Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 914210, 8 pages
http://dx.doi.org/10.1155/2013/914210
Research Article
Existence Results for a 𝑝(𝑥)-Kirchhoff-Type Equation without
Ambrosetti-Rabinowitz Condition
Libo Wang1,2 and Minghe Pei2
1
2
Institute of Mathematics, Jilin University, Chang’chun 130012, China
Department of Mathematics, Beihua University, Ji’lin 132013, China
Correspondence should be addressed to Libo Wang; wlb math@163.com
Received 25 November 2012; Accepted 28 April 2013
Academic Editor: Jaime Munoz Rivera
Copyright © 2013 L. Wang and M. Pei. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider the existence and multiplicity of solutions for the 𝑝(𝑥)-Kirchhoff-type equations without Ambrosetti-Rabinowitz
condition. Using the Mountain Pass Lemma, the Fountain Theorem, and its dual, the existence of solutions and infinitely many
solutions were obtained, respectively.
1. Introduction
Autuori et al. [30, 31] have dealt with the nonstationary Kirchhoff-type equation involving the 𝑝(𝑥)-Laplacian of the form
The Kirchhoff equation
𝜌
𝜌0
𝜕2 𝑢
𝜕2 𝑢
𝐸 𝐿 󵄨󵄨󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨󵄨
𝑑𝑥)
−
(
=0
+
∫
󵄨
󵄨
𝜕𝑡2
ℎ 2𝐿 0 󵄨󵄨󵄨 𝜕𝑥 󵄨󵄨󵄨
𝜕𝑥2
(1)
Ω
was introduced by Kirchhoff [1] in the study of oscillations
of stretched strings and plates, where 𝜌, 𝜌0 , ℎ, 𝐸, and 𝐿 are
constants. The stationary analogue of the Kirchhoff equation,
that is, (1), is as follows:
− (𝑎 + 𝑏 ∫ |∇𝑢|2 𝑑𝑥) Δ𝑢 = 𝑓 (𝑥, 𝑢) .
Ω
𝑢𝑡𝑡 − 𝑀 (∫
(2)
After the excellent work of Lions [2], problem (2) has received
more attention; see [3–10] and references therein.
The 𝑝(𝑥)-Laplace operator arises from various phenomena, for instance, the image restoration [11], the electro-rheological fluids [12], and the thermoconvective flows of nonNewtonian fluids [13, 14]. The study of the 𝑝(𝑥)-Laplace operator is based on the theory of the generalized Lebesgue space
𝐿𝑝(𝑥) (Ω) and the Sobolev space 𝑊𝑚,𝑝(𝑥) (Ω), which we called
variable exponent Lebesgue and Sobolev space. We refer the
reader to [15–19] for an overview on the variable exponent
Sobo-lev space, and to [20–29] for the study of the 𝑝(𝑥)Laplacian-type equations.
Recently, there has been an increasing interest in studying
the Kirchhoff equation involving the 𝑝(𝑥)-Laplace operator.
1
|∇𝑢|𝑝(𝑥) 𝑑𝑥) Δ 𝑝(𝑥) 𝑢
𝑝 (𝑥)
+ 𝑄 (𝑡, 𝑥, 𝑢, 𝑢𝑡 ) + 𝑓 (𝑥, 𝑢) = 0,
(3)
1
𝑢𝑡𝑡 − 𝑀 (∫
|∇𝑢|𝑝(𝑥) 𝑑𝑥) Δ 𝑝(𝑥) 𝑢
𝑝
(𝑥)
Ω
+ 𝜇|∇𝑢|𝑝(𝑥)−2 𝑢 + 𝑄 (𝑡, 𝑥, 𝑢, 𝑢𝑡 ) = 𝑓 (𝑡, 𝑥, 𝑢) .
In [32–35], applying variational method and AmbrosettiRabinowitz (AR) condition, Guowei Dai has studied the
existence and multiplicity of solutions for the 𝑝(𝑥)-Kirchhofftype equations with Dirichlet or Neumann boundary condition. In [36], by using (𝑆+ ) mapping theory and the Mountain
Pass Lemma, Fan has discussed the nonlocal 𝑝(𝑥)-Laplacian
Dirichlet problem with the nonvariational form
−𝐴 (𝑢) Δ 𝑝(𝑥) 𝑢 = 𝐵 (𝑢) 𝑓 (𝑥, 𝑢) ,
𝑢 = 0,
on 𝜕Ω,
in Ω,
(4)
2
Journal of Applied Mathematics
and the 𝑝(𝑥)-Kirchhoff-type equation with the variational
form
− 𝑎 (∫
Ω
1,𝑝(𝑥)
use the notation ‖𝑢‖ = |∇𝑢|𝑝(𝑥) for 𝑢 ∈ 𝑊0
1
|∇𝑢|𝑝(𝑥) 𝑑𝑥) Δ 𝑝(𝑥) 𝑢
𝑝 (𝑥)
= 𝑏 (∫ 𝐹 (𝑥, 𝑢) 𝑑𝑥) 𝑓 (𝑥, 𝑢) ,
Ω
𝑢 = 0,
in Ω,
(5)
on 𝜕Ω,
under (AR) condition, where 𝐴, 𝐵 are two functionals defined
𝑡
1,𝑝(𝑥)
on 𝑊0
(Ω), and 𝐹(𝑥, 𝑡) = ∫0 𝑓(𝑥, 𝑠)𝑑𝑠.
Motivated by the above works, the purpose of this paper
is to study the 𝑝(𝑥)-Kirchhoff-type equation
− (𝑎 + 𝑏 ∫
Ω
1
|∇𝑢|𝑝(𝑥) 𝑑𝑥) Δ 𝑝(𝑥) 𝑢 = 𝑓 (𝑥, 𝑢) ,
𝑝 (𝑥)
𝑢 = 0,
in Ω,
on 𝜕Ω,
(6)
without (AR) condition, where Ω is a smooth bounded
domain in R𝑁, 𝑎, 𝑏 are two positive constants, Δ 𝑝(𝑥) 𝑢 =
div(|∇𝑢(𝑥)|𝑝(𝑥)−2 ∇𝑢(𝑥)), 𝑝 ∈ 𝐶0,𝛽 (Ω) for some 𝛽 ∈ (0, 1), and
1 < 𝑝− := inf 𝑝 (𝑥) ≤ 𝑝+ := sup𝑝 (𝑥) < +∞.
Ω
1,𝑝(𝑥)
(Ω) the closure of 𝐶0∞ (Ω) in 𝑊1,𝑝(𝑥) (Ω).
Denote by 𝑊0
1,𝑝(𝑥)
|∇𝑢|𝑝(𝑥) is an equivalent norm on 𝑊0
(Ω). In this paper we
Ω
(7)
By taking the famous Mountain Pass Lemma, the Fountain
Theorem, and its dual, we obtain the existence of solutions
and infinitely many solutions for the 𝑝(𝑥)-Kirchhoff-type
equation (6) under no (AR) condition.
𝑁𝑝 (𝑥)
{
{
𝑁
𝑝 (𝑥) = { − 𝑝 (𝑥)
{
{+∞
(Ω). Define
if 𝑝 (𝑥) < 𝑁,
∗
We refer the reader to [36–38] for the elementary properties
of the space 𝑊1,𝑝(𝑥) (Ω).
Proposition 1 (see [38]). Set 𝜌(𝑢) = ∫Ω |𝑢(𝑥)|𝑝(𝑥) 𝑑𝑥. For any
𝑢 ∈ 𝐿𝑝(𝑥) (Ω), then the following are given:
(1) |𝑢|𝑝(𝑥) = 𝜆 ⇔ 𝜌(𝑢/𝜆) = 1 if 𝑢 ≠ 0;
(2) |𝑢|𝑝(𝑥) < 1(= 1; > 1) ⇔ 𝜌(𝑢) < 1(= 1; > 1);
𝑝−
𝑝+
𝑝+
𝑝−
(3) |𝑢|𝑝(𝑥) ≤ 𝜌(𝑢) ≤ |𝑢|𝑝(𝑥) if |𝑢|𝑝(𝑥) > 1;
(4) |𝑢|𝑝(𝑥) ≤ 𝜌(𝑢) ≤ |𝑢|𝑝(𝑥) if |𝑢|𝑝(𝑥) < 1;
(5) lim𝑘 → +∞ |𝑢𝑘 |𝑝(𝑥) = 0 ⇔ lim𝑘 → +∞ 𝜌(𝑢𝑘 ) = 0;
(6) lim𝑘 → +∞ |𝑢𝑘 |𝑝(𝑥) = +∞ ⇔ lim𝑘 → +∞ 𝜌(𝑢𝑘 ) = +∞.
3. Positive Energy Solution
In this section we discuss the existence of weak solutions of
1,𝑝(𝑥)
(Ω).
(6). For simplicity we write 𝑋 = 𝑊0
First, we state the assumptions on 𝑓 as follows.
(𝑓0 ) Let 𝑓 : Ω × R → R be a continuous function,
and there exist positive constants 𝑐1 , 𝑐2 such that
󵄨
󵄨󵄨
𝛼(𝑥)−1
,
󵄨󵄨𝑓 (𝑥, 𝑡)󵄨󵄨󵄨 ≤ 𝑐1 + 𝑐2 |𝑡|
2. Preliminary
We recall in this section some definitions and properties of
variable exponent Lebesgue-Sobolev space. The variable
exponent Lebesgue space 𝐿𝑝(𝑥) (Ω) is defined by
𝐿𝑝(𝑥) (Ω)
𝑝(𝑥)
= {𝑢 : 𝑢 : Ω → R is measurable, ∫ |𝑢|
Ω
(8)
(13)
where 𝛼 ∈ 𝐶(Ω) and 1 < 𝛼(𝑥) < 𝑝∗ (𝑥) for all 𝑥 ∈ Ω.
(𝑓0󸀠 ) Let 𝑓 : Ω × R → R be a continuous function,
and there exist positive constants 𝑐1 , 𝑐2 such that
󵄨
󵄨󵄨
𝛼(𝑥)−1
,
󵄨󵄨𝑓 (𝑥, 𝑡)󵄨󵄨󵄨 ≤ 𝑐1 + 𝑐2 |𝑡|
𝑑𝑥 < ∞}
(12)
if 𝑝 (𝑥) ≥ 𝑁.
(14)
where 𝛼 ∈ 𝐶(Ω) and 𝑝+ < 𝛼(𝑥) < 𝑝∗ (𝑥) for all 𝑥 ∈ Ω;
𝑡𝑓(𝑥, 𝑡) ≥ 0 for all 𝑡 > 0.
+
(𝑓1 ) Let lim𝑡 → +∞ (𝐹(𝑥, 𝑡)/|𝑡|2𝑝 ) = +∞, uniformly
𝑡
for 𝑥 ∈ Ω, where 𝐹(𝑥, 𝑡) = ∫0 𝑓(𝑥, 𝑠)𝑑𝑠.
with the norm
󵄨󵄨 𝑢 󵄨󵄨𝑝(𝑥)
|𝑢|𝐿𝑝(𝑥) = |𝑢|𝑝(𝑥) = inf {𝜎 > 0 : ∫ 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 𝑑𝑥 ≤ 1} .
Ω 󵄨𝜎󵄨
(9)
The variable exponent Sobolev space 𝑊1,𝑝(𝑥) (Ω) is defined by
𝑊1,𝑝(𝑥) (Ω) = {𝑢 ∈ 𝐿𝑝(𝑥) (Ω) : |∇𝑢| ∈ 𝐿𝑝(𝑥) (Ω)}
(10)
with the norm
‖𝑢‖𝑊1,𝑝(𝑥) = ‖𝑢‖1,𝑝(𝑥) = |𝑢|𝑝(𝑥) + |∇𝑢|𝑝(𝑥) .
(11)
(𝑓2 ) There exists 𝜃 ≥ 1 such that 𝜃𝐺(𝑥, 𝑡) ≥ 𝐺(𝑥, 𝑠𝑡)
for (𝑥, 𝑡) ∈ Ω × R and 𝑠 ∈ [0, 1], where
𝐺 (𝑥, 𝑡) = 𝑡𝑓 (𝑥, 𝑡) − 2𝑝+ 𝐹 (𝑥, 𝑡) .
(15)
+
(𝑓3 ) Let lim𝑡 → 0 (𝐹(𝑥, 𝑡)/|𝑡|𝑝 ) = 0, uniformly on 𝑥 ∈
Ω.
(𝑓3󸀠 ) There exists 𝛿 > 0, such that 𝐹(𝑥, 𝑡) ≤ 0 for 𝑥 ∈
Ω, |𝑡| < 𝛿.
Journal of Applied Mathematics
3
(𝑓4 ) Let 𝑓(𝑥, 𝑡) = −𝑓(𝑥, −𝑡) for 𝑥 ∈ Ω and 𝑡 ∈ R.
𝑞+
(𝑓5 ) Let lim𝑡 → 0 (𝐹(𝑥, 𝑡)/|𝑡| ) = 0, uniformly on 𝑥 ∈
Ω, where 𝑞 ∈ 𝐶(Ω) satisfies 1 < 𝑞(𝑥) < 𝑝(𝑥) for 𝑥 ∈
Ω.
Remark 2. Condition (𝑓2 ) was first introduced by Jeanjean
+
[39] for the case 𝑝(𝑥) = 2. Let 𝑓(𝑥, 𝑡) = 2𝑝+ |𝑡|2𝑝 −2 𝑡 ln |𝑡|,
then
+
𝐹 (𝑥, 𝑡) = |𝑡|2𝑝 ln |𝑡| −
+
1
|𝑡|2𝑝 ,
+
2𝑝
+
𝐺 (𝑥, 𝑡) = |𝑡|2𝑝 .
(16)
It is easy to see that the function 𝑓 does not satisfy (AR) condition, but it satisfies (𝑓1 )–(𝑓5 ) and (𝑓3󸀠 ).
Define 𝐼(𝑢) = 𝐽(𝑢) − Φ(𝑢), where
𝐽 (𝑢) = (𝑎 +
1
1
𝑏
∫
|∇𝑢|𝑝(𝑥) 𝑑𝑥) ∫
|∇𝑢|𝑝(𝑥) 𝑑𝑥,
2 Ω 𝑝 (𝑥)
Ω 𝑝 (𝑥)
(17)
Then 𝐼 ∈ 𝐶1 (𝑋, R).
Proposition 3 (see [38]). Assume that (𝑓0 ) hold, then the
functional 𝐽 : 𝑋 → R is sequentially weakly lower semicontinuous, Φ : 𝑋 → R is sequentially weakly continuous, and 𝐼
is sequentially weakly lower semicontinuous.
Proposition 4 (see [37]). Assume that (𝑓0 ) hold, and let 𝑢0 ∈
1,𝑝(𝑥)
(Ω) be a local minimizer (resp., a strictly local mini𝑊0
mizer) of 𝐼 in the 𝐶1 (Ω) topology. Then 𝑢0 is a local minimizer
1,𝑝(𝑥)
(resp., a strictly local minimizer) of 𝐼 in the 𝑊0
(Ω) topology.
Definition 5. We say that 𝑢 ∈ 𝑋 is a weak solution of (6), if
1
|∇𝑢|𝑝(𝑥) 𝑑𝑥) ∫ |∇𝑢|𝑝(𝑥)−2 ∇𝑢∇V 𝑑𝑥
𝑝 (𝑥)
Ω
(18)
Then we have
Ω
+
1
1
− + ) |∇𝑢|𝑝(𝑥) 𝑑𝑥
𝑝 (𝑥) 2𝑝
1
𝑏
∫
|∇𝑢|𝑝(𝑥) 𝑑𝑥
2 Ω 𝑝 (𝑥)
×∫ (
Ω
+
1
1
− + ) |∇𝑢|𝑝(𝑥) 𝑑𝑥
𝑝 (𝑥) 𝑝
(21)
1
∫ 𝐺 (𝑥, 𝑢) 𝑑𝑥}
2𝑝+ Ω
= lim {𝐼 (𝑢𝑛 ) −
𝑛→∞
1
⟨𝐼󸀠 (𝑢𝑛 ) , 𝑢𝑛 ⟩} = 𝑐.
2𝑝+
Let V𝑛 = 𝑢𝑛 /‖𝑢𝑛 ‖, then up to a subsequence we may assume
that
V𝑛 󳨀→ V
in 𝑋,
in 𝐿𝛼(𝑥) (Ω) ,
V𝑛 (𝑥) 󳨀→ V (𝑥)
Ω
󵄩
󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩 󸀠
󵄩󵄩𝑢𝑛 󵄩󵄩 󵄩󵄩𝐼 (𝑢𝑛 )󵄩󵄩󵄩󵄩 󳨀→ 0.
(20)
󵄩󵄩 󵄩󵄩
󵄩󵄩𝑢𝑛 󵄩󵄩 󳨀→ ∞,
𝐼 (𝑢𝑛 ) 󳨀→ 𝑐,
V𝑛 ⇀ V
= ∫ 𝑓 (𝑥, 𝑢) V 𝑑𝑥
(22)
a.e. 𝑥 ∈ Ω.
If V = 0, inspired by [13, 14], then we define
for any V ∈ 𝑋.
Definition 6. Let 𝑋 be a Banach space and 𝐼 ∈ 𝐶1 (𝑋, R).
Given 𝑐 ∈ R. we say that 𝐼 satisfies the Cerami 𝑐 condition
(we denote by (𝐶)𝑐 the condition), if
(i) any bounded sequence {𝑢𝑛 } ⊂ 𝑋 such that 𝐼(𝑢𝑛 ) → 𝑐
and 𝐼󸀠 (𝑢𝑛 ) → 0 has a convergent subsequence;
(ii) there exist constants 𝛿, 𝑅, 𝛽 > 0 such that
󵄩
󵄩
‖𝑢‖ 󵄩󵄩󵄩󵄩𝐼󸀠 (𝑢)󵄩󵄩󵄩󵄩 ≥ 𝛽,
Proof. From [36, Proposition 3.1], 𝐼 satisfies (i) of (𝐶) condition.
Now we check that 𝐼 satisfies (ii) of (𝐶) condition.
Arguing by contradiction, we may assume that, for some 𝑐 ∈
R,
𝑛→∞
Ω
Ω
Lemma 8. Assume that conditions (𝑓0 )–(𝑓2 ) hold. Then 𝐼
satisfies (𝐶) condition.
lim {𝑎 ∫ (
Φ (𝑢) = ∫ 𝐹 (𝑥, 𝑢) 𝑑𝑢.
(𝑎 + 𝑏 ∫
Remark 7. Although (PS) condition is stronger than (𝐶)
condition, the Deformation Theorem is still valid under (𝐶)
condition; we see that the Mountain Pass Lemma, the Fountain Theorem, and its dual are true under (𝐶) condition.
−1
∀𝑢 ∈ 𝐼 [𝑐 − 𝛿, 𝑐 + 𝛿] ,
𝐼 (𝑡𝑛 𝑢𝑛 ) = max 𝐼 (𝑡𝑢𝑛 ) .
𝑡∈[0,1]
−
For any 𝑚 > 1/2𝑝+ , let 𝑤𝑛 = (2𝑚𝑝+ )1/𝑝 V𝑛 . Since 𝑤𝑛 → 0
in 𝐿𝛼(𝑥) (Ω) and
|𝐹 (𝑥, 𝑡)| ≤ 𝑐5 + 𝑐6 |𝑡|𝛼(𝑥) ,
‖𝑢‖ ≥ 𝑅.
(19)
If 𝐼 ∈ 𝐶1 (𝑋, R) satisfies (𝐶)𝑐 condition for every 𝑐 ∈ R, then
we say that 𝐼 satisfies (𝐶) condition.
(23)
(24)
by the continuity of 𝐹(𝑥, ⋅), 𝐹(⋅, 𝑤𝑛 ) → 0 in 𝐿1 (Ω), thus,
lim ∫ 𝐹 (⋅, 𝑤𝑛 ) 𝑑𝑥 = 0.
𝑛→0 Ω
(25)
4
Journal of Applied Mathematics
1/𝑝−
Then for 𝑛 large enough, (2𝑚𝑝+ )
/‖𝑢𝑛 ‖ ∈ (0, 1) and
≥ 𝑎∫ (
Ω
𝐼 (𝑡𝑛 𝑢𝑛 ) ≥ 𝐼 (𝑤𝑛 )
= 𝑎∫
Ω
+
1 󵄨󵄨
󵄨𝑝(𝑥)
󵄨∇𝑤 󵄨󵄨 𝑑𝑥
𝑝 (𝑥) 󵄨 𝑛 󵄨
Ω
𝑝(𝑥)
1
1/𝑝− 󵄨
󵄨󵄨∇V𝑛 󵄨󵄨󵄨) 𝑑𝑥
((2𝑚𝑝+ )
󵄨
󵄨
𝑝 (𝑥)
Ω
+
2
+
− ∫ 𝐹 (𝑥, 𝑤𝑛 ) 𝑑𝑥
≥
2𝑚2 𝑏
2
(𝑝+ )
+
2
󵄨 󵄨𝑝(𝑥)
(∫ 󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨 𝑑𝑥) − ∫ 𝐹 (𝑥, 𝑤𝑛 ) 𝑑𝑥
Ω
Ω
2𝑚𝑎 2𝑚2 𝑏
+
− ∫ 𝐹 (𝑥, 𝑤𝑛 ) 𝑑𝑥.
2
𝑝+
Ω
(𝑝+ )
That is, 𝐼(𝑡𝑛 𝑢𝑛 ) → ∞. From 𝐼(0) = 0 and 𝐼(𝑢𝑛 ) → 𝑐, we
know that 𝑡𝑛 ∈ (0, 1) and
1 󵄨󵄨
󵄨𝑝(𝑥)
󵄨
󵄨𝑝(𝑥)
󵄨∇𝑡 𝑢 󵄨󵄨 𝑑𝑥) ∫ 󵄨󵄨󵄨∇𝑡𝑛 𝑢𝑛 󵄨󵄨󵄨 𝑑𝑥
𝑝 (𝑥) 󵄨 𝑛 𝑛 󵄨
Ω
1
1
(𝐼 (𝑡𝑛 𝑢𝑛 ) − + ⟨𝐼󸀠 (𝑡𝑛 𝑢𝑛 ) , 𝑡𝑛 𝑢𝑛 ⟩)
𝜃
2𝑝
=
1
𝐼 (𝑡 𝑢 ) → ∞.
𝜃 𝑛 𝑛
𝑎∫ (
Ω
1
1 󵄨󵄨
󵄨𝑝(𝑥)
) 󵄨∇𝑢 󵄨󵄨 𝑑𝑥
−
𝑝 (𝑥) 2𝑝+ 󵄨 𝑛 󵄨
+
1 󵄨󵄨
𝑏
󵄨𝑝(𝑥)
∫
󵄨∇𝑢 󵄨󵄨 𝑑𝑥
2 Ω 𝑝 (𝑥) 󵄨 𝑛 󵄨
1 󵄨
1
󵄨𝑝(𝑥)
− + ) 󵄨󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 𝑑𝑥
×∫ (
𝑝
Ω 𝑝 (𝑥)
1
+ + ∫ 𝐺 (𝑥, 𝑢𝑛 ) 𝑑𝑥
2𝑝 Ω
This contradicts (21).
If V ≠ 0, from (20), when ‖𝑢𝑛 ‖ ≥ 1,
𝑎
1
𝑏
𝑐 + 𝑜 (1)
− 󵄩 󵄩2𝑝+
+ +
2
−
𝑝− 󵄩󵄩󵄩𝑢 󵄩󵄩󵄩𝑝
󵄩󵄩𝑢𝑛 󵄩󵄩
2(𝑝 )
󵄩 𝑛󵄩
󵄩 󵄩
Ω
Therefore, from (𝑓2 ), we have
(28)
Then from (𝑓1 ) we have
− ∫ 𝑓 (𝑥, 𝑡𝑛 𝑢𝑛 ) 𝑢𝑛 𝑑𝑥
𝑑 󵄨󵄨󵄨
= ⟨𝐼󸀠 (𝑡𝑛 𝑢𝑛 ) , 𝑡𝑛 𝑢𝑛 ⟩ = 𝑡𝑛 󵄨󵄨󵄨 𝐼 (𝑡𝑢𝑛 ) = 0.
𝑑𝑡 󵄨󵄨𝑡=𝑡𝑛
𝐺 (𝑥, 𝑡𝑛 𝑢𝑛 )
1
𝑑𝑥
∫
+
2𝑝 Ω
𝜃
𝑎 󵄩󵄩 󵄩󵄩𝑝+
𝑏 󵄩󵄩 󵄩󵄩2𝑝+
󵄩󵄩𝑢𝑛 󵄩󵄩 +
󵄩𝑢𝑛 󵄩󵄩 − (𝑐 + 𝑜 (1)) ≥ ∫Ω 𝐹 (𝑥, 𝑢𝑛 ) 𝑑𝑥.
2󵄩
−
𝑝
2(𝑝− )
(29)
󵄨
󵄨𝑝(𝑥)
𝑎 ∫ 󵄨󵄨󵄨∇𝑡𝑛 𝑢𝑛 󵄨󵄨󵄨 𝑑𝑥
Ω
Ω
1
1
󵄨
󵄨𝑝(𝑥)
−
) 𝑡𝑝(𝑥) 󵄨󵄨∇𝑢 󵄨󵄨 𝑑𝑥
𝑝 (𝑥) 𝑝+ 𝑛 󵄨 𝑛 󵄨
=
(26)
+ 𝑏 (∫
1 𝑝(𝑥) 󵄨󵄨
𝑏
󵄨𝑝(𝑥)
𝑡 󵄨∇𝑢 󵄨󵄨 𝑑𝑥
∫
2𝜃 Ω 𝑝 (𝑥) 𝑛 󵄨 𝑛 󵄨
Ω
2𝑚𝑎
󵄨 󵄨𝑝(𝑥)
∫ 󵄨󵄨∇V 󵄨󵄨 𝑑𝑥
𝑝+ Ω 󵄨 𝑛 󵄨
+
𝐺 (𝑥, 𝑡𝑛 𝑢𝑛 )
1
𝑑𝑥
∫
2𝑝+ Ω
𝜃
×∫ (
Ω
1 󵄨
1
󵄨𝑝(𝑥)
− + ) 󵄨󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 𝑑𝑥
𝑝 (𝑥) 𝑝
1
1
𝑎
󵄨
󵄨𝑝(𝑥)
−
) 𝑡𝑝(𝑥) 󵄨󵄨∇𝑢 󵄨󵄨 𝑑𝑥
∫ (
𝜃 Ω 𝑝 (𝑥) 2𝑝+ 𝑛 󵄨 𝑛 󵄨
≥
𝑝(𝑥)
1
𝑏
1/𝑝− 󵄨
󵄨󵄨∇V𝑛 󵄨󵄨󵄨) 𝑑𝑥)
((2𝑚𝑝+ )
+ (∫
󵄨
󵄨
2 Ω 𝑝 (𝑥)
≥
1 󵄨󵄨
𝑏
󵄨𝑝(𝑥)
∫
󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 𝑑𝑥
2 Ω 𝑝 (𝑥)
×∫ (
2
1 󵄨󵄨
𝑏
󵄨𝑝(𝑥)
+ (∫
󵄨󵄨∇𝑤𝑛 󵄨󵄨󵄨 𝑑𝑥) − ∫ 𝐹 (𝑥, 𝑤𝑛 ) 𝑑𝑥
2 Ω 𝑝 (𝑥)
Ω
= 𝑎∫
1 󵄨󵄨
1
󵄨𝑝(𝑥)
−
) 󵄨∇𝑢 󵄨󵄨 𝑑𝑥
𝑝 (𝑥) 2𝑝+ 󵄨 𝑛 󵄨
(27)
𝐹 (𝑥, 𝑢 )
≥ ∫ 󵄩 󵄩2𝑝𝑛+ 𝑑𝑥
Ω 󵄩
󵄩󵄩𝑢𝑛 󵄩󵄩󵄩
𝐹 (𝑥, 𝑢 ) 󵄨 󵄨2𝑝+
+ ∫ ) 󵄨 󵄨2𝑝+𝑛 󵄨󵄨󵄨V𝑛 󵄨󵄨󵄨 𝑑𝑥
= (∫
󵄨󵄨𝑢𝑛 󵄨󵄨
V𝑛 ≠ 0
V𝑛 =0
󵄨 󵄨
=∫
V𝑛 ≠ 0
(30)
𝐹 (𝑥, 𝑢𝑛 ) 󵄨󵄨 󵄨󵄨2𝑝+
󵄨󵄨 󵄨󵄨2𝑝+ 󵄨󵄨V𝑛 󵄨󵄨 𝑑𝑥.
󵄨󵄨𝑢𝑛 󵄨󵄨
For 𝑥 ∈ Θ := {𝑥 ∈ Ω : V(𝑥) ≠ 0}, |𝑢𝑛 (𝑥)| → +∞. By (𝑓1 ) we
have
𝐹 (𝑥, 𝑢𝑛 ) 󵄨󵄨 󵄨󵄨𝑝+
󵄨󵄨 󵄨󵄨𝑝+ 󵄨󵄨V𝑛 󵄨󵄨 󳨀→ +∞.
󵄨󵄨𝑢𝑛 󵄨󵄨
(31)
Journal of Applied Mathematics
5
Note that the Lebesgue measure of Θ is positive; using the
Fatou Lemma, we have
𝐹 (𝑥, 𝑢𝑛 ) 󵄨󵄨 󵄨󵄨2𝑝+
󵄨V 󵄨 𝑑𝑥 󳨀→ +∞.
(32)
2𝑝+ 󵄨 𝑛 󵄨
V𝑛 ≠ 0 󵄨󵄨󵄨𝑢 󵄨󵄨󵄨
𝑛
󵄨 󵄨
This contradicts (30).
The technique used in this lemma was first introduced by
[39, 40].
∫
Theorem 9. Assume that conditions (𝑓0 )–(𝑓2 ) and (𝑓3 ) (or
(𝑓3󸀠 )) hold. Then (6) has a nontrivial solution with positive
energy.
Proof. From Lemma 8, 𝐼 satisfies (𝐶) condition. Let us show
that the functional 𝐼 has a Mountain-Pass-type geometry.
Note that 𝐼(0) = 0. By (𝑓3 ), there exists 𝛿 > 0, and for any
𝑢 ∈ 𝑋 with |𝑢|𝐿∞ (Ω) < 𝛿,
1
𝐼 (𝑢) = 𝑎 ∫
|∇𝑢|𝑝(𝑥) 𝑑𝑥
Ω 𝑝 (𝑥)
+
𝑏
𝑎
2𝑝+
− ∫ 𝐹 (𝑥, 𝑢) 𝑑𝑥 > 0.
‖𝑢‖𝑝 +
2 ‖𝑢‖
+
𝑝+
Ω
(𝑝 )
This shows that 0 is a strictly local minimizer of 𝐼 in the 𝐶(Ω)
topology, and hence 0 is a strictly local minimizer of 𝐼 in
the 𝐶1 (Ω) topology. By [37, Theorem 1.1], 0 is a strictly local
1,𝑝(𝑥)
minimizer of 𝐼 in the 𝑊0
(Ω) topology. Thus there exists
𝑟 > 0 such that 𝐼(𝑢) > 0 for every 𝑢 ∈ 𝑋 \ {0} with ‖𝑢‖ ≤ 𝑟.
We claim that inf ‖𝑢‖=𝑟 𝐼(𝑢) > 0. To prove this claim,
arguing by contradiction, assume that there exists a sequence
{𝑢𝑛 } ⊂ 𝑋 with ‖𝑢𝑛 ‖ = 𝑟 such that 𝐼(𝑢𝑛 ) → 0 as 𝑛 → ∞. We
may assume that 𝑢𝑛 ⇀ 𝑢0 in 𝑋. Since 𝐼 is sequentially weakly
lower semicontinuous, we have that 𝐼(𝑢0 ) = 0, and hence
𝑢0 = 0. Since Φ is sequentially weakly continuous, then we
have that Φ(𝑢𝑛 ) → Φ(0) = 0, and hence 𝐽(𝑢𝑛 ) = 𝐼(𝑢𝑛 ) +
Φ(𝑢𝑛 ) → 0. It follows from this that 𝑢𝑛 → 0 in 𝑋 which contradicts with ‖𝑢𝑛 ‖ = 𝑟.
Let 𝑦 ∈ 𝑋 with 𝑦 > 0 in Ω and ‖𝑦‖ = 1. By (𝑓0 ) and (𝑓1 ),
for 𝑠 ≥ 1 we have
1 󵄨󵄨
󵄨𝑝(𝑥)
𝐼 (𝑠𝑦) = 𝑎 ∫
󵄨󵄨∇𝑠𝑦󵄨󵄨󵄨 𝑑𝑥
Ω 𝑝 (𝑥)
𝑏 2𝑝+
𝑎 𝑝+
𝑠 +
𝑠
2
𝑝−
(𝑝− )
+
󵄨 󵄨2𝑝+
− 𝑐1 𝑠2𝑝 ∫ 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 𝑑𝑥 + 𝑐2 󳨀→ −∞ as 𝑠 󳨀→ +∞.
Ω
(34)
We set 𝑒 = 𝑠𝑦. Then for 𝑠 large, we obtain
𝐼 (𝑒) < 0.
𝑋 = span {𝑒𝑗 : 𝑗 = 1, 2, . . .},
𝑋∗ = span {𝑒𝑗∗ : 𝑗 = 1, 2, . . .},
1,
⟨𝑒𝑖 , 𝑒𝑗∗ , ⟩ = {
0,
(36)
𝑖 = 𝑗,
𝑖 ≠ 𝑗.
For convenience, we write 𝑋𝑗 = span{𝑒𝑗 }, 𝑌𝑘 = ⊕𝑘𝑗=1 𝑋𝑗 , 𝑍𝑘 =
⊕∞
𝑗=𝑘 𝑋𝑗 .
(37)
Then lim𝑘 → +∞ 𝛽𝑘 = 0.
Proposition 11 (Fountain Theorem). Assume that 𝐼 ∈
𝐶1 (𝑋, R) is an even functional. If, for any 𝑘 ∈ N, there exists
𝜌𝑘 > 𝑟𝑘 > 0 such that
(𝐴 1 ) 𝑎𝑘 = max𝑢∈𝑌𝑘 ,‖𝑢‖=𝜌𝑘 𝐼(𝑢) ≤ 0,
(𝐴 2 ) 𝑏𝑘 = inf 𝑢∈𝑍𝑘 ,‖𝑢‖=𝑟𝑘 𝐼(𝑢) → +∞ as 𝑘 → ∞,
(𝐴 3 ) 𝐼 satisfies (𝐶)𝑐 condition for every 𝑐 > 0,
then 𝐼 has an unbounded sequence of critical values.
Proposition 12 (Dual Fountain Theorem). Assume that 𝐼 ∈
𝐶1 (𝑋, R) is an even functional. If, for any 𝑘 ≥ 𝑘0 , there exists
𝜌𝑘 > 𝑟𝑘 > 0 such that
(𝐵1 ) 𝑎𝑘 = inf 𝑢∈𝑍𝑘 ,‖𝑢‖=𝜌𝑘 𝐼(𝑢) ≥ 0,
(𝐵2 ) 𝑏𝑘 = max𝑢∈𝑌𝑘 ,‖𝑢‖=𝑟𝑘 𝐼(𝑢) < 0,
(𝐵3 ) 𝑑𝑘 = inf 𝑢∈𝑍𝑘 ,‖𝑢‖≤𝜌𝑘 𝐼(𝑢) → 0 as 𝑘 → ∞,
(𝐵4 ) 𝐼 satisfies (𝑐)∗𝑐 condition for every 𝑐 ∈ [𝑑𝑘0 ,0 ],
then 𝐼 has a sequence of negative critical values converging to 0.
2
1 󵄨󵄨
󵄨𝑝(𝑥)
+ 𝑏(∫
󵄨󵄨∇𝑠𝑦󵄨󵄨󵄨 𝑑𝑥) − ∫ 𝐹 (𝑥, 𝑠𝑦) 𝑑𝑥
Ω 𝑝 (𝑥)
Ω
‖𝑒‖ > 𝑟,
Since 𝑋 is a reflexive and separable Banach space, then there
exists {𝑒𝑗 } ⊂ 𝑋 and {𝑒𝑗∗ } ⊂ 𝑋∗ such that
𝛽𝑘 = sup {|𝑢|𝛼(𝑥) : ‖𝑢‖ = 1, 𝑢 ∈ 𝑍𝑘 } .
(33)
≤
4. Infinitely Many Solutions
Lemma 10 (see [21]). If 𝛼 ∈ 𝐶(Ω), 1 < 𝛼(𝑥) < 𝑝∗ for any 𝑥 ∈
Ω, denote
2
1
𝑏
+ (∫
|∇𝑢|𝑝(𝑥) 𝑑𝑥) − ∫ 𝐹 (𝑥, 𝑢) 𝑑𝑥
2 Ω 𝑝 (𝑥)
Ω
≥
Hence by the famous Mountain Pass Lemma, problem (6) has
a nontrivial weak solution with positive energy.
(35)
Theorem 13. Assume that the conditions (𝑓0󸀠 ), (𝑓1 )–(𝑓4 ) hold.
Then (6) has infinitely many solutions {𝑢𝑘 } such that 𝐼(𝑢𝑘 ) →
∞ as 𝑘 → ∞.
Proof. By conditions (𝑓0󸀠 ), (𝑓1 ), and (𝑓3 ), for any 𝜀 > 0, there
exists 𝐶𝜀 such that
+
+
𝐹 (𝑥, 𝑢) ≥ 𝐶𝜀 |𝑢|2𝑝 − 𝜀|𝑢|𝑝 ,
∀ (𝑥, 𝑢) ∈ Ω × R.
(38)
6
Journal of Applied Mathematics
Proof . By conditions (𝑓0󸀠 ), (𝑓1 ), and (𝑓5 ), for any 𝜀 > 0, there
exists 𝐶𝜀 such that
For 𝑢 ∈ 𝑌𝑘 , when ‖𝑢‖ > 1,
𝐼 (𝑢) = 𝑎 ∫
Ω
1
|∇𝑢|𝑝(𝑥) 𝑑𝑥
𝑝 (𝑥)
+
𝐼 (𝑢) = 𝑎 ∫
Ω
+
+
𝑏
𝑎
‖𝑢‖𝑝 +
‖𝑢‖2𝑝
2
−
𝑝
2(𝑝− )
2𝑝+
𝑝+
− 𝐶𝜀 |𝑢|2𝑝+ + 𝜀|𝑢|𝑝+ 󳨀→ −∞
as ‖𝑢‖ 󳨀→ +∞.
(39)
On the other hand, by
that
≤
and (𝑓3 ), there exists 𝐶𝜀 > 0 such
∀ (𝑥, 𝑢) ∈ Ω × R.
(41)
1
|∇𝑢|𝑝(𝑥) 𝑑𝑥
𝑝 (𝑥)
𝑏𝑘 :=
−
|𝐹 (𝑥, 𝑢)| ≤ 𝜀|𝑢|𝑞 + 𝐶𝜀 |𝑢|𝛼(𝑥) ,
Ω
(42)
−
−
𝑎
‖𝑢‖𝑝 − 𝑐𝛽𝑘 ‖𝑢‖𝛼 .
+
2𝑝
−
− 1/(𝛼 −𝑝 )
If we choose 𝑟𝑘 := (𝑎/4𝑐𝑝+ 𝛽𝑘𝛼 )
then, for 𝑢 ∈ 𝑍𝑘 with ‖𝑢‖ = 𝑟𝑘 ,
→ ∞ as 𝑘 → ∞,
≥
+
−
𝑎
𝑞+
‖𝑢‖𝑝 − 𝑐𝐶𝜀 |𝑢|𝛼𝛼− − 𝑐𝜀|𝑢|𝑞+
+
𝑝
≥
+
𝑎
𝑞+
‖𝑢‖𝑝 − 𝑐|𝑢|𝑞+
+
2𝑝
≥
+
+
𝑎
𝑞+
‖𝑢‖𝑝 − 𝑐𝛽𝑘 ‖𝑢‖𝑞 .
2𝑝+
If we choose 𝜌𝑘 := (4𝑐𝑝+ 𝛽𝑘 /𝑎)
for 𝑢 ∈ 𝑍𝑘 with ‖𝑢‖ = 𝜌𝑘 ,
𝐼 (𝑢) ≥
:= 𝑏𝑘 ,
(47)
1
|∇𝑢|𝑝(𝑥) 𝑑𝑥
𝑝 (𝑥)
𝑞+
𝑝− /(𝛼− −𝑝− )
𝑎
𝑎
𝐼 (𝑢) ≥ + (
− )
4𝑝 4𝑐𝑝+ 𝛽𝑘𝛼
∀ (𝑥, 𝑢) ∈ Ω × R.
2
1
𝑏
+ (∫
|∇𝑢|𝑝(𝑥) 𝑑𝑥) − ∫ 𝐹 (𝑥, 𝑢) 𝑑𝑥
2 Ω 𝑝 (𝑥)
Ω
−
−
𝑎
≥ + ‖𝑢‖𝑝 − 𝑐|𝑢|𝛼𝛼−
2𝑝
−
(46)
Let 𝛽𝑘 := sup𝑢∈𝑍𝑘 ,‖𝑢‖=𝑟𝑘 |𝑢|𝑞− , then 𝛽𝑘 → 0 as 𝑘 → ∞. For
𝑢 ∈ 𝑍𝑘 , when ‖𝑢‖ and 𝜀 small enough,
𝐼 (𝑢) = 𝑎 ∫
−
−
𝑏
𝑎
𝑝+
2𝑝−
− 𝐶𝜀 |𝑢|𝛼𝛼− − 𝜀|𝑢|𝑝+
‖𝑢‖𝑝 +
2 ‖𝑢‖
+
𝑝+
2(𝑝 )
max 𝐼 (𝑢) < 0.
𝑢∈𝑌𝑘 ,‖𝑢‖=𝑟𝑘
On the other hand, by (𝑓5 ), there exists 𝐶𝜀 > 0 such that
2
1
𝑏
+ (∫
|∇𝑢|𝑝(𝑥) 𝑑𝑥) − ∫ 𝐹 (𝑥, 𝑢) 𝑑𝑥
2 Ω 𝑝 (𝑥)
Ω
≥
𝑞−
as ‖𝑢‖ 󳨀→ +∞.
(40)
Let 𝛽𝑘 := sup𝑢∈𝑍𝑘 ,‖𝑢‖=𝜌𝑘 |𝑢|𝛼− . From Lemma 10, 𝛽𝑘 → 0 as
𝑘 → ∞. For 𝑢 ∈ 𝑍𝑘 , when ‖𝑢‖ ≤ 1 and 𝜀 small enough,
≥
(45)
Then for some 𝑟𝑘 > 0 large enough,
+
Ω
+
+
𝑎
𝑏
‖𝑢‖𝑝 +
‖𝑢‖2𝑝
2
−
𝑝
(𝑝− )
2𝑝+
max 𝐼 (𝑢) ≤ 0.
|𝐹 (𝑥, 𝑢)| ≤ 𝜀|𝑢|𝑝 + 𝐶𝜀 |𝑢|𝛼(𝑥) ,
𝐼 (𝑢) = 𝑎 ∫
1
|∇𝑢|𝑝(𝑥) 𝑑𝑥
𝑝 (𝑥)
− 𝐶𝜀 |𝑢|2𝑝+ + 𝜀|𝑢|𝑞− 󳨀→ −∞
𝑢∈𝑌𝑘 ,‖𝑢‖=𝜌𝑘
(𝑓0󸀠 )
(44)
2
1
𝑏
+ (∫
|∇𝑢|𝑝(𝑥) 𝑑𝑥) − ∫ 𝐹 (𝑥, 𝑢) 𝑑𝑥
2 Ω 𝑝 (𝑥)
Ω
Then for some 𝜌𝑘 > 0 large enough,
𝑎𝑘 :=
∀ (𝑥, 𝑢) ∈ Ω × R.
For 𝑢 ∈ 𝑌𝑘 , when ‖𝑢‖ is large enough,
2
1
𝑏
+ (∫
|∇𝑢|𝑝(𝑥) 𝑑𝑥) − ∫ 𝐹 (𝑥, 𝑢) 𝑑𝑥
2 Ω 𝑝 (𝑥)
Ω
≤
+
𝐹 (𝑥, 𝑢) ≥ 𝐶𝜀 |𝑢|2𝑝 − 𝜀|𝑢|𝑞 ,
(43)
which implies that 𝑏𝑘 := inf 𝑢∈𝑍𝑘 ,‖𝑢‖=𝑟𝑘 𝐼(𝑢) ≥ 𝑏𝑘 → +∞ as
𝑘 → +∞.
Assume that conditions (𝑓0󸀠 ), (𝑓1 ), (𝑓2 ), (𝑓4 ), and
Theorem 14.
(𝑓5 ) hold. Then (6) has infinitely many solutions {𝑢𝑘 } such that
𝐼(𝑢𝑘 ) → 0 as 𝑘 → ∞.
1/(𝑝+ −𝑞+ )
+
+ 𝑞
𝑞+ 4𝑐𝑝 𝛽𝑘
𝑐𝛽𝑘 (
𝑎
(48)
→ 0 as 𝑘 → ∞, then,
𝑞+ /(𝑝+ −𝑞+ )
)
:= 𝑎𝑘 ,
(49)
which implies that 𝑎𝑘 := inf 𝑢∈𝑍𝑘 ,‖𝑢‖=𝜌𝑘 𝐼(𝑢) ≥ 𝑎𝑘 → 0 as 𝑘 →
+∞.
Furthermore, if 𝑢 ∈ 𝑍𝑘 with ‖𝑢‖ ≤ 𝜌𝑘 , then
𝑞− 𝑞−
𝐼 (𝑢) ≥ −𝑐𝛽𝑘 𝜌𝑘 󳨀→ 0
as 𝑘 󳨀→ ∞,
(50)
which implies that 𝑑𝑘 = inf 𝑢∈𝑍𝑘 ,‖𝑢‖≤𝜌𝑘 𝐼(𝑢) → 0 as 𝑘 → ∞.
Journal of Applied Mathematics
Acknowledgment
This paper is supported by the National Natural Science Foun
dation of China (11126339, 11201008).
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