλb )′(αu+ + βu−), βu−〉

)
. (2.2)



6 H. PU, S. LI, S. LIANG, D. D. REPOVŠ EJDE-2021/19

For brevity, we define the quantities

A+(u) =

∫
RN
|∇u+|2 dx, A−(u) =

∫
RN
|∇u−|2 dx, B(u) =

∫
RN

∆u+∆u− dx.

Lemma 2.1. Assume that (A1)–(A5) hold. Then for any u ∈ E with u± 6= 0,
there is the unique maximum point pair of positive numbers (αu, βu) such that
αuu

+ + βuu
− ∈ N λ

b .

Proof. Our proof consists in verifying three claims.

Claim 1. There exists a pair of positive numbers (αu, βu) such that αuu
++βuu

− ∈
N λ
b , for any u ∈ E with u± 6= 0. Note that

〈(Iλb )′(αu+ + βu−), αu+〉

=

∫
RN

∆(αu+ + βu−)∆αu+dx+

∫
RN
|∇αu+|2 dx+

∫
RN

V (x)|αu+|2dx

+ b

∫
RN
|∇(αu+ + βu−)|2dx

∫
RN
|∇αu+|2dx

− λ
∫
RN

f(αu+)αu+dx−
∫
RN
|αu+|2

∗∗
dx

and

〈(Iλb )′(αu+ + βu−), βu−〉

=

∫
RN

∆(αu+ + βu−)∆βu−dx+

∫
RN
|∇βu−|2 dx+

∫
RN

V (x)|βu−|2dx

+ b

∫
RN
|∇(αu+ + βu−)|2dx

∫
RN
|∇βu−|2dx

− λ
∫
RN

f(βu−)βu−dx−
∫
RN
|βu−|2

∗∗
dx.

By a direct computation we obtain that

〈(Iλb )′(αu+ + βu−), αu+〉

= α2‖u+‖2 + α2β2bA+(u)A−(u) + α4b
(
A+(u)

)2
+ αβB(u)− λ

∫
RN

f(αu+)αu+dx−
∫
RN
|αu+|2

∗∗
dx

(2.3)

and
〈(Iλb )′(αu+ + βu−), βu−〉

= β2‖u−‖2 + α2β2bA+(u)A−(u) + β4b
(
A−(u)

)2
+ αβB(u)− λ

∫
RN

f(βu−)βu−dx−
∫
RN
|βu−|2

∗∗
dx.

(2.4)

By assumptions (A2) and (A3), we have∫
RN

f(αu+)αu+dx ≤ ε
∫
RN
|αu+|2 dx+ Cε

∫
RN
|αu+|p dx. (2.5)

Choose ε > 0 small enough such that (1 − λεCε) > 0, which together with (2.5)
and (2.3), yields

〈(Iλb )′(αu+ + βu−), αu+〉 ≥ (1− λεCε)α2‖u+‖2 + α2β2bA+(u)A−(u)



EJDE-2021/19 NODAL SOLUTIONS OF FOURTH-ORDER KIRCHHOFF EQUATIONS 7

+ α4b
(
A+(u)

)2 − λCε ∫
RN
|αu+|p dx−

∫
RN
|αu+|2

∗∗
dx.

Since 2∗∗ > 4, we have 〈(Iλb )′(αu+ + βu−), αu+〉 > 0 for a small enough α and for
all β ≥ 0.

Similarly, according to (2.5) and (2.4), we get 〈(Iλb )′(αu+ + βu−), βu−〉 > 0, for
small enough β and all α ≥ 0. Hence, there exists r > 0 such that

〈(Iλb )′(ru+ + βu−), ru+〉 > 0 and 〈(Iλb )′(αu+ + ru−), ru−〉 > 0, (2.6)

for all α, β ≥ 0.
On the other hand, by (A3) and (A4), we have

f(t)t > 0, t 6= 0; F (t) ≥ 0, t ∈ R. (2.7)

Now, choose R > r. For sufficiently large R, and by (2.3), (2.4), (2.7), we have

〈(Iλb )′(Ru+ + βu−), Ru+〉 < 0 and 〈(Iλb )′(αu+ +Ru−), Ru−〉 < 0, (2.8)

for all α, β ∈ [r,R]. Invoking Miranda’s theorem [22], together with (2.6) and (2.8),
we can conclude that there exists (αu, βu) ∈ R+×R+ such that W (αu, βu) = (0, 0),
i.e., αuu

+ + βuu
− ∈ N λ

b .

Claim 2. The pair (αu, βu) is unique.
• Case u ∈ N λ

b . Then we have

〈(Iλb )′(u), u+〉 = 0 and 〈(Iλb )′(u), u−〉 = 0,

that is,
‖u+‖2 +B(u) + bA+(u)

(
A+(u) +A−(u)

)
= λ

∫
RN

f(u+)u+dx+

∫
RN
|u+|2

∗∗
dx

(2.9)

and
‖u−‖2 +B(u) + bA−(u)

(
A+(u) +A−(u)

)
= λ

∫
RN

f(u−)u−dx+

∫
RN
|u−|2

∗∗
dx.

(2.10)

By Claim 1, we know that there exists at least one positive pair (α0, β0) satisfying
α0u

+ + β0u
− ∈ N λ

b .
Next we show that (α0, β0) = (1, 1) is the unique pair of numbers. Without loss

of generality, let us assume that α0 ≤ β0. It follows from (2.8) that

α2
0

(
‖u+‖2 +B(x)

)
+ α4

0bA
+(u)

(
A+(u) +A−(u)

)
= λ

∫
RN

f(α0u
+)α0u

+ dx+

∫
RN
|α0u

+|2
∗∗
dx.

(2.11)

If α0 < 1, then from (2.9), (2.11) and (A5), we have

0 < [(α0)−2 − 1]
(
‖u+‖2 +B(u)

)
≤ λ

∫
RN

(f(x, α0u
+)

(α0u+)3
− f(u+)

(u+)3

)
(u+)4 dx

+ [(α0)2∗∗−4 − 1]

∫
RN
|u+|2

∗∗
dx < 0,

(2.12)

which is a contradiction. Hence, 1 ≤ α0 ≤ β0.
Adopting a similar approach, we can see that β0 ≤ 1, which implies that α0 =

β0 = 1.



8 H. PU, S. LI, S. LIANG, D. D. REPOVŠ EJDE-2021/19

• Case u /∈ N λ
b . Assume there exist two other pairs of positive numbers (α1, β1)

and (α2, β2) such that

σ1 = α1u
+ + β1u

− ∈ N λ
b and σ2 = α2u

+ + β2u
− ∈ N λ

b .

Then

σ2 =
(α2

α1

)
α1u

+ +
(β2

β1

)
β1u
− =

(α2

α1

)
σ+

1 +
(β2

β1

)
σ−1 ∈ N λ

b .

Since σ1 ∈ N λ
b , it is clear that

α2

α1
=
β2

β1
= 1,

which means that α1 = α2, β1 = β2.

Claim 3. The pair (αu, βu) is the unique maximum point of the function ϕ on
R+ × R+. We know from the above that (αu, βu) is the unique critical point of ϕ
on R+ × R+. By definition and (2.5), we have

ϕ(α, β) = Iλb (αu+ + βu−)

=
α2

2
‖u+‖2 +

β2

2
‖u−‖2 + αβB(u) +

α4b

4

(
A+(u)

)2
+
β4b

4

(
A−(u)

)2
+
α2β2b

2
A+(u)A−(u)− λ

∫
RN

F (αu+) dx− λ
∫
RN

F (βu−) dx

− α2∗∗

2∗∗

∫
RN
|u+|2

∗∗
dx− β2∗∗

2∗∗

∫
RN
|u−|2

∗∗
dx

<
α2

2
‖u+‖2 +

β2

2
‖u−‖2 + αβB(u) +

α4b

4

(
A+(u)

)2
+
β4b

4

(
A−(u)

)2
+
α2β2b

2
A+(u)A−(u)− α2∗∗

2∗∗

∫
RN
|u+|2

∗∗
dx− β2∗∗

2∗∗

∫
RN
|u−|2

∗∗
dx,

as |(α, β)| → ∞. This implies that lim|(α,β)|→∞ ϕ(α, β) = −∞, because 2∗∗ > 4.
Hence, it suffices to show that the maximum point cannot be achieved on the
boundary of R+ × R+.

We carry out the proof by contradiction. Assuming (0, β̄) is the global maximum
point of ϕ with β̄ ≥ 0, we have

ϕ(α, β̄) =
α2

2
‖u+‖2 +

β̄2

2
‖u−‖2 + αβ̄B(u) +

α4b

4

(
A+(u)

)2
+
β̄4b

4

(
A−(u)

)2
+
α2β̄2b

2
A+(u)A−(u)− λ

∫
RN

F (αu+) dx− λ
∫
RN

F (β̄u−) dx

− α2∗∗

2∗∗

∫
RN
|u+|2

∗∗
dx− β̄2∗∗

2∗∗

∫
RN
|u−|2

∗∗
dx.

Hence, it is clear that

ϕ′α(α, β̄) = α‖u+‖2 + β̄B(u) + α3b
(
A+(u)

)2
+ αβ̄2bA+(u)A−(u)

− λ
∫
RN

f(αu+)u+dx− α2∗∗−1

∫
RN
|u+|2

∗∗
dx > 0,

for small enough α. This means that ϕ is an increasing function with respect to
α if α is small enough, which is a contradiction. In a similar way, we can deduce



EJDE-2021/19 NODAL SOLUTIONS OF FOURTH-ORDER KIRCHHOFF EQUATIONS 9

that ϕ cannot achieve its global maximum at (α, 0) with α ≥ 0. Thus, we have
completed the proof. �

Lemma 2.2. Assume that (A1) —(A5) hold. Then for any u ∈ E with u± 6= 0
such that 〈(Iλb )′(u), u±〉 ≤ 0, the unique maximum point pair of ϕ on R+ × R+

satisfies 0 < αu, βu ≤ 1.

Proof. Without loss of generality, we may assume that αu ≥ βu > 0. Since αuu
+ +

βuu
− ∈ N λ

b , we have

α2
u‖u+‖2 + αuβuB(u) + α2

uβ
2
ubA

+(u)A−(u) + α4
ub
(
A+(u)

)2
= λ

∫
RN

f(αuu
+)αuu

+dx+

∫
RN
|αuu+|2

∗∗
dx.

(2.13)

Furthermore, since 〈(Iλb )′(u), u+〉 ≤ 0, we have

‖u+‖2 +B(u) + b
(
A+(u)

)2
+ bA+(u)A−(u) ≤ λ

∫
RN

f(u+)u+dx+

∫
RN
|u+|2

∗∗
dx.

Then by (2.13), we have

[(αu)−2 − 1]
(
‖u+‖2 +B(u)

)
≥ λ

∫
RN

(
f(αuu

+)

(αuu+)3
− f(u+)

(u+)3

)
(u+)4 dx+ [(αu)2∗∗−4 − 1]

∫
RN
|u+|2

∗∗
dx.

(2.14)

Obviously, the left hand side of (2.14) is negative for αu > 1 whereas the right hand
side is positive, which is a contradiction. Therefore 0 < αu, βu ≤ 1. �

Lemma 2.3. Suppose that cλb = infu∈Nλb I
λ
b (u). Then limλ→∞ cλb = 0.

Proof. For every u ∈ N λ
b , we have 〈(Iλb )′(u), u〉 = 0, thus

‖u+‖2 + ‖u−‖2 + 2B(u) + b
(
A+(u) +A−(u)

)2
= λ

∫
RN

f(u)u dx+

∫
RN
|u|2

∗∗
dx.

Then, by (2.5), we have

‖u‖2 ≤ λ
∫
RN

f(u±)u± dx+

∫
RN
|u±|2

∗∗
dx

≤ λε
∫
RN
|u±|2dx+ λCε

∫
RN
|u±|p dx+

∫
RN
|u±|2

∗∗
dx.

(2.15)

Choose ε small so that λε
∫
RN |u

±|2dx ≤ 1
2‖u
±‖2. Then we can claim that there

exists ρ > 0 such that

‖u±‖2 ≥ ρ for all u ∈ N λ
b , (2.16)

since 4 < 2∗∗. Next, by (A5), we have for t 6= 0 that

F(t) := tf(t)− 4F (t) ≥ 0,

and F(t) is increasing when t > 0, and decreasing when t < 0. Therefore,

Iλb (u) = Iλb (u)− 1

4
〈(Iλb )′(u), u〉

=
1

4
‖u‖2 + (

1

4
− 1

2∗∗
)

∫
RN
|u|2

∗∗
dx+

λ

4

∫
RN

[f(u)u− 4F (u)] dx

≥ 1

4
‖u‖2 ≥ ρ

4
> 0.

(2.17)



10 H. PU, S. LI, S. LIANG, D. D. REPOVŠ EJDE-2021/19

So we have Iλb (u) > 0 for all u ∈ N λ
b , which means that cλb = infu∈Nλb I

λ
b (u) is

well-defined.
Fix u ∈ E with u± 6= 0. According to Lemma 2.1, for each λ > 0, there exist

αλ, βλ > 0 such that αλu
+ + βλu

− ∈ N λ
b . Therefore,

0 ≤ cλb = inf
u∈Nλb

Iλb (u)

≤ Iλb (αλu
+ + βλu

−)

≤ 1

2
‖αλu+ + βλu

−‖2 +
b

4

(∫
RN
|∇(αλu

+ + βλu
−)|2dx

)2

=
α2
λ

2
‖u+‖2 +

β2
λ

2
‖u−‖2 + αλβλB(u) +

α4
λb

4

(
A+(u)

)2
+
β4
λb

4

(
A−(u)

)2
+
α2
λβ

2
λb

2
A+(u)A−(u).

It suffices to prove that αλ → 0 and βλ → 0, as λ→∞. Let

T = {(αλ, βλ) ∈ R+ × R+ : W (αλ, βλ) = (0, 0), λ > 0},

where W is defined as in (2.2). Then

α2∗∗

λ

∫
RN
|u+|2

∗∗
dx+ β2∗∗

λ

∫
RN
|u−|2

∗∗
dx

≤ α2∗∗

λ

∫
RN
|u+|2

∗∗
dx+ β2∗∗

λ

∫
RN
|u−|2

∗∗
dx

+ λ

∫
RN

f(αλu
+)αλu

+dx+ λ

∫
RN

f(βλu
−)βλu

−dx

= ‖αλu+ + βλu
−‖2 + b

(
α2
λA

+(u) + β2
λA
−(u)

)2
.

Therefore, T is bounded, since 4 < 2∗∗. Let {λn} ⊂ (0,∞) be such that λn →∞,
as n→∞. Then there exist α0 and β0 such that (αλn , βλn)→ (α0, β0), as n→∞.

Now we claim that α0 = β0 = 0. Assume, to the contrary, that α0 > 0 or β0 > 0.
Since αλnu

+ + βλnu
− ∈ N λn

b , then for any n ∈ N, we have

‖αλnu+ + βλnu
−‖2 + b

(
α2
λnA

+(u) + β2
λnA

−(u)
)2

= λn

∫
RN

f(αλnu
+ + βλnu

−)(αλnu
+ + βλnu

−) dx

+

∫
RN
|αλnu+ + βλnu

−|2
∗∗
dx.

(2.18)

Then, invoking αλnu
+ → α0u

+, βλnu
− → β0u

− in E and the Lebesgue dominated
convergence theorem, we have∫

RN
f(αλnu

+ + βλnu
−)(αλnu

+ + βλnu
−) dx

→
∫
RN

f(α0u
+ + β0u

−)(α0u
+ + β0u

−) dx > 0,

as n → ∞. This contradicts (2.18), given that λn → ∞, as n → ∞ and that
{αλnu+ + βλnu

−} is bounded in E. Therefore, α0 = β0 = 0, which implies
limλ→∞ cλb = 0. �



EJDE-2021/19 NODAL SOLUTIONS OF FOURTH-ORDER KIRCHHOFF EQUATIONS 11

Lemma 2.4. There exists λ∗ > 0 such that the infimum cλb is achieved for all
λ ≥ λ∗.

Proof. According to the definition of cλb , there exists a sequence {un} ⊂ N λ
b such

that limn→∞ Iλb (un) = cλb . Clearly, {un} is bounded in E. By Lemma 2.1 and the
properties of Lp space, up to a subsequence, we have

u±n ⇀ u± in E,

u±n → u± in Lp(RN ) for p ∈ [2, 2∗∗),

u±n → u± a.e. in RN .

In view of Lemma 2.1, we also have

Iλb (αu+
n + βu−n ) ≤ Iλb (un),

for all α, β ≥ 0. So, by the Brézis-Lieb lemma, Fatou’s lemma and the weak lower
semicontinuity of norm, we can conclude that

lim inf
n→∞

Iλb (αu+
n + βu−n )

≥ α2

2
lim
n→∞

(‖u+
n − u+‖2 + ‖u+‖2) +

β2

2
lim
n→∞

(‖u−n − u−‖2 + ‖u−‖2)

+
α4b

4

[
lim
n→∞

∫
RN
|∇u+

n −∇u+|2 dx+

∫
RN
|∇u+|2 dx

]2
+
β4b

4

[
lim
n→∞

∫
RN
|∇u−n −∇u−|2 dx+

∫
RN
|∇u−|2 dx

]2
− α2∗∗

2∗∗

[
lim
u
ndersetn→∞

∫
RN
|u+
n − u+|2

∗∗
dx+ lim

n→∞

∫
RN
|u+|2

∗∗
dx
]

− β2∗∗

2∗∗

[
lim
n→∞

∫
RN
|u−n − u−|2

∗∗
dx+ lim

n→∞

∫
RN
|u−|2

∗∗
dx
]

− λ
∫
RN

F (αu+) dx− λ
∫
RN

F (βu−) dx

+
α2β2b

2
lim inf
n→∞

∫
RN
|∇u+

n |2dx
∫
RN
|∇u−n |2dx

≥ Iλb (αu+ + βu−) +
α2

2
A1 +

α4b

4
A2

3 +
α4b

2
A3A

+(u)− α2∗∗

2∗∗
B1

+
β2

2
A2 +

β4b

4
A2

4 +
β4b

2
A4A

−(u)− β2∗∗

2∗∗
B2,

where

A1 = lim
n→∞

‖u+
n − u+‖2, A2 = lim

n→∞
‖u−n − u−‖2,

A3 = lim
n→∞

∫
RN
|∇u+

n −∇u+|2dx, A4 = lim
n→∞

∫
RN
|∇u−n −∇u−|2dx,

B1 = lim
n→∞

∫
RN
|u+
n − u+|2

∗∗
dx, B2 = lim

n→∞

∫
RN
|u−n − u−|2

∗∗
dx.



12 H. PU, S. LI, S. LIANG, D. D. REPOVŠ EJDE-2021/19

That is,

cλb ≥ Iλb (αu+ + βu−) +
α2

2
A1 +

α4b

4
A2

3 +
α4b

2
A3A

+(u)− α2∗∗

2∗∗
B1

+
β2

2
A2 +

β4b

4
A2

4 +
β4b

2
A4A

−(u)− β2∗∗

2∗∗
B2,

(2.19)

for all α, β ≥ 0.

Step 1: u± 6= 0. We carry out our proof by contradiction. Assume that u+ = 0.
Lettong β = 0 in (2.19) we have

cλb ≥
α2

2
A1 +

α4b

4
A2

3 −
α2∗∗

2∗∗
B1 := φ(α), (2.20)

for all α ≥ 0.

Case 1: B1 = 0. If A1 = 0, then u+
n → u+ in E. By (2.15), we obtain ‖u±‖ > 0,

which contradicts our assumption. If A1 > 0, then by (2.20), we have cλb ≥ α2

2 A1

for all α ≥ 0, which contradicts Lemma 2.3.

Case 2: B1 > 0. From the definition of S and Lemma 2.3, there exists λ∗ > 0
such that

cλb <
2

N
S−2/N (2.21)

for all λ ≥ λ∗. According to the Sobolev embedding and the fact that B1 > 0, we
obtain A1 > 0. By (2.20), we have

2

N
S−2/N ≤ 2

N

[A 2∗∗
2

1

B1

] 2
2∗∗−2

≤ max
α≥0

{α2

2
A1 −

α2∗∗

2∗∗
B1

}
≤ max

α≥0

{α2

2
A1 +

α4b

4
A2

3 −
α2∗∗

2∗∗
B1

}
≤ cλb ,

which is a contradiction. Hence, we can conclude that u+ 6= 0. Similarly, we get
that u− 6= 0.

Step 2: B1 = B2 = 0. Given that the proof of B2 = 0 is analogous, we just prove
B1 = 0. By contradiction, assume B1 > 0.

Case 1: B2 > 0. Since B1, B2 > 0, we get A1, A2 > 0. Clearly, φ(α) > 0 for α
small enough, where φ(α) is given by (2.20), and φ(α) < 0 for α sufficiently large.
Therefore, by continuity of φ(α), there exists ᾱ > 0 such that

ᾱ2

2
A1 +

ᾱ4b

4
A2

3 −
ᾱ2∗∗

2∗∗
B1 = max

α≥0

{α2

2
A1 +

α4b

4
A2

3 −
α2∗∗

2∗∗
B1

}
.

Similarly, there exists β̄ > 0 such that

β̄2

2
A2 +

β̄4b

4
A2

4 −
β̄2∗∗

2∗∗
B2 = max

β≥0

{β2

2
A2 +

β4b

4
A2

4 −
β2∗∗

2∗∗
B2

}
.

In view of the compactness of [0, ᾱ] × [0, β̄] and the continuity of φ, there exists
(αu, βu) ∈ [0, ᾱ]× [0, β̄] such that

ϕ(αu, βu) = max
(α,β)∈[0,ᾱ]×[0,β̄]

ϕ(α, β),

where ϕ is defined as in Lemma 2.1.



EJDE-2021/19 NODAL SOLUTIONS OF FOURTH-ORDER KIRCHHOFF EQUATIONS 13

Now we prove that (αu, βu) ∈ (0, ᾱ) × (0, β̄). Note that if β is small enough,
then we have

ϕ(α, 0) = Iλb (αu+) < Iλb (αu+) + Iλb (βu−) ≤ Iλb (αu+ + βu−) = ϕ(α, β),

for all α ∈ [0, ᾱ]. Thus, there exists β0 ∈ [0, β̄] such that ϕ(α, 0) ≤ ϕ(α, β0), for all
α ∈ [0, ᾱ]. That is, (αu, βu) /∈ [0, ᾱ] × {0}. With a similar method, we can show
that (αu, βu) /∈ {0} × [0, β̄].

It is obvious that

α2

2
A1 +

α4b

4
A2

3 +
α4b

2
A3A

+(u)− α2∗∗

2∗∗
B1 > 0, α ∈ (0, ᾱ] (2.22)

and

β2

2
A2 +

β4b

4
A2

4 +
β4b

2
A4A

−(u)− β2∗∗

2∗∗
B2 > 0, β ∈ (0, β̄]. (2.23)

Thus we obtain

2

N
S−2/N ≤ ᾱ2

2
A1 +

ᾱ4b

4
A2

3 −
ᾱ2∗∗

2∗∗
B1 +

ᾱ4b

2
A3A

+(u)

+
β2

2
A2 +

β4b

4
A2

4 +
β4b

2
A4A

−(u)− β2∗∗

2∗∗
B2

and

2

N
S−2/N ≤ β̄2

2
A2 +

β̄4b

4
A2

4 −
β̄2∗∗

2∗∗
B2 +

β̄4b

2
A4A

−(u)

+
α2

2
A1 +

α4b

4
A2

3 +
α4b

2
A3A

+(u)− α2∗∗

2∗∗
B1,

for all α ∈ [0, ᾱ], β ∈ [0, β̄].
From the these inequalities and (2.19), we obtain ϕ(ᾱ, β) ≤ 0, ϕ(α, β̄) ≤ 0 for all

α ∈ [0, ᾱ], β ∈ [0, β̄]. Therefore, (αu, βu) /∈ {ᾱ} × [0, β̄] and (αu, βu) /∈ [0, ᾱ]× {β̄},
which means (αu, βu) ∈ (0, ᾱ)× (0, β̄). It follows that (αu, βu) is a critical point of
ϕ.

So, αuu
+ + βuu

− ∈ N λ
b . By (2.19), we have

cλb ≥ Iλb (αuu
+ + βuu

−) +
α2
u

2
A1 +

α4
ub

4
A2

3 +
α4
ub

2
A3A

+(u)− α2∗∗

u

2∗∗
B1

+
β2
u

2
A2 +

β4
ub

4
A2

4 +
β4
ub

2
A4A

−(u)− β2∗∗

u

2∗∗
B2 > Iλb (αuu

+ + βuu
−) ≥ cλb ,

which is a contradiction. Therefore B1 = 0.

Case 2: B2 = 0. In this case, we can maximize in [0, ᾱ]× [0,∞). It is possible to
show that there exists β0 ∈ [0,∞) satisfying

Iλb (αuu
+ + βuu

−) ≤ 0 for all (α, β) ∈ [0, ᾱ]× [β0,∞).

Then there is (αu, βu) ∈ [0, ᾱ]× [0,∞) such that

ϕ(αu, βu) = max
(α,β)∈[0,ᾱ]×[0,∞)

ϕ(α, β).

We claim that (αu, βu) ∈ (0, ᾱ)× (0,∞). Indeed, ϕ(α, 0) < ϕ(α, β) for α ∈ [0, ᾱ]
and β small enough, while ϕ(0, β) < ϕ(α, β) for β ∈ [0,∞) and α sufficiently small,
which implies (αu, βu) /∈ [0, ᾱ]× {0} and (αu, βu) /∈ {0} × [0,∞).



14 H. PU, S. LI, S. LIANG, D. D. REPOVŠ EJDE-2021/19

Note that

2

N
S−2/N ≤ ᾱ2

2
A1+

ᾱ4b

4
A2

3−
ᾱ2∗∗

2∗∗
B1+

ᾱ4b

2
A3A

+(u)+
β2

2
A2+

β4b

4
A2

4+
β4b

2
A4A

−(u),

for every β ∈ [0,∞). Therefore, we have ϕ(ᾱ, β) ≤ 0 for all β ∈ [0,∞), which means
(αu, βu) /∈ {ᾱ}× [0,∞). Based on the above, we get (αu, βu) ∈ (0, ᾱ)× (0,∞), that
is, (αu, βu) is an inner maximizer of ϕ in [0, ᾱ]× [0,∞). Therefore, αuu

+ +βuu
− ∈

N λ
b . In that case, by (2.22), we have

cλb ≥ Iλb (αuu
+ + βuu

−) +
ᾱ2

2
A1 +

ᾱ4b

4
A2

3 −
ᾱ2∗∗

2∗∗
B1

+
ᾱ4b

2
A3A

+(u) +
β2

2
A2 +

β4b

4
A2

4 +
β4b

2
A4A

−(u)

> Iλb (αuu
+ + βuu

−) ≥ cλb ,

which is a contradiction. Hence, we have B1 = B2 = 0.

Step 3: cλb is achieved. Given u± 6= 0, according to Lemma 2.1, there exists
αu, βu > 0 such that û := αuu

+ + βuu
− ∈ N λ

b . Moreover, 〈(Iλb )′(u), u±〉 ≤ 0. By
Lemma 2.2, we have 0 < αu, βu ≤ 1.

Combining un ∈ N λ
b and Lemma 2.1, we obtain

Iλb (αuu
+
n + βuu

−
n ) ≤ Iλb (u+

n + u−n ) = Iλb (un).

Taking into consideration B1 = B2 = 0 and the semicontinuity of the norm, we
obtain

cλb ≤ Iλb (û)

= Iλb (û)− 1

4
〈(Iλb )′(û), û〉

=
1

4
‖û‖2 + (

1

4
− 1

2∗∗
)

∫
RN
|û|2

∗∗
dx+

λ

4

∫
RN

[f(û)û− 4F (û)] dx

≤ 1

4
‖u‖2 + (

1

4
− 1

2∗∗
)

∫
RN
|u|2

∗∗
dx+

λ

4

∫
RN

[f(u)u− 4F (u)] dx

≤ lim inf
n→∞

[
Iλb (un)− 1

4
〈(Iλb )′(un), un〉

]
≤ cλb .

Hence, we can conclude that αu = βu = 1, and cλb is achieved by ub := u+ + u− ∈
N λ
b . �

3. Proofs of main results

Proof of Theorem 1.2. Thanks to Lemma 2.4, we only need to prove that the min-
imizer ub for cλb is indeed a nodal solution to problem (1.1).

Because ub ∈ N λ
b , we have 〈(Iλb )′(ub), u

+
b 〉 = 〈(Iλb )′(ub), u

−
b 〉 = 0. In view of

Lemma 2.1, for (α, β) ∈ (R+ × R+)\(1, 1), we have

Iλb (αu+
b + βu−b ) < Iλb (u+

b + u−b ) = cλb . (3.1)

Now we proceed by contradiction. Suppose (Iλb )′(ub) 6= 0, then there exist δ > 0
and θ > 0 such that

‖(Iλb )′(v)‖ ≥ θ for all ‖v − ub‖ ≤ 3δ.



EJDE-2021/19 NODAL SOLUTIONS OF FOURTH-ORDER KIRCHHOFF EQUATIONS 15

Choose τ ∈ (0,min{1/2, δ√
2‖ub‖

}), and define

D := (1− τ, 1 + τ)× (1− τ, 1 + τ),

g(α, β) := αu+
b + βu−b for all (α, β) ∈ D.

By (3.1), we have

c̄λ := max
∂D

(Iλb ◦ g) < cλb . (3.2)

Let ε := min{(cλb − c̄λ)/2, θδ/8} and Sδ := B(ub, δ). By [32, Lemma 2.3], there
exists a deformation η ∈ C([0, 1]×D,D) such that

(a) η(1, v) = v if v /∈ (Iλb )−1([cλb − 2ε, cλb + 2ε] ∩ S2δ),

(b) η(1, (Iλb )c
λ
b+ε ∩ Sδ) ⊂ (Iλb )c

λ
b−ε,

(c) Iλb (η(1, v)) ≤ Iλb (v) for all v ∈ E.

Clearly,

max
(α,β)∈D̄

Iλb (η(1, g(α, β))) < cλb . (3.3)

Therefore we claim that η(1, g(D)) ∩ N λ
b 6= ∅ , which contradicts the definition of

cλb .
We define h(α, β) := η(1, g(α, β)),

Φ0(α, β) : = (〈(Iλb )′(g(α, β)), u+
b 〉, 〈(I

λ
b )′(g(α, β)), u−b 〉)

= (〈(Iλb )′(αu+
b + βu−b ), u+

b 〉, 〈(I
λ
b )′(αu+

b + βu−b ), u−b 〉)

and

Φ1(α, β) :=
( 1

α
〈(Iλb )′(h(α, β)), (h(α, β))+〉, 1

β
〈(Iλb )′(h(α, β)), (h(α, β))−〉

)
.

With an approach similar to [14], we use degree theory to obtain deg(Φ0, D, 0) =
1. Then by (3.2), we obtain

g(α, β) = h(α, β) on ∂D,

as a result of which, we have deg(Φ1, D, 0) = deg(Φ0, D, 0) = 1. Hence, Φ1(α0, β0) =
0 for some (α0, β0) ∈ D so that

η(1, g(α0, β0)) = h(α0, β0) ∈ N λ
b ,

which contradicts (3.3). Hence, (Iλb )′(ub) = 0, which implies ub is a critical point
of Iλb . Thus, we can deduce that ub is a nodal solution to problem (1.1). �

By Theorem 1.2, we obtain a least energy nodal solution ub to problem (1.1),
contributing to the establishment of Theorem 1.3, where we shall prove that the
energy of ub is strictly larger than twice the ground state energy.

Proof of Theorem 1.3. As in the proof of Lemma 2.3, there exists λ∗1 > 0 such that
for all λ ≥ λ∗1, and for each b > 0, there exists vb ∈Mλ

b such that Iλb (vb) = c∗ > 0.
By standard arguments (see [10, Corollary 2.13]), the critical points of the functional
Iλb on Mλ

b are critical points of Iλb in E, so we obtain (Iλb )′(vb) = 0. That is, vb is
a ground state solution to problem (1.1).

As stated in Theorem 1.2, ub is known as a least energy nodal solution to problem
(1.1), which changes sign only once when λ ≥ λ∗.

Let λ∗∗ = max{λ∗, λ∗1} and assume ub = u+
b + u−b . Adopting the same approach

as in Lemma 2.1, we claim there exist αu+
b
> 0 and βu−

b
> 0 such that αu+

b
u+
b ∈Mλ

b



16 H. PU, S. LI, S. LIANG, D. D. REPOVŠ EJDE-2021/19

and βu−
b
u−b ∈ Mλ

b . Then, by Lemma 2.2, we obtain αu+
b
, βu−

b
∈ (0, 1). Hence,

thanks to Lemma 2.1, we have

2c∗ ≤ Iλb (αu+
b
u+
b ) + Iλb (βu−

b
u−b ) ≤ Iλb (αu+

b
u+
b + βu−

b
u−b ) < Iλb (u+

b + u−b ) = cλb .

It follows that c∗ > 0 cannot be achieved by a nodal function. �

We complete this section with the proof of Theorem 1.4. In the sequel, we regard
b > 0 as a parameter in problem (1.1).

Proof of Theorem 1.4. In 3 steps, we analyze the convergence property of ub as
b→ 0, where ub is the least energy nodal solution obtained in Theorem 1.2.

Step 1. For any sequence {bn}, we prove that {ubn} is bounded in E, if bn ↘ 0.
Let χ ∈ C∞0 (RN ) be a nonzero function with χ± 6= 0 fixed. Analogous to the
argument in Lemma 2.1, for any b ∈ [0, 1], there exists a pair of positive numbers
(λ1, λ2) independent of b, such that

〈(Iλb )′(λ1χ
+ + λ2χ

−), λ1χ
+〉 < 0 and 〈(Iλb )′(λ1χ

+ + λ2χ
−), λ2χ

−〉 < 0.

Then according to Lemma 2.2, for any b ∈ [0, 1], there exists a unique pair
(αχ(b), βχ(b)) ∈ (0, 1]× (0, 1] such that χ := αχ(b)λ1χ

+ +βχ(b)λ2χ
− ∈ N λ

b . There-
fore, by (2.5), it follows that, for any b ∈ [0, 1],

Iλb (ub) ≤ Iλb (χ) = Iλb (χ)− 1

4
〈(Iλb )′(χ), χ〉

=
1

4
‖χ‖2 + (

1

4
− 1

2∗∗
)

∫
RN
|χ|2

∗∗
dx+

λ

4

∫
RN

[f(χ)χ− 4F (χ)] dx

≤ 1

4
‖χ‖2 + (

1

4
− 1

2∗∗
)

∫
RN
|χ|2

∗∗
dx+

λ

4

∫
RN

(
C1|χ|2 + C2|χ|p

)
dx

≤ 1

4
‖λ1χ

+‖2 + (
1

4
− 1

2∗∗
)

∫
RN
|λ1χ

+|2
∗∗
dx

+
λ

4

∫
RN

(
C1|λ1χ

+|2 + C2|λ1χ
+|p
)
dx

+
1

4
‖λ2χ

−‖2 + (
1

4
− 1

2∗∗
)

∫
RN
|λ2χ

−|2
∗∗
dx

+
λ

4

∫
RN

(
C1|λ2χ

−|2 + C2|λ2χ
−|p
)
dx

:= C∗,

where C∗ is a positive constant independent of b. Thus, as n→∞, it follows that

C∗ + 1 ≥ Iλbn(ubn) = Iλbn(ubn)− 1

4
〈(Iλbn)′(ubn), ubn〉 ≥

1

4
‖ubn‖2,

that is, {ubn} is bounded in E.

Step 2. In this step, we prove that problem (1.10) possesses one nodal solution u0.
Since {ubn} is bounded in E, thanks to Step 1, up to a subsequence, there exists
u0 ∈ E such that

ubn ⇀ u0 inE,

ubn → u0 in Lp(RN ) for p ∈ [2, 2∗∗),

ubn → u0 a.e. inRN .
(3.4)



EJDE-2021/19 NODAL SOLUTIONS OF FOURTH-ORDER KIRCHHOFF EQUATIONS 17

Given that {ubn} is a weak solution to (1.1) with b = bn, we have∫
RN

(∆u∆φ+∇u∇φ+ V (x)uφ) dx+ bn

∫
RN
|∇u|2dx

∫
RN
∇u∇φdx

= λ

∫
RN

f(u)φdx+

∫
RN
|u|2

∗∗−2uφ dx

(3.5)

for all φ ∈ C∞0 (RN ).
Combing (3.4), (3.5) and Step 1, we find that∫
RN

(∆u0∆φ+∇u0∇φ+ V (x)u0φ) dx+ bn

∫
RN
|∇u0|2dx

∫
RN
∇u0∇φdx

= λ

∫
RN

f(u0)φdx+

∫
RN
|u0|2

∗∗−2u0φdx

(3.6)

for all φ ∈ C∞0 (RN ), which in turn implies that u0 is a weak solution to (1.10).
Analogous to the process of Lemma 2.3, we obtain that u±0 6= 0. Thus, we have
completed the proof of this step.

Step 3. In this step, we prove that problem (1.10) possesses a least energy nodal
solution v0, and that there exists a unique pair (αbn , βbn) ∈ R+ × R+ satisfying
αbnv

+
0 + βbnv

−
0 ∈ N λ

bn
. Also we prove that (αbn , βbn)→ (1, 1) as n→∞.

Similar to the proof of Theorem 1.2, we can reach the conclusion that prob-
lem (1.10) possesses a least energy nodal solution v0, where Iλ0 (v0) = c0 and
(Iλ0 )′(v0) = 0. Then, in view of Lemma 2.1, we can obtain with ease the existence
and uniqueness of the pair (αbn , βbn) such that αbnv

+
0 + βbnv

−
0 ∈ N λ

bn
. Besides,

we know αbn > 0 and βbn > 0. To complete the proof, we just establish that
(αbn , βbn)→ (1, 1) as n→∞. Actually, given that αbnv

+
0 + βbnv

−
0 ∈ N λ

bn
, we have

α2
bn‖v

+
0 ‖2 + αbnβbn

∫
RN

∆v+
0 ∆v−0 dx

+ α2
bnbn

∫
RN
|∇v+

0 |2dx
(
α2
bn

∫
RN
|∇v+

0 |2dx+ β2
bn

∫
RN
|∇v−0 |2dx

)
= λ

∫
RN

f(αbnv
+
0 )αbnv

+
0 dx+

∫
RN
|αbnv+

0 |2
∗∗
dx

(3.7)

and

β2
bn‖v

−
0 ‖2 + αbnβbn

∫
RN

∆v+
0 ∆v−0 dx

+ β2
bnbn

∫
RN
|∇v−0 |2dx

(
β2
bn

∫
RN
|∇v−0 |2dx+ α2

bn

∫
RN
|∇v+

0 |2dx
)

= λ

∫
RN

f(βbnv
−
0 )βbnv

−
0 dx+

∫
RN
|βbnv−0 |2

∗∗
dx.

(3.8)

a Since bn ↘ 0, we conclude that the sequences {αbn} and {βbn} are bounded.
Assume, up to a subsequence, αbn → α0 and βbn → β0. Then by (3.7) and (3.8),
we have

α2
0‖v+

0 ‖2+α0β0

∫
RN

∆v+
0 ∆v−0 dx = λ

∫
RN

f(α0v
+
0 )α0v

+
0 dx+

∫
RN
|α0v

+
0 |2

∗∗
dx (3.9)

and

β2
0‖v+

0 ‖2 + α0β0

∫
RN

∆v+
0 ∆v−0 dx = λ

∫
RN

f(β0v
−
0 )β0v

−
0 dx+

∫
RN
|β0v

−
0 |2

∗∗
dx.

(3.10)



18 H. PU, S. LI, S. LIANG, D. D. REPOVŠ EJDE-2021/19

Noticing that v0 is a nodal solution to problem (1.10), we obtain

‖v+
0 ‖2 +

∫
RN

∆v+
0 ∆v−0 dx = λ

∫
RN

f(v+
0 )v+

0 dx+

∫
RN
|v+

0 |2
∗∗
dx, (3.11)

‖v+
0 ‖2

∫
RN

∆v+
0 ∆v−0 dx = λ

∫
RN

f(v−0 )v−0 dx+

∫
RN
|v−0 |2

∗∗
dx. (3.12)

Therefore, from (3.9)-(3.12), we can easily obtain that (α0, β0) = (1, 1), and thus
Step 3 follows.

We can now complete the proof of Theorem 1.4. We claim that u0 obtained in
Step 2 is a least energy solution to problem (1.10). In fact, according to Step 3 and
Lemma 2.1, we see that

Iλ0 (v0) ≤ Iλ0 (u0) = lim
n→∞

Iλbn(ubn)

≤ lim
n→∞

Iλbn(αbnv
+
0 + βbnv

−
0 )

= lim
n→∞

Iλ0 (v+
0 + v−0 )

= Iλ0 (v0),

which yields completest the proof of Theorem 1.4. �

Acknowledgments. H. Pu was supported by the Graduate Scientific Research
Project of Changchun Normal University (SGSRPCNU No. 2020-51). S. Liang was
supported by the Foundation for China Postdoctoral Science Foundation (Grant no.
2019M662220), Scientific research projects for Department of Education of Jilin
Province, China (JJKH20210874KJ), Natural Science Foundation of Changchun
Normal University (No. 2017-09). D. D. Repovš was supported by the Slovenian
Research Agency (No. P1-0292, N1-0114, N1-0083, N1-0064, and J1-8131). We
want to thank the anonymous referees for their comments and suggestions.

References

[1] C. O. Alves, A. B. Nóbrega; Nodal ground state solution to a biharmonic equation via dual

method, J. Differential Equations, 260 (2016), 5174–5201.
[2] C. O. Alves, M. A. S. Souto; Existence of least energy nodal solution for a Schrödinger-

Poisson system in bounded domains, Z. Angew. Math. Phys., 65 (2014), 1153–1166.
[3] T. Bartsch, Z. Liu, T. Weth; Sign changing solutions of superlinear Schrödinger equations,

Comm. Partial Differential Equations, 29 (2004), 25–42.

[4] T. Bartsch, Z. Q. Wang; Existence and multiplicity results for some superlinear elliptic

problems on RN , Comm. Partial Differential Equations, 20 (1995), 1725–1741.
[5] T. Bartsch, T. Weth; Three nodal solutions of singularly perturbed elliptic equations on

domains without topology, Ann Inst H Poincaré Anal Non Linéaire, 22 (2005), 259–281.
[6] G. F. Carrier; On the nonlinear vibration problem of the elastic string, Quart. Appl. Math.,

3 (1945), 157–165.

[7] S. Chen, B. Zhang, X. Tang; Existence and non-existence results for Kirchhoff-type problems
with convolution nonlinearity, Adv. Nonlinear Anal., 9 (2020), 148–167.

[8] Y. B. Deng, W. Shuai; Sign-changing multi-bump solutions for Kirchhoff-type equations in

R3, Discrete Contin. Dyn. Syst. Ser. A, 38 (2018), 3139–3168.
[9] A. Fiscella, P. Pucci, B.L. Zhang; p-fractional Hardy-Schrödinger-Kirchhoff systems with

critical nonlinearities, Adv. Nonlinear Anal., 8 (2019), 1111–1131.

[10] X. M. He, W. M. Zou; Ground states for nonlinear Kirchhoff equations with critical growth,
Ann. Mat. Pura Appl., 193 (2014), 473–500.

[11] G. Kirchhoff; Mechanik, Teubner, Leipzig, 1883.

[12] F. Y. Li, C. Gao, X. Zhu; Existence and concentration of sign-changing solutions to Kirchhoff-
type system with Hartree-type nonlinearity, J. Math. Anal. Appl., 448 (2017), 60–80.



EJDE-2021/19 NODAL SOLUTIONS OF FOURTH-ORDER KIRCHHOFF EQUATIONS 19

[13] S. Liang, P. Pucci, B. Zhang; Multiple solutions for critical Choquard-Kirchhoff type equa-

tions, Adv. Nonlinear Anal., 10 (2021), 400–419.

[14] S. Liang, V. D. Rădulescu; Least-energy nodal solutions of critical Kirchhoff problems with
logarithmic nonlinearity, Anal. Math. Phys., 10:45 (2020), 1–31.

[15] S. Liang, S. Shi; Soliton solutions to Kirchhoff type problems involving the critical growth in

RN , Nonlinear Anal., 81 (2013), 31–41.
[16] S. Liang, J. Zhang; Existence of solutions for Kirchhoff type problems with critical nonlin-

earity in R3, Nonlinear Anal. Real World Applications, 17 (2014), 126–136.

[17] S. Liang, J. Zhang; Existence and multiplicity of solutions for fourth-order elliptic equations
of Kirchhoff type with critical growth in RN , J. Math. Phys., 57 (2016), 111505.

[18] S. Liang, J. Zhang; Multiplicity of solutions for the noncooperative Schrödinger-Kirchhoff

system involving the fractional p−laplacian in RN , Z. Angew. Math. Phys., 68:63 (2017),
1–18.

[19] J. L. Lions; On some questions in boundary value problems of mathematical physics Sci-
enceDirect, North-Holland Mathematics Studies, 30 (1978), 284–346.

[20] S. Lu; Signed and sign-changing solutions for a Kirchhoff-type equation in bounded domains,

J. Math. Anal. Appl., 432 (2015), 965–982.
[21] T. F. Ma, J. E. Munoz Rivera; Positive solutions for a nonlinear nonlocal elliptic transmission

problem, Appl. Math. Lett., 16 (2003), 243–248.

[22] C. Miranda; Un’osservazione su un teorema di Brouwer, Boll Un Mat Ital, 3 (1940), 5–7.
[23] G. Molica Bisci, V.D. Rădulescu, R. Servadei; Variational Methods for Nonlocal Fractional

Problems, Encyclopedia of Mathematics and its Applications, vol. 162, Cambridge University

Press, Cambridge, 2016.
[24] D. Oplinger; Frequency response of a nonlinear stretched string, J. Acoust. Soc. Am., 32

(1960), 1529–1538.

[25] P. Pucci, M. Xiang, B. Zhang; Existence and multiplicity of entire solutions for fractional
p–Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27–55.

[26] W. Shuai; Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains,
J. Differ. Equations, 259 (2015), 1256–1274.

[27] J. Sun, L. Li, M. Cencelj, B. Gabrovšek; Infinitely many sign-changing solutions for Kirchhoff

type problems in R3, Nonlinear Anal., 186 (2018), 33–54.
[28] X. H. Tang and B. Cheng; Ground state sign-changing solutions for Kirchhoff type problems

in bounded domains, J. Differ. Equations, 261 (2016), 2384–2402.

[29] D.B. Wang; Least energy sign-changing solutions of Kirchhoff-type equation with critical
growth, J. Math. Phys., 61 (2020), 011501.

[30] F. Wang, M. Avci, Y. An; Existence of solutions for fourth order elliptic equations of Kirch-

hoff type, J. Math. Anal. Appl., 409 (2014), 140–146.
[31] T. Weth; Energy bounds for entire nodal solutions of autonomous superlinear equations,

Calc. Var. Partial Differential Equations, 27 (2006), 421-437.

[32] M. Willem; Minimax Theorems, Birkhäuser, Boston, 1996.
[33] M. Q. Xiang, B.L. Zhang, V. D. Rădulescu; Multiplicity of solutions for a class of quasilinear

Kirchhoff system involving the fractional p-Laplacian, Nonlinearity, 29 (2016), 3186–3205.

[34] M. Q. Xiang, M. Q. Zhang, V. D. Rădulescu; Superlinear Schrödinger-Kirchhoff type problems
involving the fractional p-Laplacian and critical exponent, Adv. Nonlinear Anal., 9 (2020),

690–709.
[35] M. Q. Zhang, V. D. Rădulescu, L. Wang; Existence results for Kirchhoff-type superlinear

problems involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A., 149 (2019),
1061–1081.

[36] W. Zhang, X. Tang, B. Cheng, J. Zhang; Sign-changing solutions for fourth order elliptic

equations with Kirchhoff-type, Comm. Pure Appl. Anal., 15 (6) (2016), 2161–2177.

[37] Z. T. Zhang, K. Perera; Sign changing solutions of Kirchhoff type problems via invariant sets
of descentow, J. Math. Anal. Appl., 317 (2006), 456–463.

Hongling Pu

College of Mathematics, Changchun Normal University, Changchun 130032, China
Email address: pauline phl@163.com



20 H. PU, S. LI, S. LIANG, D. D. REPOVŠ EJDE-2021/19

Shiqi Li

College of Mathematics, Changchun Normal University, Changchun 130032, China

Email address: lishiqi59@126.com

Sihua Liang

College of Mathematics, Changchun Normal University, Changchun 130032, China
Email address: liangsihua@163.com

Dušan D. Repovš

Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana
& Institute of Mathematics, Physics and Mechanics, Ljubljana, 1000, Slovenia

Email address: dusan.repovs@guest.arnes.si


	1. Introduction
	2. Some technical lemmas
	3. Proofs of main results
	Acknowledgments

	References