# Nontrivial Solutions of a Class of Kirchhoff Type Problems via Local Linking Method

Authors: Anmin Mao; Lin Zhang
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The existence of nontrivial solutions of Kirchhoff type problems is proved by using the local linking method, and the new results do not require classical compactness conditions.


---

## Body

## 1. Introduction

This paper is concerned with the existence of nontrivial solutions of the following nonlocal Kirchhoff type problem:(1)
-
a
+
b
∫
Ω
‍
∇
u
2
Δ
u
=
f
x
,
u
in
Ω
;
u
=
0
on
⁡
∂
Ω
,
where Ω is a smooth bounded domain in R
n (n
=
1,2), a
,
b
>
0, and f
(
x
,
u
): Ω
¯
×
R is a continuous real function. Similar nonlocal problems model several biological systems where u describes a process which depends on the average of itself, for example, that of the population density; see [1].Problem (1) is related to the stationary analogue of the equation
(2)
u
t
t
-
a
+
b
∫
Ω
‍
∇
u
2
Δ
u
=
g
x
,
u
proposed by Kirchhoff [2] as an extension of the classical D’Alembert wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. Some early classical studies of Kirchhoff equations were those of Bernstein [3] and Pohožaev [4]. However, (2) received great attention only after Lions [5] proposed an abstract framework for the problem. Some interesting results can be found in [6] and the references theorem. In recent years, Alves et al. [7] and Ma and Rivera [8] have obtained positive solutions of such problems by variational methods.Later Perera and Zhang [9, 10] obtained nontrivial solutions of Kirchhoff type problems with asymptotically 4-linear term via Yang index. In [9] using the invariant sets of descent flow, Perera and Zhang got a positive, a negative, and a sign-changing solution under the 4-sublinear case, asymptotically 4-linear case, and 4-superlinear case. In [10], the authors considered the 4-superlinear case:
(3)
(
A
R
)
:
∃
ν
>
4
:
ν
F
(
x
,
t
)
≤
t
f
(
x
,
t
)
,
t
large
,
where F
(
x
,
t
)
=
∫
0
t
‍
f
(
x
,
s
)
d
s, which implies
(4)
F
(
x
,
t
)
≥
C
t
ν
-
1
.
In [11], Mao and Zhang used minimax methods and invariant sets of decent flow to prove 4-superlinear Kirchhoff type problems without the PS condition and got multiple solutions. In [12], the paper studies the problem (1) by means of the Morse theory and local linking.Motivated by [12–15] and by the new information concerning local linking, we study Kirchhoff type problems 4-superlinear growth as |
u
|
→
∞. We make the following assumptions.(
S
1
)
There existδ
>
0, k
≥
1, c
>
0, and 2
<
α
<
4 such that
(5)
c
u
α
+
λ
k
2
u
2
<
F
(
x
,
u
)
<
λ
k
+
1
2
u
2
for
u
<
δ
.
(
S
2
)
(
F
(
x
,
u
)
/
u
4
)
→
+
∞ as |
u
|
→
+
∞ uniformly in x
∈
Ω.
(
S
3
)
There existr
>
4 and μ
>
r
-
4 such that
(6)
limsup
⁡
|
u
|
→
∞
F
(
x
,
u
)
u
r
<
∞
uniformly
for
a
.
e
.
x
∈
Ω
,
(7)
liminf
⁡
u
→
∞
f
x
,
u
u
-
4
F
x
,
u
u
μ
>
0
uniformly
for
a
.
e
.
x
∈
Ω
.We are ready to state our main results.Theorem 1.
Assume that (S
1)–(S
3) hold. Then problem (1) has at least one nontrivial solution.Remark 2.
Most of the results on the existence and multiplicity of solutions of (1) were obtained under the above superlinear condition (A
R) with or without the evenness assumption. Roughly speaking the role of (A
R) is to ensure the boundedness of all (P
S) (or (P
S
*)) sequences for the corresponding functional. However, there are many functions which are superlinear but it is not necessary to satisfy (A
R) even if 1
<
θ
≤
2. For example,
(8)
f
x
,
u
=
μ
u
μ
-
2
u
+
(
μ
-
1
)
u
μ
-
3
u
sin
2
u
+
u
μ
-
1
sin
2
u
,
u
∈
R
∖
0
,
where μ
>
2. Then it is easy to check that (A
R) does not hold even for any θ
>
μ
-
1
>
1. On the other hand, in order to verify (A
R), it is usually an annoying task to compute the primitive function of f and sometimes it is almost impossible. For example,
(9)
f
x
,
u
=
u
u
1
+
e
1
+
sin
u
α
+
cos
⁡
u
α
,
u
∈
R
,
where α
>
0.The aim of this paper is to deal with superlinear problems which do not satisfy (A
R). (
S
2
) is weaker than (
A
R
) used in [10]. Without (A
R), it becomes more complicated. We do not know in our situations whether the Palais-Smale sequence is bounded. We replace (P
S) (or (P
S
*)) sequences with weaker Cerami sequence or (
C
)
*-sequence (the definition given in the next section).Remark 3.
Compared with the method of invariant sets of descent flow and Morse theory, Yang index, our method is more simple and direct.


## 2. Preliminaries

LetX be a real Banach space with a direct sum decomposition:
(10)
X
=
X
1
⊕
X
2
.
Consider two sequences of subspace:
(11)
X
0
1
⊂
X
1
1
⊂
⋯
⊂
X
1
,
X
0
2
⊂
X
1
2
⊂
⋯
⊂
X
2
such that X
j
=
⋃
n
∈
N
‍
X
n
j, j
=
1,2, where dim
⁡
X
n
j
<
∞, j
=
1,2, n
∈
N.For every multi-indexα
=
(
α
1
,
α
2
)
∈
N
×
N, we denote by X
α the space
(12)
X
α
1
1
⊕
X
α
2
2
.
We can know
(13)
α
≤
β
⟺
α
1
≤
β
1
,
α
2
≤
β
2
.
A sequence {
α
n
}
⊂
N
2 is admissible if, for every α
∈
N
2, there is m
∈
N such that
(14)
n
≥
m
⟹
α
n
≥
α
.
For every φ
:
X
→
R we denote by φ
α the functional φ restricted to X
α.Definition 4 (see [14]).
Letφ
∈
C
1
(
X
,
R
). The function φ satisfies (
C
)
* condition if every sequence {
u
α
n
} such that α
n is admissible and
(15)
u
α
n
∈
X
α
n
,
sup
⁡
n
φ
u
α
n
<
∞
,
φ
α
n
′
(
u
α
n
)
⟶
0
contains a subsequence which converges to a critical point of φ.Definition 5 (see [15]).
LetX be Banach space with a direct sum decomposition X
=
X
1
⊕
X
2. The functional φ
∈
C
1
(
X
,
R
) has a local linking at 0 if, for some r
>
0,
(16)
φ
u
≥
0
u
∈
X
1
,
u
≤
r
,
φ
u
≤
0
u
∈
X
2
,
u
≤
r
.Lemma 6 (see [14]).
LetX be Banach space with a direct sum decomposition X
=
X
1
⊕
X
2. Suppose that φ
∈
C
1
(
X
,
R
) satisfies the following assumptions.(
I
1
)
φ has a local linking at 0 with respect to (
X
1
,
X
2
).
(
I
2
)
φ satisfies (
C
)
* condition.
(
I
3
)
φ maps bounded sets into bounded sets.
(
I
4
)
For everym
∈
N, φ
(
u
)
→
-
∞ as u
∈
X
m
1
⊕
X
2 and u
→
∞.
Thenφ has at least one nontrivial critical point.Next we letH
:
H
0
1
(
Ω
) be the Sobolev space equipped with the inner product and the norm
(17)
〈
u
,
v
〉
=
∫
Ω
‍
∇
u
∇
v
,
u
=
u
,
u
1
/
2
.Denote by0
<
λ
1
<
λ
2
<
⋯ the distinct Dirichlet eigenvalues of -
Δ on Ω, and denote by φ
1
,
φ
2
,
φ
3
,
… the eigenfunction corresponding to the eigenvalues; then
(18)
λ
1
(
Ω
)
:
=
inf
⁡
u
∈
H
,
|
u
|
2
=
1
∇
u
2
2
is achieved by φ
1
>
0. Let
(19)
E
j
:
=
⨁
i
≤
j
ker
⁡
(
-
Δ
-
λ
i
)
,
j
∈
N
.
Recall that a function u
∈
H is called a weak solution of (1) if
(20)
(
a
+
b
u
2
)
∫
Ω
‍
∇
u
·
∇
v
=
∫
Ω
‍
f
(
x
,
u
)
v
,
∀
v
∈
H
.
Weak solutions are the critical points of C
1 functional:
(21)
φ
(
u
)
:
=
a
2
u
2
+
b
4
u
4
-
∫
Ω
‍
F
(
x
,
u
)
,
∀
u
∈
H
.
Then
(22)
φ
′
u
v
=
a
+
b
u
2
∫
Ω
‍
∇
u
·
∇
v
-
∫
Ω
‍
f
x
,
u
v
,
∀
u
,
v
∈
H
.


## 3. The Existence of the Solutions

Lemma 7.
LetF satisfies (S
1); the function φ has a local linking with respect to H
=
X
1
⊕
X
2, where X
1
=
E
k, X
2
=
(
X
1
)
⊥.Proof.
LetX
1
=
E
k, X
2
=
(
X
1
)
⊥. If u
∈
X
1, then u
2
≤
λ
k
|
u
|
2
2; and if u
∈
X
2, then u
2
≥
λ
k
+
1
|
u
|
2
2. If u
∈
X
1, from (
S
1
), for the given δ, there exists a ρ
1
=
δ
/
c
1
>
0 such that u
≤
ρ
1
⇒
|
u
|
≤
c
1
u
≤
δ and since dim
⁡
X
1 is finite c
|
u
|
α
α
≥
c
2
u
α; by (
S
1
), we have
(23)
φ
u
=
a
2
u
2
+
b
4
u
4
-
∫
Ω
F
x
,
u
d
x
≤
a
2
u
2
+
b
4
u
4
-
a
λ
k
2
u
2
2
-
c
u
α
α
≤
b
4
u
4
-
c
2
u
α
.
Noting that 2
<
α
<
4, we can choose ρ
2
≤
ρ
1 small enough such that φ
(
u
)
<
0 for u
≤
ρ
2.
Foru
∈
X
2, from (
S
1
), (
S
3
), we have
(24)
F
x
,
u
≤
a
λ
k
+
1
2
u
2
+
c
3
u
r
;
then
(25)
φ
u
=
a
2
u
2
+
b
4
u
4
-
∫
Ω
F
x
,
u
d
x
>
a
2
u
2
+
b
4
u
4
-
a
λ
k
+
1
2
u
2
2
-
c
3
u
r
r
≥
b
4
u
4
-
c
4
u
r
.
Forr
>
4, choosing ρ
3 small enough such that (
b
/
4
)
u
4
-
c
4
u
r
>
0 for u
≤
ρ
3, we have φ
(
u
)
>
0 for u
∈
X
2, u
≤
ρ
3.
Chooseρ
=
min
⁡
{
ρ
2
,
ρ
3
}; it is very clear that φ has a local linking at 0 with respect to H
=
X
1
⊕
X
2.Lemma 8.
Suppose (S
3) holds. Then function φ satisfies condition (
C
)
*; that is, for every sequence {
u
α
n
}
⊂
X
α
n
,
{
u
α
n
} has a convergent subsequence if {
φ
(
u
α
n
)
} is bounded and (
1
+
u
n
)
φ
α
n
′
(
u
α
n
)
→
0 as n
→
∞, where X
α
n
=
X
α
n
1
⊕
X
2, X
α
n
1
=
span
⁡
¯
{
e
0
,
e
1
,
…
,
e
n
}, and (
e
i
)
i
≥
0 is a Hilbert basis for X
1.Proof.
Suppose that{
u
α
n
}
⊂
X
α
n
,
φ
(
u
α
n
) is bounded and (
1
+
u
α
n
)
φ
′
(
u
α
n
)
→
0 as n
→
∞; then there exists a constant M
>
0 such that
(26)
|
φ
(
u
α
n
)
|
≤
M
,
1
+
u
α
n
φ
′
(
u
α
n
)
≤
M
for every n
∈
N. On the one hand, by (6), there is constant c
5
>
0 such that
(27)
lim
⁡
|
u
0
|
→
∞
sup
⁡
|
u
|
>
u
0
F
(
x
,
u
)
u
r
<
c
5
;
then there exists δ
1
>
0, u
0
>
δ
1, such that
(28)
sup
⁡
|
u
|
>
u
0
F
(
x
,
u
)
u
r
<
c
5
,
F
(
x
,
u
)
u
r
<
c
5
when
u
≥
δ
1
.
Hence
(29)
F
x
,
u
≤
c
5
u
r
∀
u
≥
δ
1
,
a
.
e
.
x
∈
Ω
.
We know
(30)
F
x
,
u
≤
M
1
∀
u
≤
δ
1
,
a
.
e
.
x
∈
Ω
.
So for all u
∈
R
n
(
n
=
1,2
) and a.e. x
∈
Ω, we obtain
(31)
F
(
x
,
u
)
≤
c
5
u
r
+
M
1
.
It follows from (26) and (31)(32)
b
4
u
α
n
4
=
φ
u
α
n
+
∫
Ω
‍
F
x
,
u
α
n
d
x
-
a
2
u
α
n
2
≤
φ
u
α
n
+
∫
Ω
‍
F
x
,
u
α
n
d
x
<
M
+
M
1
Ω
+
c
5
∫
Ω
‍
u
α
n
r
d
x
.
On the other hand, by (7),
(33)
lim
⁡
|
u
0
|
→
∞
inf
⁡
|
u
|
>
u
0
f
(
x
,
u
)
u
-
4
F
(
x
,
u
)
u
μ
>
0
;
then there are constants c
6
>
0 and δ
2
>
0, u
0
>
δ
2, such that
(34)
inf
⁡
u
>
u
0
f
x
,
u
u
-
4
F
x
,
u
u
μ
>
c
6
,
f
x
,
u
u
-
4
F
x
,
u
u
μ
>
c
6
when
u
≥
δ
2
.
Hence
(35)
f
x
,
u
u
-
4
F
x
,
u
≥
c
6
u
μ
>
0
∀
u
≥
δ
2
,
a
.
e
.
x
∈
Ω
.
By F
∈
C
1, one has
(36)
f
x
,
u
u
-
4
F
x
,
u
≤
M
2
for
every
u
≤
δ
2
,
a
.
e
.
x
∈
Ω
.
Hence we obtain that
(37)
f
(
x
,
u
)
u
-
4
F
(
x
,
u
)
≥
c
6
u
μ
-
M
2
-
c
6
δ
2
μ
for all u
∈
R
n and a.e. x
∈
Ω. Then we have
(38)
5
M
≥
4
φ
u
α
n
-
φ
′
u
α
n
,
u
α
n
=
2
a
u
α
n
2
+
b
u
α
n
4
-
4
∫
Ω
‍
F
x
,
u
α
n
-
a
u
α
n
2
-
b
u
α
n
4
+
∫
Ω
‍
f
x
,
u
α
n
u
α
n
=
∫
Ω
‍
f
x
,
u
α
n
u
α
n
-
4
F
x
,
u
α
n
d
x
+
a
u
α
n
2
≥
c
6
∫
Ω
‍
u
α
n
μ
d
x
-
M
2
Ω
+
a
u
α
n
2
-
c
6
δ
2
μ
.
So ∫
Ω
‍
|
u
α
n
|
μ
d
x is bounded. If μ
≥
r, by (32) and Hölder inequality,
(39)
∫
Ω
‍
u
α
n
r
d
x
≤
Ω
(
μ
-
r
)
/
μ
∫
Ω
‍
u
α
n
μ
d
x
r
/
μ
.
Thus we get that {
u
α
n
} is bounded. If μ
≤
r, we obtain
(40)
∫
Ω
‍
u
α
n
r
d
x
=
∫
Ω
‍
u
α
n
r
-
μ
·
u
α
n
μ
≤
u
α
n
∞
r
-
μ
∫
Ω
‍
u
α
n
μ
d
x
≤
c
7
u
α
n
r
-
μ
∫
Ω
‍
u
α
n
μ
d
x
.
Then by (32) and r
-
μ
<
4, we know that {
u
α
n
} is bounded too. Hence {
u
α
n
} is bounded. Since {
u
α
n
} is bounded, for a subsequence, {
u
α
n
} converges to some u
α weakly in H and strongly in L
s
(
Ω
), 1
≤
s
<
+
∞, and a.e. in Ω:
(41)
φ
′
u
α
n
u
α
-
u
α
n
=
a
+
b
u
α
n
2
∫
Ω
‍
∇
u
α
n
·
∇
u
α
-
u
α
n
-
∫
Ω
‍
f
x
,
u
α
n
u
α
-
u
n
⟶
0
.
Noting that p
∈
(
1
,
+
∞
), q
=
p
/
(
p
-
1
),
(42)
∫
Ω
f
x
,
u
α
n
u
α
-
u
α
n
≤
∫
Ω
f
x
,
u
α
n
q
1
/
q
×
∫
Ω
u
α
-
u
α
n
p
1
/
p
≤
c
7
·
u
α
-
u
α
n
p
⟶
0
as n
→
+
∞. In addition,
(43)
a
+
b
u
α
n
2
∫
Ω
‍
∇
u
α
n
·
∇
u
α
-
u
n
=
a
+
b
u
α
n
2
u
α
n
,
u
α
-
u
α
n
,
u
α
n
=
a
+
b
u
α
n
2
×
u
α
n
,
u
α
-
u
α
,
u
α
+
u
α
,
u
α
-
u
α
n
,
u
α
n
.
Since {
u
α
n
}
⇀
u
α, then (
u
α
n
,
u
α
)
-
(
u
α
,
u
α
)
=
(
u
α
n
-
u
α
,
u
α
)
→
0. The combination of (32)–(44) implies that
(44)
u
α
n
,
u
α
n
⟶
u
α
,
u
α
,
j
⟶
+
∞
.
Thus, {
u
α
n
}
→
u
α in H.Lemma 9.
φ maps bounded sets into bounded sets.Proof.
From(
S
3
), we know
(45)
F
x
,
u
≤
c
5
u
r
+
M
1
,
so
(46)
φ
u
=
a
2
u
2
+
b
4
u
4
-
∫
Ω
‍
F
x
,
u
d
x
≤
a
2
u
2
+
b
4
u
4
+
∫
Ω
‍
M
1
+
c
5
u
r
≤
a
2
u
2
+
b
2
u
4
+
M
1
Ω
+
c
8
u
r
.
Hence φ maps bounded sets into bounded sets.Lemma 10.
For everym
∈
N, φ
(
u
)
→
-
∞, u
→
∞, u
∈
X
m
1
⊕
X
2.Proof.
By condition(
S
2
) and (2), there exist enough constants M
3
>
0, M
4
>
0, for all (
x
,
u
)
∈
Ω
×
R
n; we have
(47)
F
x
,
u
≥
M
3
u
4
-
M
4
.
So, for all u
0
∈
X
m
1
⊕
X
2, u
0
=
1, one has
(48)
φ
s
u
0
=
a
2
s
u
0
2
+
b
4
s
u
0
4
-
∫
Ω
F
x
,
s
u
0
d
x
≤
a
2
s
2
+
b
4
s
4
-
M
3
s
4
∫
Ω
u
0
4
+
M
4
Ω
≤
a
2
s
2
+
b
4
s
4
-
M
3
c
9
s
4
+
M
4
Ω
≤
a
2
s
2
+
s
4
b
4
-
M
3
c
9
+
M
4
Ω
⟶
-
∞
.
Hence, for every m
∈
N, ∀
u
∈
X
m
1
⊕
X
2, we have
(49)
φ
(
u
)
⟶
-
∞
as
u
⟶
∞
.
Then the conclusion is obtained.Proof of Theorem1.
By Lemmas7–10 and Lemma 6, we complete the proof.



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*Source: 573616-2014-12-10.xml*