Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 44, pp. 1–15.

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

MULTIPLE POSITIVE SOLUTIONS FOR BIHARMONIC

EQUATION OF KIRCHHOFF TYPE INVOLVING

CONCAVE-CONVEX NONLINEARITIES

FENGJUAN MENG, FUBAO ZHANG, YUANYUAN ZHANG

Abstract. In this article, we study the multiplicity of positive solutions for

the biharmonic equation of Kirchhoff type involving concave-convex nonlin-

earities,

∆2u−
(
a+ b

∫
RN
|∇u|2dx

)
∆u+ V (x)u = λf1(x)|u|q−2u+ f2(x)|u|p−2u.

Using the Nehari manifold, Ekeland variational principle, and the theory of

Lagrange multipliers, we prove that there are at least two positive solutions,

one of which is a positive ground state solution.

1. Introduction

In this article, we are concerned with the multiplicity of positive solutions for
the biharmonic equations of Kirchhoff type

∆2u−
(
a+ b

∫
RN
|∇u|2dx

)
∆u+ V (x)u = f(x, u) in RN , (1.1)

with u ∈ H2(RN ).
This problem is often referred to be nonlocal because of the presence of the term∫

RN |∇u|
2dx∆u, which implies the problem is no longer a pointwise identity. This

phenomenon provokes some mathematical difficulties, which make the study of such
a class of problem particularly interesting. So there are many papers presented to
study the nonlocal problems. We refer the reader to [2, 10, 13, 15, 16, 18, 20, 22, 30].

The motivation of this paper is from the studies on the dynamical system. Prob-
lem (1.1) is related to the stationary analog of the dissipative evolutionary equation

utt + h(ut) + ∆2u−
(
a+ b

∫
RN
|∇u|2dx

)
∆u+ f(x, u) = 0. (1.2)

This equation arises as an evolutionary mathematical model in various systems for
relevant physical applications, see [20] and the references therein.

The understanding of the asymptotic behavior of dynamical systems generated
by dissipative evolutionary equation is an important problem of modern mathemat-
ical physics. The main method to treat this problem for a dissipative system is to
consider the existence of a global attractor and analyze the structure of the global

2010 Mathematics Subject Classification. 35J35, 35J40, 35J91.
Key words and phrases. Biharmonic equation; ground state solution; Nehari manifold;

concave-convex nonlinearity.
c©2020 Texas State University.

Submitted March 11, 2019. Published May 19, 2020.

1



2 F. MENG, F. ZHANG, Y. ZHANG EJDE-2020/44

attractor, which is an invariant compact set and attracts all bounded subsets in
some space.

Recently, the authors in [19, 21, 33, 35] proved the existence of the multiple
equilibrium points in the global attractors for the symmetric dynamical systems
by estimating the lower bound of Z2 index of two disjoint subsets of the global
attractor for which one subset is located in the area where the Lyapunov function
F is positive and the other subset is located in the area where the Lyapunov function
F is negative. By the way, a fixed point, or a stationary point, or an equilibrium
point for a semigroup of an evolutionary equation corresponds to the solution of
the related stationary equation [24].

To have a better understanding of the asymptotic behavior of the dissipative
system (1.2) for future studies, our aim in the present paper is to find the multi-
plicity of positive solutions for the corresponding stationary equation (1.1) based
on the variational methods.

For a = 1, b = 0 in (1.1), we obtain the fourth-order elliptic equation

∆2u−∆u+ V (x)u = f(x, u) in RN , (1.3)

with u ∈ H2(RN ). The existence and multiplicity of positive, negative, sign-
changing and high energy solutions of (1.3) have been the subject of extensive
mathematical studies in recent years, see [1, 4, 31, 32] and the references therein.

Without the term ∆2u, system (1.1) becomes the equation

−
(
a+ b

∫
RN
|∇u|2dx

)
∆u+ V (x)u = f(x, u) in RN . (1.4)

The solvability of the Kirchhoff-type Equation (1.4) has been widely studied by
various authors. For example, Ma and Muñoz Rivera [18] investigated the existence
of positive solutions of such problems by using variational methods. Perera and
Zhang [22] obtained a nontrivial solution of (1.4) via Yang index and critical group.
He and Zou [13] studied (1.4) the existence of infinitely many solutions by using
the local minimum methods and the fountain theorems. Chen, Kuo and Wu [10]
considered problem (1.4) with concave and convex nonlinearities by using Nehari
manifold and fibering map methods, and the existence of multiple positive solutions
were obtained. Li and Ye [16] considered (1.4) with super linear nonlinearities by
using a monotonicity trick and a new version of global compactness lemma, and
obtained the existence of positive ground state solutions. For other important
results, see [2, 15, 30] and the references therein.

By using the mountain pass techniques and the truncation method, Wang, Avci
and An [26] considered the existence of nontrivial solutions for problem (1.1) in
bounded domain. By using variational methods and the truncation method, Wang,
An and An [25] studied the positive solutions of (1.1) with V (x) = 1. Very recently,
Khoutir and Chen [14] obtained the existence of ground state solutions and a least
energy sign-changing solution of (1.1) by using the variational methods and the
Nehari method with f(x, u) = |u|p−2u, Wang, Ru and An [27] investigated the
existence of nontrivial solutions of (1.1) via Galerkin method with V (x) = 1.

Concerning the concave-convex nonlinearity, there is a considerable literature
that takes into account different type of problem, see for instance, the pioneering
paper by Ambrosetti, Brezis and Cerami [3] for the elliptic problems in bounded
domain, Wu [29] for the elliptic problem in unbounded domain, Liu and Wang [17]
and Cao and Xu [9] for the Schrödinger equations, Zhang, Xu and Zhang [34] for



EJDE-2020/44 BIHARMONIC EQUATION OF KIRCHHOFF TYPE 3

the Choquard equation. However, very little work has been done for problem (1.1)
in RN

Inspired by the above-mentioned papers, we are focus on system (1.1) in the case
of f(x, u) involving a combination of convex and concave terms,

∆2u−
(
a+ b

∫
RN
|∇u|2dx

)
∆u+ V (x)u = λf1(x)|u|q−2u+ f2(x)|u|p−2u, (1.5)

with u ∈ H2(RN ), where N ≤ 7, a, b are positive constants, λ > 0 is a parameter
1 < q < 2, 4 < p < 2∗ (2∗ =∞ if N ≤ 4 and 2∗ = 2N

N−4 if N = 5, 6, 7).

We assume that V (x) and f(x) satisfy the following hypotheses:

(H1) V ∈ C(RN ,R), infx∈RN V (x) ≥ a0 > 0, where a0 is a constant. Moreover,
for every M > 0, meas{x ∈ RN : V (x) ≤ M} < ∞, where meas denotes
the Lebesgue measure in RN ;

(H2) f1 ∈ Lq
∗
(RN ) ∩ C(RN ) \ {0}, where q∗ = p/(p− q);

(H3) f2 ∈ C(RN ) ∩ L∞(RN ) and f2(x) > 0 for almost every x ∈ RN .

Our main result reads as follows.

Theorem 1.1. Suppose the conditions (H1)–(H3) hold. Let

λ0 =
(p− 2)S

q/2
p

(p− q)|f1|q∗

[ (2− q)Sp/2p

(p− q)|f2|∞

] 2−q
p−2

,

with Sp defined in (2.1) below. Then we have:

(1) for each 0 < λ < λ0, problem (1.5) has at least two positive solutions, one
of which has negative energy;

(2) if 0 < λ < q
p−2λ0, the solution corresponding to the negative energy is

a positive ground state solution and the other one corresponds to positive
energy.

When we restrict the space dimension to N ≤ 7, because 4 < p < 2∗ = 2N
N−4 ,

then 2N
N−4 > 4, hence N < 8. In the present study, we mainly focus on the case

p > 4, whereas the different solution structure will appear if p < 4 as in [10], which
we will investigate in the future study.

Since problem (1.5) is defined in RN which is unbounded, the lack of compactness
of the Sobolev embedding becomes more delicate by using variational techniques.
To overcome the lack of compactness, the condition (V) , which was first introduced
by Bartsch and Wang in [5], is always assumed to preserve the compactness of
embedding of the working space. From [11], we know that under the assumption
(H1), the continuous embedding E ↪→ Ls(RN ) is compact for 2 ≤ s < 2∗, where E
is denoted in Section 2.1.

Since the functional of (1.5) is concave-convex, it may have several critical points
in the direction of nontrivial u. Hence the standard method of Nehari manifold is
invalid. Motivated by [8, 10, 28, 29], we will make a partition of the Nehari manifold
and figure out two non-degenerate submanifolds, and then consider minimization
problems in the two submanifolds respectively to obtain one positive energy solution
and one negative energy solution.

The remainder of this paper is organized as follows. After presenting some
preliminary results in Section 2, we give the proof of our main result in Section 3.



4 F. MENG, F. ZHANG, Y. ZHANG EJDE-2020/44

2. Preliminaries

In the sequel, we shall use the following notation:
• H = H2(RN ) := {u ∈ L2(RN ) : |∇u|,∆u ∈ L2(RN} is the usual Sobolev space
endowed with the scalar product and norm

〈u, v〉H =

∫
RN

(∆u∆v +∇u∇v + uv)dx, ‖u‖H = 〈u, u〉1/2H .

In Ls(RN ), we define the norm

|u|s = (

∫
RN
|u|sdx)1/s for 0 < s ≤ ∞.

• E = {u ∈ H2(RN ) :
∫
RN V (x)u2dx <∞}, with inner in E and norm

〈u, v〉 =

∫
RN

(∆u∆v + a∇u∇v + V (x)uv)dx, ‖u‖ = 〈u, u〉1/2.

• Denote by Sp the best Sobolev constant for the embedding E ↪→ Lp(RN ) which
is given by

Sp = inf
E\{0}

‖u‖2

(
∫
RN |u|pdx)2/p

> 0 . (2.1)

Then we have

|u|p ≤ S−1/2p ‖u‖, ∀u ∈ E. (2.2)

• C,Ci denote different positive constants whose exact valued is inessential.
The energy functional corresponding to (1.5) is

Iλ(u) =
1

2

∫
RN

(|∆u|2 + a|∇u|2 + V (x)u2)dx+
b

4
|∇u|42

− λ

q

∫
RN

f1(x)|u|qdx− 1

p

∫
RN

f2(x)|u|pdx, u ∈ E.

By the assumptions (H1)–(H3), one has that Iλ(u) ∈ C1(E,R) and for any u, v ∈ E,

〈I ′λ(u), v〉 =

∫
RN

(∆u∆v + a∇u∇v + V (x)uv)dx+ b

∫
RN
|∇u|2dx

∫
RN
∇u∇v dx

− λ
∫
RN

f1(x)|u|q−2uv dx−
∫
RN

f2(x)|u|p−2uv dx.

It is well-known that u is a solution of system (1.5) if and only if u ∈ E is a critical
point of Iλ.

It is easy to verify that the energy functional Iλ is not bounded from below
on E. However, it is convenient to consider the functional restricted to a natural
constraint, the Nehari manifold

Nλ = {u ∈ E \ {0} : 〈I ′λ(u), u〉 = 0}.

Thus u ∈ Nλ, if and only if

‖u‖2 + b|∇u|42 = λ

∫
RN

f1(x)|u|qdx+

∫
RN

f2(x)|u|pdx.

Obviously, Nλ contains every nontrivial solution of (1.5). The following result is
readily established.

Lemma 2.1. The energy functional Iλ is coercive and bounded from below on Nλ.



EJDE-2020/44 BIHARMONIC EQUATION OF KIRCHHOFF TYPE 5

Proof. By Hölder’s inequality and (2.1), we have∫
RN

f1(x)|u|qdx ≤
(∫

RN
|f1(x)|q

∗
dx
)1/q∗(∫

RN
|u|pdx

)q/p
= |f1|q∗ |u|qp ≤ |f1|q∗S−q/2p ‖u‖q.

(2.3)

For u ∈ Nλ, noting (2.3), we can conclude that

Iλ(u) = Iλ(u)− 1

4
〈I ′λ(u), u〉

=
1

4
‖u‖2 − λ(

1

q
− 1

4
)

∫
RN

f1(x)|u|qdx+ (
1

4
− 1

p
)

∫
RN

f2(x)|u|pdx

≥ 1

4
‖u‖2 − λ(

1

q
− 1

4
)|f1|q∗S−q/2p ‖u‖q.

Combining 1 < q < 2 < 4 < p < 2∗, we know that Iλ is coercive and bounded from
below on Nλ. �

The Nehari manifold Nλ is closely linked to the behavior of fibering maps which
is given by Ku(t) = Iλ(tu) for t > 0. The fibering map has been introduced by
Drábek and Pohozaev in [12] and are also discussed in Brown and Zhang [8] and
Brown and Wu [7]. If u ∈ E, we have

Ku(t) =
1

2
t2‖u‖2 +

bt4

4
|∇u|42 −

λtq

q

∫
RN

f1(x)|u|qdx− tp

p

∫
RN

f2(x)|u|pdx;

K ′u(t) = t‖u‖2 + bt3|∇u|42 − λtq−1
∫
RN

f1(x)|u|qdx− tp−1
∫
RN

f2(x)|u|pdx;

K ′′u(t) = ‖u‖2 + 3bt2|∇u|42 − λ(q − 1)tq−2
∫
RN

f1(x)|u|qdx

− (p− 1)tp−2
∫
RN

f2(x)|u|pdx.

It is easy to see that for u ∈ E \ {0} and t > 0,K ′u(t) = 0 if and only if tu ∈ Nλ,
i.e., positive critical points of Ku correspond to points on the Nehari manifold. In
particular, K ′u(1) = 0 if and only if u ∈ Nλ. Since Ku(t) ∈ C2(R+,R), we split
Nλ into three parts corresponding to local minima, points of inflection and local
maxima.

N+
λ = {u ∈ Nλ : K ′′u(1) > 0},
N 0
λ = {u ∈ Nλ : K ′′u(1) = 0},
N−λ = {u ∈ Nλ : K ′′u(1) < 0}.

For each u ∈ Nλ, we have

K ′′u(1) = ‖u‖2 + 3b|∇u|42 − λ(q − 1)

∫
RN

f1(x)|u|qdx− (p− 1)

∫
RN

f2(x)|u|pdx

(2.4)

= K ′′u(1)− (q − 1)〈I ′λ(u), u〉

= (2− q)‖u‖2 + b(4− q)|∇u|42 − (p− q)
∫
RN

f2(x)|u|pdx (2.5)

= K ′′u(1)− (p− 1)〈I ′λ(u), u〉



6 F. MENG, F. ZHANG, Y. ZHANG EJDE-2020/44

= (2− p)‖u‖2 + b(4− p)|∇u|42 + λ(p− q)
∫
RN

f1(x)|u|qdx. (2.6)

We now derive some basic properties of N 0
λ , N+

λ , N−λ .

Lemma 2.2. If λ ∈ (0, λ0), then N 0
λ = ∅

Proof. Suppose the contrary, then there exists a λ∗ ∈ (0, λ0) such that N 0
λ∗ 6= ∅.

Hence there at least exists a u0 ∈ N 0
λ∗ satisfying K ′′u0

(1) = 0. From (2.5) and the
Hölder and Sobolev inequalities, we have

(2− q)‖u0‖2 + b(4− q)|∇u0|42 = (p− q)
∫
RN

f2(x)|u0|pdx

≤ (p− q)|f2|∞S−p/2p ‖u0‖p,
(2.7)

which implies

(2− q)‖u0‖2 ≤ (p− q)|f2|∞S−p/2p ‖u0‖p,
hence, we derive that

‖u0‖ ≥
( (2− q)Sp/2p

(p− q)|f2|∞

) 1
p−2

. (2.8)

Similarly, using (2.6) and Hölder and Sobolev inequalities, we have

(p− 2)‖u0‖2 + b(p− 4)|∇u0|42 = λ∗(p− q)
∫
RN

f1(x)|u|qdx

≤ λ∗(p− q)|f1|q∗S−q/2p ‖u0‖q,
(2.9)

thus

(p− 2)‖u0‖2 ≤ λ∗(p− q)|f1|q∗S−q/2p ‖u0‖q,
which implies

‖u0‖ ≤
(λ∗(p− q)|f1|q∗

(p− 2)S
q/2
p

) 1
2−q

. (2.10)

Combining with (2.8) and (2.10), we deduce that

λ∗ ≥ (p− 2)S
q/2
p

(p− q)|f1|q∗

[ (2− q)Sp/2p

(p− q)|f2|∞

] 2−q
p−2

= λ0

which contradicts the assumptions. The proof is complete. �

to have a better understanding of the Nehari manifold Nλ and the fibering maps
Ku(t), we consider the function hb(t) : R+ → R defined by

hb(t) = t2−q‖u‖2 + bt4−q|∇u|42 − tp−q
∫
RN

f2(x)|u|pdx .

Then

K ′u(t) = tq−1
(
hb(t)− λ

∫
RN

f1(x)|u|qdx
)
.

Clearly, tu ∈ Nλ if and only if hb(t) = λ
∫
RN f1(x)|u|qdx, tu ∈ N+

λ (or N−λ ) if and
only if h′b(t) > 0 (or < 0).



EJDE-2020/44 BIHARMONIC EQUATION OF KIRCHHOFF TYPE 7

For u ∈ E\{0} with
∫
RN f2(x)|u|pdx > 0, it is obviously that hb(0) = 0, hb(t) > 0

for t is small enough and hb(t)→ −∞ as t→∞. Note that 1 < q < 2, 4 < p < 2∗,
and from

h′b(t) = tp−q−1
(
(2− q)t2−q‖u‖2 + b(4− q)t4−p|∇u|42 − (p− q)

∫
RN

f2(x)|u|pdx
)

= 0,

we can infer that there is a unique tb,max > 0 such that hb(t) achieves its maximum
at tb,max, increasing for t ∈ [0, t−b,max) and decreasing for t ∈ (t+b,max,∞) with

limt→∞ hb(t) = −∞ where t−b,max ≤ tb,max ≤ t+b,max.

Lemma 2.3. Suppose that 0 < λ < λ0 and u ∈ E \ {0}. Then

(i) if
∫
RN f1(x)|u|qdx ≤ 0, then there at least exists a t− > tb,max such that

t−u ∈ N−λ , and

Iλ(t−u) = sup
t≥0

Iλ(tu);

(ii) if
∫
RN f1(x)|u|qdx > 0, then there at least exists a t− > tb,max and a t+ <

tb,max, i.e. 0 < t+ < tb,max < t− such that t+u ∈ N+
λ and t−u ∈ N−λ ,

satisfying

Iλ(t+u) = inf
tb,max≥t≥0

Iλ(tu), Iλ(t−u) = sup
t≥tb,max

Iλ(tu).

Proof. Denote h0(t)
.
= hb(t)|b=0. Note that b > 0, we have

hb(t) > h0(t) = t2−q‖u‖2 − tp−q
∫
RN

f2(x)|u|pdx.

It is easy to see that h0(t) has a unique critical point

t0,max =
( (2− q)‖u‖2

(p− q)
∫
RN f2(x)|u|pdx

) 1
p−2 ,

and

h0(t0,max) =
( (2− q)‖u‖2

(p− q)
∫
RN f2(x)|u|pdx

) 2−q
p−2 ‖u‖2

−
( (2− q)‖u‖2

(p− q)
∫
RN f2(x)|u|pdx

) p−q
p−2

∫
RN

f2(x)|u|pdx

= ‖u‖q
( ‖u‖p∫

RN f2(x)|u|pdx

) 2−q
p−2 (2− q

p− q
) 2−q
p−2

p− 2

p− q

≥ ‖u‖q
( ‖u‖p

|f2|∞S−p/2p ‖u‖p

) 2−q
p−2 (2− q

p− q
) 2−q
p−2

p− 2

p− q

= ‖u‖q
[ (2− q)Sp/2p

(p− q)|f2|∞

] 2−q
p−2 p− 2

p− q
> 0.

Hence

hb(tb,max) ≥ hb(t0,max) > h0(t0,max) > 0.

From 0 < λ < λ0,

λ

∫
RN

f1(x)|u|qdx ≤ λ
(∫

RN
|f1(x)|q

∗
dx
)1/q∗(∫

RN
|u|pdx

)q/p
≤ λ|f1|q∗ |u|qp ≤ λ|f1|q∗S−q/2p ‖u‖q



8 F. MENG, F. ZHANG, Y. ZHANG EJDE-2020/44

≤ ‖u‖q
[ (2− q)Sp/2p

(p− q)|f2|∞

] 2−q
p−2 p− 2

p− q
≤ h0(t0,max) < hb(tb,max).

The rest of the proof is similar to the one in [29, Lemma 2.6], we omit it here. �

From Lemma 2.3, we can see that for 0 < λ < λ0, N+
λ 6= ∅ and N−λ 6= ∅.

Combining with Lemma 2.1, we define

α+
λ = inf

u∈N+
λ

Iλ(u), α−λ = inf
u∈N−λ

Iλ(u).

In the following part, we will drive some basic properties of α+
λ , α

−
λ .

Lemma 2.4. For given λ0 in Theorem 1.1, we have

(i) α+
λ < 0 for 0 < λ < λ0;

(ii) α−λ > 0, for 0 < λ < q
p−2λ0.

Proof. (i) For each u ∈ N+
λ , K ′′u(1) > 0. From (2.6), we have

λ(p− q)
∫
RN

f1(x)|u|qdx > (p− 2)‖u‖2 + b(p− 4)|∇u|42.

Then for each u ∈ N+
λ ,

Iλ(u) = Iλ(u)− 1

p
〈I ′λ(u), u〉

=
p− 2

2p
‖u‖2 +

p− 4

4p
b|∇u|42 − λ

p− q
pq

∫
RN

f1(x)|u|qdx

<
p− 2

2p
‖u‖2 +

p− 4

4p
b|∇u|42 −

1

pq
((p− 2)‖u‖2 + (p− 4)b|∇u|42)

=
(p− 2)(q − 2)

2pq
‖u‖2 +

(p− 4)(q − 4)

4pq
b|∇u|42 < 0,

it follows that α+
λ = infu∈N+

λ
Iλ(u) < 0.

(ii) Let u ∈ N−λ , by (2.5) and similar to the proof of (2.8), we have

‖u‖ ≥
( (2− q)Sp/2p

(p− q)|f2|∞

) 1
p−2

. (2.11)

Then

Iλ(u) = Iλ(u)− 1

4
< I ′λ(u), u >

=
1

4
‖u‖2 − λ(

1

q
− 1

4
)

∫
RN

f1(x)|u|qdx+ (
1

4
− 1

p
)

∫
RN

f2(x)|u|pdx

≥ 1

4
‖u‖2 − λ(

1

q
− 1

4
)|f1|q∗S−1/2p ‖u‖q

= ‖u‖q
(1

4
‖u‖2−q − λ(

1

q
− 1

4
)|f1|q∗S−1/2p

)
≥
( (2− q)Sp/2p

(p− q)|f2|∞

) q
p−2
(1

4

( (2− q)Sp/2p

(p− q)|f2|∞

) 2−q
p−2 − λp− q

4q
|f1|q∗S−1/2p

)
.

Thus, if 0 < λ < q
p−2λ0, we have Iλ(u) > c0 for some c0 > 0, which implies that

α−λ = infu∈N−λ
Iλ(u) > 0. This completes the proof. �



EJDE-2020/44 BIHARMONIC EQUATION OF KIRCHHOFF TYPE 9

Lemma 2.5. If 0 < λ < λ0, then N−λ is closed in E.

Proof. Let un ∈ N−λ such that un → u in E. We need to prove u ∈ N−λ . Note that
〈I ′λ(un), un〉=0 and

〈I ′λ(un), un〉 − 〈I ′λ(u), u〉
= 〈I ′λ(un)− I ′λ(u), u〉 − 〈I ′λ(un), un − u〉 → 0, as n→∞,

(2.12)

we have 〈I ′λ(u), u〉 = 0, which implies u ∈ Nλ.
For any u ∈ N−λ , by (2.6), we obtain that N−λ is bound away from 0.
By (2.5), it follows that K ′′un(1) → K ′′u(1), noting that K ′′un(1) < 0, we have

K ′′u(1) ≤ 0. By Lemma 2.3, for λ < λ0, K ′′u(1) < 0. Therefore u ∈ N−λ . �

The following lemma aims to find the critical point of Iλ on the whole space
from the minimizer for Iλ on Nehari manifold.

Lemma 2.6. For λ ∈ (0, λ0), if u0 is a local minimizer for Iλ on Nλ, then I ′λ(u0) =
0 in H−2(RN ), where H−2(RN ) is the dual space of H2(RN ).

Proof. From Lemma 2.3, we know that u0 /∈ N 0
λ . The rest of the proof is essentially

the same as that in [8], see also in [6, 28] we omit it here. �

By the above lemma, we know that the problem of finding solutions of (1.5) can
be translated into that finding minimizers of Iλ on Nλ.

3. Proof of the main result

We first establish a lemma for locally compactness.

Lemma 3.1. Under assumptions (H1)–(H3), Iλ satisfies the (PS)c condition with
c ∈ R on N+

λ (or N−λ ), i.e. if un ∈ N+
λ (or N−λ ) such that Iλ(un) → c and

I ′λ(un)→ 0, then there exists a convergent subsequence of un.

Proof. Assume that un ∈ N+
λ (or N−λ ) such that Iλ(un)→ c and I ′λ(un)→ 0. By

Lemma 2.1, we infer that un is bounded in E. Up to a subsequence, we may assume
that

un ⇀ u in E, un → u in Ls(Rn), s ∈ [2, 2∗).

It follows that

b
(∫

RN
|∇u|2dx−

∫
RN
|∇un|2dx

)∫
RN
∇u∇(un − u)dx→ 0 as n→∞. (3.1)

By using twice the Hölder inequality, the corresponding exponents are ( p
p−q ,

p
q ) and

(q, q
q−1 ) respectively, we obtain∣∣λ ∫

RN
f1(x)(|un|q−2un − |u|q−2u)(un − u)dx

∣∣
≤ λ

(∫
RN
|f1(x)|q

∗
dx
)1/q∗(∫

RN
||un|q−2un − |u|q−2u|p/q|un − u|p/qdx

)q/p
≤ λC|f1|q∗

(
|un|q−1p + |u|q−1p

)
|un − u|p → 0, as n→∞,

where C is a positive constant. Similarly,∣∣ ∫
RN

f2(x)(|un|p−2un − |u|p−2u)(un − u)dx
∣∣→ 0, as n→∞.



10 F. MENG, F. ZHANG, Y. ZHANG EJDE-2020/44

Then

o(1) = 〈I ′λ(un)− I ′λ(u), un − u〉

= ‖un − u‖2 + b

∫
RN
|∇un|2dx

∫
RN
|∇(un − u)|2dx

− b
(∫

RN
|∇u|2dx−

∫
RN
|∇un|2dx

)∫
RN
∇u∇(un − u)dx

− λ
∫
RN

f1(x)(|un|q−2un − |u|q−2u)(un − u)dx

−
∫
RN

f2(x)(|un|p−2un − |u|p−2u)(un − u)dx

= ‖un − u‖2 + b

∫
RN
|∇un|2dx

∫
RN
|∇(un − u)|2dx+ o(1)

Thus un → u in E. �

Now we use an idea in [23] to extract a (PS)α+
λ

sequence from the minimizing

sequence of the energy functional Iλ on Nehari manifold Nλ.

Lemma 3.2. Suppose that u ∈ N+
λ , there exist ε = ε(u) > 0 and a differentiable

function ψ+ : Bε(0)→ R+ := (0,+∞) such that

(1) ψ+(0) = 1;
(2) ψ+(w)(u− w) ∈ N+

λ ,∀w ∈ Bε(0);

(3) 〈(ψ+)(0), w〉 = L(u,w)
K′′u (1)

, where

L(u,w) = 2(u,w) + 4b

∫
RN
|∇u|2∇u∇w dx− q

∫
RN

f1(x)|u|q−2uw dx

− p
∫
RN

f2(x)|u|p−2uw dx.

Moreover, for any C1, C2 > 0, there exists C > 0 such that if C1 ≤ ‖u‖ ≤ C2, then
|〈(ψ+)′(0), w〉| ≤ C‖w‖.

Proof. Define a C1 mapping J : R+ × E → R by J(t, w) = K ′u−w(t), that is

J(t, w) = t‖u− w‖2 + bt3(

∫
RN
|∇u− w|2dx)2 − λtq−1

∫
RN

f1(x)|u− w|qdx

− tp−1
∫
RN

f2(x)|u− w|pdx.

Note that u ∈ N+
λ , then J(1, 0) = 0 and Jt(1, 0) = K ′′u(1) > 0. Applying the

implicit function theorem at point (1, 0), we obtain that there exist ε = ε(u) > 0
and a differentiable function ψ+ : Bε(0)→ R+ := (0,+∞) such that

ψ+(0) = 1, J(ψ+(w), w) = 0, ∀w ∈ Bε(0).

Next, we prove that ψ+(u−w) ∈ N+
λ for all w ∈ Bε(0). Indeed, by u ∈ N+

λ and

the set N−λ ∪N 0
λ is closed, we can get dist(u,N−λ ∪N 0

λ ) > 0. Note that ψ+(w)(u−w)
is continuous with respect to w, choose ε = ε(u) > 0 small enough, such that

‖ψ+(w)(u− w)− u‖ < 1

2
dist(u,N−λ ∪N

0
λ ), ∀w ∈ Bε(0).



EJDE-2020/44 BIHARMONIC EQUATION OF KIRCHHOFF TYPE 11

Hence

‖ψ+(w)(u− w)−N−λ ∪N
0
λ‖ ≥ dist(u,N−λ ∪N

0
λ )− dist(ψ+(w)(u− w), u)

≥ 1

2
dist(u,N−λ ∪N

0
λ ) > 0,

thus ψ+(w)(u− w) ∈ N+
λ , for all w ∈ Bε(0).

By the differentiability of the implicit function theorem, we have

〈(ψ+)′(0), w〉 = −〈Jw(1, 0), w〉
Jt(1, 0)

.

Note that L(u,w) = −〈Jw(1, 0), w〉, and K ′′u(1) = Jt(1, 0), therefore, we have

〈(ψ+)(0), w〉 =
L(u,w)

K ′′u(1)
.

Finally, we verify that there exists δ > 0 such that K ′′u(1) ≥ δ > 0 with C1 ≤
‖u‖ ≤ C2, u ∈ N+

λ , where C1, C2 > 0. We will prove that by contradiction.

Otherwise, if there exists a sequence {un} ∈ N+
λ with C1 ≤ ‖un‖ ≤ C2, satisfying

K ′′un(1) ≤ δn for any δn sufficiently small and δn → 0 as n → ∞. From (2.5), we
have

(2− q)‖un‖2 + b(4− q)
(∫

RN
|∇un|2dx

)2
= (p− q)

∫
RN

f2(x)|un|pdx+ o(δn),

where o(δn) → 0 as n → ∞. Noting that 1 < q < 2, 4 < p < 2∗, C1 ≤ ‖un‖ ≤ C2

and (2.7), we have

(2− q)‖un‖2 ≤ (p− q)|f2|∞S−p/2p ‖un‖p + o(δn),

and hence

‖un‖ ≥
( (2− q)Sp/2p

(p− q)|f2|∞

) 1
p−2

+ o(δn). (3.2)

From (2.6), we have

(p− 2)‖un‖2 + b(p− 4)
(∫

RN
|∇un|2dx

)2
= λ(p− q)

∫
RN

f1(x)|un|qdx+ o(δn).

In view of (2.9), we have

(p− 2)‖un‖2 ≤ λ(p− q)|f1|q∗S−q/2p ‖un‖q + o(δn),

which implies

‖un‖ ≤
(λ0(p− q)|f1|q∗

(p− 2)S
q/2
p

) 1
2−q

+ o(δn), (3.3)

Combing this with (3.2) and (3.3), as n→∞, we deduce a contradiction.
Therefore, if there exists C > 0 such that if C1 ≤ ‖u‖ ≤ C2, then |〈(ψ+)′(0), w〉| ≤

C‖w‖. This ends the proof of Lemma 3.2. �

In the same way, to extract a (PS)α−λ
sequence from the minimizing sequence of

problem, we establish the following lemma.

Lemma 3.3. Suppose that u ∈ N−λ , there exist ε = ε(u) > 0 and a differentiable
function ψ− : Bε(0)→ R+ := (0,+∞) such that

(1) ψ−(0) = 1;
(2) ψ−(w)(u− w) ∈ N−λ for all w ∈ Bε(0);



12 F. MENG, F. ZHANG, Y. ZHANG EJDE-2020/44

(3) 〈(ψ−)(0), w〉 = L(u,w)
K′′u (1)

, where

L(u,w) = 2(u,w) + 4b

∫
RN
|∇u|2∇u∇w dx− q

∫
RN

f1(x)|u|q−2uw dx

− p
∫
RN

f2(x)|u|p−2uw dx.

Moreover, for any C1, C2 > 0, there exists C > 0 such that if C1 ≤ ‖u‖ ≤ C2, then
|〈(ψ−)′(0), w〉| ≤ C‖w‖.

We are now ready to construct the (PS)α+
λ

(or (PS)α−λ
) sequence from the

minimizing sequence of the energy functional Iλ on the Nehari manifold N+
λ (or

N−λ ).

Lemma 3.4. Suppose (H1)–(H3) hold, and 0 < λ < λ0, then there exists a sequence
un ∈ N+

λ such that Iλ(un)→ α+
λ and I ′λ(un)→ 0 as n→∞.

Proof. By Lemma 2.1 and the Ekeland Variational Principle on N+
λ ∪ N 0

λ , there

exists a minimizing sequence {un} ⊂ N+
λ ∪N 0

λ such that

inf
u∈N+

λ ∪N
0
λ

Iλ(u) ≤ Iλ(un) < inf
u∈N+

λ ∪N
0
λ

Iλ(u) +
1

n
,

Iλ(v) ≥ Iλ(un)− 1

n
‖v − un‖, ∀v ∈ N+

λ ∪N
0
λ .

Observe that N 0
λ = ∅, then we have infu∈N+

λ ∪N
0
λ
Iλ(u) = infu∈N+

λ
Iλ(u) = α+

λ .

Thus Iλ(un) → α+
λ , and we may assume that un ∈ N+

λ . By Lemma 2.4, we know

that α+
λ < 0.

To finish the proof, we only need to verify that I ′λ(un) → 0. Applying Lemma
3.2 with un to obtain the function ψ+

n : Bεn(0)→ R+ such that

ψ+
n (w)(un − w) ∈ N+

λ , ∀w ∈ Bεn(0).

By the continuity of ψ+
n (w) and ψ+

n (0) = 1, without loss of generality, we can
assume εn is sufficiently small such that 1/2 ≤ ψ+

n (w) ≤ 3/2 for ‖w‖ ≤ εn. From
ψ+
n (w)(un − w) ∈ N+

λ and (b), we have

Iλ(ψ+
n (w)(un − w))− Iλ(un) ≥ − 1

n
‖ψ+

n (w)(un − w)− un‖,

and by the mean value theorem, we have

〈I ′λ(un), ψ+
n (w)(un − w)− un〉+ o(‖ψ+

n (w)(un − w)− un‖)

≥ − 1

n
‖ψ+

n (w)(un − w)− un‖.

Consequently,

ψ+
n (w)〈I ′λ(un), w〉+ (1− ψ+

n (w))〈I ′λ(un), un〉

≤ 1

n
‖(ψ+

n (w)− 1)un − ψ+
n (w)w‖+ o

(
‖ψ+

n (w)(un − w)− un‖
)
.

By the choice of εn and 1
2 ≤ ψ+

n (w) ≤ 3
2 , we infer that there exists C3 > 0 such

that

|〈I ′λ(un), w〉| ≤ 1

n
‖〈(ψ+

n )′(0), w〉un‖+
C3

n
‖w‖+ o

(
|〈(ψ+

n )′(0), w〉|(‖un‖+ ‖w‖)
)
.



EJDE-2020/44 BIHARMONIC EQUATION OF KIRCHHOFF TYPE 13

For un ∈ N+
λ , we claim that infn∈N ‖un‖ ≥ C1 > 0, where C1 is a constant.

Otherwise, Iλ(un) would converge to zero, which contradict with Iλ(un)→ α+
λ < 0.

Moreover, by Lemma 2.1 we know that Iλ is coercive on N+
λ , ‖un‖ is bounded in

E. Thus, there exists C2 > 0 such that 0 < C1 ≤ ‖un‖ ≤ C2. From Lemma 3.2,
|〈(ψ+

n )′(0), w〉| ≤ C‖w‖. Hence

|〈I ′λ(un), w〉| ≤ C

n
‖w‖+

C

n
‖w‖+ o(‖w‖),

‖I ′λ(un)‖ = sup
w∈E\{0}

|〈I ′λ(un), w〉|
‖w‖

≤ C

n
+ o(1).

Then ‖I ′λ(un)‖ → 0 as n→∞. Thus, {un} ⊂ N+
λ is a (PS)α+

λ
sequence for Iλ on

E. Similarly, we can construct the (PS)α−λ
sequence. �

Lemma 3.5. Suppose (H1)–(H3) hold, and 0 < λ < λ0, then there exists a sequence
un ∈ N−λ such that Iλ(un)→ α−λ and I ′λ(un)→ 0 as n→∞.

Now, we are in a position to give the proof of our main result.

Proof of Theorem 1.1. Firstly, we consider the minimization problem

α+
λ = inf

u∈N+
λ

Iλ(u).

By Lemma 3.4, there exists un ∈ N+
λ such that Iλ(un)→ α+

λ and I ′λ(un)→ 0. From
Lemma 3.1, there exists a strongly convergent subsequence of {un}, still denoted
by {un}, satisfying un → w1 in E. From the proof of Lemma 3.4 we know that
there exist C1, C2 > 0 such that 0 < C1 ≤ ‖un‖ ≤ C2, then 0 < C1 ≤ ‖w1‖ ≤ C2.
Thus w1 6= 0. Next we prove w1 ∈ N+

λ . Indeed, by (2.5), it follows that K ′′un(1)→
K ′′w1

(1). From K ′′un(1) > 0, we have K ′′w1
(1) ≥ 0. By Lemma 2.3, we know that

K ′′w1
(1) > 0. Hence

w1 ∈ N+
λ , I(w1) = lim

n→∞
Iλ(un) = inf

u∈N+
λ

Iλ(u).

Thus w1 is a nontrivial solution of (1.5) by Lemma 2.6. Since Iλ(w1) = Iλ(|w1|)
and |w1| ∈ N+

λ , we may assume that w1 is a positive solution of (1.5). Therefore,
we find a positive solution of (1.5).

Secondly, we consider the minimization problem α−λ = infu∈N−λ
Iλ(u). Similar

to the above proof, we can also find a positive solution w2 ∈ N−λ .
From the above proof, we know if 0 < λ < λ0, then problem(1.5) has at least

two positive solutions w1 ∈ N+
λ and w2 ∈ N−λ . Combining with Lemma 2.4 (i),

we have Iλ(w1) < 0. Moreover, by Lemma 2.4 (ii), if 0 < λ < q
p−2λ0, for any

u ∈ N−λ , Iλ(u) > 0, then Iλ(w2) > 0. Hence if 0 < λ < q
p−2λ0, then Iλ(w1) =

infu∈Nλ Iλ(u), w1 is a positive ground state solution of (1.5). �

Acknowledgement. This work was supported by the NSFC (11671077, 11701230,
11801228), by the Qinglan Project, Jiangsu Overseas visiting scholar Program for
University Prominent Young Middle aged Teachers and Presidents, and by the
Natural Science Foundation of Jiangsu Province (BK20170308).



14 F. MENG, F. ZHANG, Y. ZHANG EJDE-2020/44

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Fengjuan Meng
School of Mathematics and Physics, Jiangsu University of Technology, Changzhou

213001, China

Email address: fjmeng@jsut.edu.cn

Fubao Zhang (corresponding author)

School of Mathematics, Southeast University, Nanjing 210096, China
Email address: 101009933@seu.edu.cn

Yuanyuan Zhang
School of Business, Jiangsu University of Technology, Changzhou 213001, China

Email address: zyylhh1227@163.com


	1. Introduction
	2. Preliminaries
	3. Proof of the main result
	Acknowledgement 

	References