Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 749059, 14 pages
doi:10.1155/2012/749059
Research Article
Multiplicity of Solutions for
a Class of Fourth-Order Elliptic Problems with
Asymptotically Linear Term
Qiong Liu and Dengfeng Lü
School of Mathematics and Statistics, Hubei Engineering University, Hubei, Xiaogan 432000, China
Correspondence should be addressed to Dengfeng Lü, hhldf@sina.com
Received 10 January 2012; Accepted 5 April 2012
Academic Editor: Turgut Öziş
Copyright q 2012 Q. Liu and D. Lü. 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 study the following fourth-order elliptic equations: Δ2 u aΔu fx, u, x ∈ Ω, u Δu 0, x ∈ ∂Ω, where Ω ⊂ RN is a bounded domain with smooth boundary ∂Ω and fx, u
is asymptotically linear with respect to u at infinity. Using an equivalent version of Cerami’s
condition and the symmetric mountain pass lemma, we obtain the existence of multiple solutions
for the equations.
1. Introduction and Main Results
In this paper, we will investigate the existence of multiple solutions to the following fourthorder elliptic boundary value problem:
Δ2 u aΔu fx, u,
u Δu 0,
x ∈ Ω,
x ∈ ∂Ω,
1.1
where Ω ⊂ RN is a bounded domain with smooth boundary ∂Ω, Δ2 is the biharmonic
operator, a < λ1 λ1 is the first eigenvalue of −Δ in H01 Ω is a parameter. We assume that
fx, u satisfies the following hypotheses.
f1 fx, u ∈ CΩ × R, R.
f2 lim|u| → 0 fx, u/u 0 uniformly for x ∈ Ω.
2
Journal of Applied Mathematics
f3 lim|u| → ∞ fx, u/u uniformly for x ∈ Ω, where ∈ 0, ∞ is a constant, or
∞, and there exists C > 0, q ∈ 2, 2∗ such that
fx, u ≤ C 1 |u|q−1 ,
1.2
where 2∗ 2N/N − 4.
f4 fx, u is odd in u.
f5 lim|u| → ∞ fx, uu − 2Fx, u ∞ uniformly for x ∈ Ω, where Fx, u u
fx, tdt.
0
f6 fx, u/u is nondecreasing with respect to u ≥ 0, for a.e. x ∈ Ω.
Problem 1.1 is usually used to describe some phenomena appeared in different
physical, engineering and other sciences. In recent years, there are many results for the fourthorder elliptic equations. In 1
, Lazer and McKenna considered the fourth-order problem:
Δ2 u aΔu d u 1 − 1 ,
u Δu 0,
x ∈ ∂Ω,
x ∈ Ω,
1.3
where u max{u, 0} and d ∈ R. They pointed out that this type of nonlinearity furnishes a
model to study traveling waves in suspension bridges. They also presented a mathematical
model for the bridge that takes account of the fact that the coupling provided by the stays
connecting the suspension cable to the deck of the road bed is fundamentally nonlinear see
1–3
. Since then, more general nonlinear fourth-order elliptic boundary value problems
have been studied. Problem 1.1 and 1.3 have been studied extensively in recent years,
we refer the reader to 4–14
.
For problem 1.3, Lazer and McKenna 2
proved the existence of 2k − 1 solutions
when N 1 and d > λk λk − c λk is the sequence of the eigenvalues of −Δ in H01 Ω by the
global bifurcation method. In 4
, Tarantello found a negative solution when d > λ1 λ1 − c by
a degree argument. For Problem 1.1, when fx, u bgx, u, the existence of two or three
nontrivial solutions has been obtained in 5, 6
for gx, u under certain conditions by using
variational methods. In 7
, positive solutions of problem 1.1 were got when f satisfies
the local superlinearity and sublinearity. When f is asymptotically linear at infinity, the
existence of three nontrivial solutions has been obtained in 8
by using variational method,
and the existence of a nontrivial solution has been obtained in 9
by using the mountain pass
theorem. For more similar problems, we refer to 10–20
and the references therein.
In this paper, we prove a new existence result about a multiple solutions of problem
1.1 under the assumption that fx, u is asymptotically linear with respect to u at infinity.
In this case, the Ambrosetti-Rabinowitz condition AR condition for short does not hold,
hence it is difficult to verify the classical PSc condition. To overcome this difficulty, by using
an equivalent version of Cerami’s condition and the symmetric mountain pass lemma see
21
, we obtain the existence of multiple solutions for problem 1.1. To the best of our
knowledge, our main results are new. Before stating the main results, we give some notations.
Journal of Applied Mathematics
3
Set E H 2 Ω ∩ H01 Ω, then E is a Hilbert space with the following inner product and
the norm:
u, vE Ω
ΔuΔv − a∇u∇vdx,
uE u, u1/2
E .
1.4
The corresponding energy functional of problem 1.1 is defined on E by
Iu 1
2
Fx, udx,
|Δu|2 − a|∇u|2 dx −
Ω
Ω
1.5
u
where Fx, u 0 fx, tdt. From f1 –f3 , it is easy to see that I ∈ C1 E, R, it is well known
that the weak solutions of problem 1.1 are the critical points of the energy functional Iu.
Our main results are stated as follows.
Theorem 1.1. Assume that fx, u satisfies assumptions f1 –f4 , and Λk is given by 2.8. Then
the following hold.
i If ∈ Λk , ∞ is not an eigenvalue of problem 2.4, then problem 1.1 has at least k
pairs of nontrivial solutions in E.
ii Suppose that f5 is satisfied, then the conclusion of (i) holds even if is an eigenvalue of
problem 2.4.
iii If ∞, and f6 holds, then problem 1.1 has infinitely many nontrivial solutions in
E.
2. Preliminaries
In this section, we give some preliminary results which will be used to prove our main results.
Throughout this paper, we will denote by |Ω| the Lebesgue measure of Ω, Bρ {u ∈
E : uE < ρ}. C will denote various positive constants, → respectively denotes strong
respectively weak convergence. om 1 denote om 1 → 0 as m → ∞. Ls Ω, 1 ≤ s < ∞
denote Lebesgue spaces, the norm Ls is denoted by | · |s for 1 ≤ s < ∞. The dual space of a
Banach space E will be denoted by E−1 .
First, we recall an equivalent version of Cerami’s condition as follows see 22
.
Definition 2.1. Let E be a Banach space. I ∈ C1 E, R is said to satisfy condition C at level
c ∈ R Cc for short, if the following fact is true: any sequence {um } ⊂ E, which satisfies
Ium −→ c,
1 um E I um E−1 −→ 0,
m −→ ∞
2.1
possesses a convergent subsequence in E.
Next, we will state an abstract symmetric mountain pass lemma. For this purpose, we
should first introduce the definition of genus see 23–25
.
4
Journal of Applied Mathematics
Definition 2.2. Let E be a real Banach space and A a subset of E. A is said to be symmetric if
u ∈ A implies −u ∈ A. For a closed symmetric set A which does not contain the origin, we
define a genus γA of A by the smallest integer k such that there exists an odd continuous
mapping from A to Rk \ {0}. If there does not exist such a k, we define γA ∞. Moreover,
we set γ∅ 0.
0 {u ∈ E :
Let E be an infinite dimensional real Banach space, I ∈ C1 E, R, A
∗
0 }, Γm Iu ≥ 0}, Γ {h0 0, h is an odd homeomorphism of E and hB1 ⊂ A
{K ⊂ E : K is compact, symmetric with respect to the origin, and for any h ∈ Γ∗ ,
∅, define
there holds γK ∩ h∂B1 ≥ m}. If Γm /
bm inf max Iu.
2.2
K⊂Γm u∈K
Now, we recall an abstract symmetric mountain pass lemma, which can be found in
26, 27
.
Lemma 2.3. Let e1 , e2 , . . . , em , . . . be linearly independent in E, and Ei span{e1 , e2 , . . . , ei }, i 1, 2, . . . , m, . . .. Suppose that I ∈ C1 E, R satisfies I0 0, I−u Iu, and Cc condition for
c ≥ 0. Furthermore, there exists ρ > 0, α > 0 such that Iu > 0 in Bρ \ {0} and Iu|∂Bρ ≥ α. Then,
0 is bounded, then Γm /
0 is
∅ and bm ≥ α > 0 is a critical value of I. Moreover, if Emi ∩ A
if Em ∩ A
bounded for all i 1, . . . r, and
bm1 · · · bmr b,
2.3
0 is bounded for all m, then
then γKb ≥ r, where Kb {u ∈ E : Iu b, I u 0}. If Em ∩ A
Iu possesses infinitely many critical values.
Let us consider the eigenvalue problem:
Δ2 u aΔu Λu,
u Δu 0,
x ∈ Ω,
2.4
x ∈ ∂Ω.
Set
Φu Ω
2
2
|Δu| − a|∇u| dx,
Ψu Ω
|u|2 dx.
2.5
For a < λ1 , Φu and Ψu are well defined. Furthermore, Φu, Ψu ∈ C1 E, R, and a real
value Λ is an eigenvalue of problem 2.4 if and only if there exists u ∈ E \ {0} such that
Φ u ΛΨ u. At this point, let us set
N {u ∈ E : Ψu 1}.
2.6
Then N /
∅ and N is a C1 manifold in E. It follows from the standard Lagrange multiples
arguments that eigenvalues of 2.4 correspond to critical values of Φ|N , and Φ satisfies the
Journal of Applied Mathematics
5
PS condition on N. Thus a sequence of critical values of Φ|N comes from the LjusternikSchnirelmann critical point theory on C1 manifolds. For any k ∈ N, set
Γk A ⊂ N : A is compact, symmetric and γA ≥ k .
2.7
Λk : inf max Φu
2.8
Then values:
A∈Γk u∈A
are critical values and hence are eigenvalues of problem 2.4. Moreover, 0 < Λ1 < Λ2 ≤ Λ3 ≤
· · · ≤ Λk ≤ · · · → ∞.
We prove some properties of functional Iu in the following lemma.
Lemma 2.4. For the functional Iu defined by 1.5, if assumptions f1 and f6 hold, and for any
{um } ⊂ E with I um , um → 0 as m → ∞, then there is a subsequence, still denoted by {um },
such that
Itum ≤
1 t2
Ium 2m
2.9
holds for all t > 0, m ∈ N .
Proof. This lemma is essentially due to 27, 28
. For the sake of completeness, we prove it
here.
By I um , um → 0 as m → ∞, for a suitable subsequence, we may assume that
1 − < I um , um um 2E −
m
Ω
fx, um um dx <
1
,
m
∀m.
2.10
We claim that for any t > 0 and m ∈ N ,
Itum <
t2
2m
Ω
1
fx, um um − Fx, um dx.
2
2.11
Indeed, for any t > 0, at fixed x ∈ Ω and m ∈ N , we set
ht t2
fx, um um − Fx, tum ,
2
2.12
then
h t tfx, um um − fx, tum um
1
tum fx, um − fx, tum t
≥ 0,
0 < t ≤ 1,
≤ 0,
t ≥ 1,
by f6 ,
2.13
6
Journal of Applied Mathematics
hence
ht ≤ h1
∀t > 0.
2.14
Therefore,
t2
2
Itum um E −
Fx, tum dx
2
Ω
t2 1
fx, um um dx −
Fx, tum dx by 2.10
≤
2 m
Ω
Ω
2
t
t2
fx, um um − Fx, tum dx
≤
2m
2
Ω
t2
≤
2m
Ω
2.15
1
fx, um um − Fx, um dx by 2.14
2
and our claim 2.11 is proved.
On the other hand,
1
2
Ium um E −
Fx, um dx
2
Ω
1
1
≥
fx, um um dx −
Fx, um dx,
− 2
m
Ω
Ω
2.16
that is,
Ω
1
1
fx, um um − Fx, um dx ≤
Ium .
2
2m
2.17
Combining 2.11 and 2.17 we have that
Itum ≤
1 t2
Ium ,
2m
∀t > 0, m ∈ N .
2.18
The proof is completed.
3. Proof of the Main Results
We begin with the following lemma.
Lemma 3.1. Let c ≥ 0. Assume that fx, u satisfies assumptions f1 –f3 . Then the following
hold.
i Iu satisfies Cc condition if < ∞ in assumption f3 , and is not an eigenvalue of
problem 2.4.
Journal of Applied Mathematics
7
ii If < ∞ is an eigenvalue of problem 2.4 and f5 holds, then Iu satisfies Cc condition.
iii If ∞, and f6 holds, then Iu satisfies Cc condition.
Proof. Suppose that {um } ⊂ E is a Cc sequence, that is, as m → ∞, we have
Ium −→ c ≥ 0,
1 um E I um E−1 −→ 0,
3.1
in E−1 .
3.2
It is easy to see that 3.2 implies that as m → ∞, there hold
um 2E
Ω
fx, um um dx om 1,
Δum Δϕ − a∇um ∇ϕ dx −
Ω
−
3.3
Ω
fx, um ϕdx om 1,
∀ϕ ∈ E.
3.4
By Sobolev compact embedding, to show that Iu satisfies Cc condition, it suffices to show
the boundedness of Cc sequence in E for each case.
i Suppose that 0 < < ∞ and is not an eigenvalue of problem 2.4. Arguing by
contradiction, we suppose that there exists a subsequence, still denoted by {um }, such that as
m → ∞, there holds um E → ∞. Define
⎧
fx, um x
⎪
⎪
,
⎨
um x
pm x ⎪
⎪
⎩
0,
um x /
0,
3.5
um x 0.
Then from assumptions f1 –f3 , there exists M > 0 such that
0 ≤ pm x ≤ M.
3.6
Let
wm um
.
um E
3.7
Obviously, wm is bounded in E. Going if necessary to a subsequence, we can assume that
wm w,
weakly in E,
wm −→ w,
a.e. in Ω,
wm −→ w,
strongly in L Ω, ∀s ∈ 2, 2 .
3.8
s
∗
8
Journal of Applied Mathematics
It is easy to show that w /
≡ 0. In fact, if w ≡ 0, then from 3.3, 3.6, 3.8 and the definitions
of pm and wm , as m → ∞, we have
1 wm 2E Ω
pm x|wm |2 dx om 1 ≤ M
Ω
|wm |2 dx om 1 −→ 0,
3.9
which is a contradiction.
From 3.6, there exists hx ∈ L∞ Ω with 0 ≤ hx ≤ M such that, up to a subsequence, as m → ∞, there holds
weakly∗ in L∞ Ω.
pm x hx,
3.10
Then from 3.8 it follows that
pm xwm hxw weakly in L2 Ω,
pm x|wm |2 dx −→
hx|w|2 dx.
Ω
3.11
Ω
On the other hand, from 3.3, 3.4, 3.5, and 3.7, we have
Ω
Δwm Δϕ − a∇wm ∇ϕ dx Ω
2
2
Ω
pm xwm ϕdx om 1,
∀ϕ ∈ E.
3.12
|Δwm | − a|∇wm | dx Ω
pm x|wm |2 dx om 1.
3.13
It follows from 3.11–3.13 that
wm 2E
Ω
Ω
hx|w|2 dx om 1,
Δwm Δϕ−a∇wm ∇ϕ dx 3.14
Ω
hxwϕdxom 1,
∀ϕ ∈ E.
Therefore 3.14 implies that w satisfies
Ω
ΔwΔϕ − a∇w∇ϕ dx Ω
hxwϕdx,
∀ϕ ∈ E.
3.15
Let
Ω0 {x ∈ Ω : wx 0},
Ω {x ∈ Ω : wx > 0},
Ω− {x ∈ Ω : wx < 0}.
3.16
Journal of Applied Mathematics
9
Then um x → ∞ as m → ∞ if x ∈ Ω , and um x → −∞ as m → ∞ if x ∈ Ω− . From
assumption f3 , hx ≡ for all x ∈ Ω ∪ Ω− . Thus 3.15 implies that w satisfies
Ω0
ΔwΔϕ−a∇w∇ϕ−hxwϕ dx
Ω ∪Ω−
ΔwΔϕ−a∇w∇ϕ−hxwϕ dx 0,
∀ϕ ∈ E.
3.17
Therefore
Ω
ΔwΔϕ − a∇w∇ϕ dx Ω
wϕdx,
∀ϕ ∈ E.
3.18
This means that is an eigenvalue of problem 2.4, which contradicts our assumption, so
{um } is bounded in E.
ii Suppose ∈ 0, ∞ is an eigenvalue of problem 2.4, we need the additional
assumption f5 .
From assumption f5 , there exists T0 > 0 such that
fx, uu − 2Fx, u ≥ 0,
∀|u| ≥ T0 , x ∈ Ω,
3.19
fx, um um − 2Fx, um dx ≥ −C0 .
3.20
and there exists C0 C0 T0 > 0 such that
{|um |≤T0 }
Furthermore, under assumptions f1 –f3 , there exists M > 0 such that
fx, u ≤ M|u|,
|Fx, u| ≤
M 2
|u| ,
2
∀x ∈ Ω.
3.21
Let K 2c C0 2MSN/2 , where M > 0 is given by 3.21, S > 0 is the best Sobolev constant
such that
2∗
Ω
|u| dx
2/2∗
≤S
Ω
|Δu|2 − a|∇u|2 dx,
∀u ∈ E.
3.22
From assumption f5 , there exists T T K > T0 > 0 such that
fx, uu − 2Fx, u ≥ K,
∀|u| ≥ T, x ∈ Ω.
3.23
For the above T > 0 and each m ≥ 1, set
Am {x ∈ Ω : |um x| ≥ T },
Bm {x ∈ Ω : |um x| ≤ T }.
3.24
10
Journal of Applied Mathematics
From estimates 3.20, 3.1, 3.3, and 3.23, we get
2c om 1 fx, um um − 2Fx, um dx
Ω
fx, um um − 2Fx, um dx − C0
≥
3.25
Am
≥ K|Am | − C0 ,
where |Am | denotes the measure of Am .
On the other hand, for any fixed r > 2, from 3.1 and 3.3, we have
1 1
1
2
−
Fx, um − fx, um um dx c om 1.
um E −
2 r
r
Ω
3.26
Since Ω is bounded and f ∈ CΩ × R, R, there exists a constant C CΩ, f, T such that
1
Fx, um − fx, um um dx ≤ C,
Bm
r
∀x ∈ Ω.
3.27
Then, from 3.21–3.26, Hölder inequality and Sobolev inequality, we have
1 1
1
−
Fx, um − fx, um um dx
um 2E − C −
2 r
r
Am
1 1
1
1
≥
−
fx, um um − fx, um um dx
um 2E − C −
2 r
r
Am 2
1 1
1 1
−
−
fx, um um dx
um 2E − C −
2 r
2 r
Am
1 1
1 1
−
−
≥
M
|um |2 dx
um 2E − C −
2 r
2 r
Am
1 1
1 1
−
−
≥
M|um |22∗ |Am |2/N
um 2E − C −
2 r
2 r
2/N
1 1
1 1
2
2 2c C0 o
−
−
MSum E ≥
1
um E − C −
m
2 r
2 r
K 1 1 1
1 1
−
−
≥
MSum 2E · om 1,
um 2E − C −
2 2 r
2 r
c om 1 ≥
that is, {um } is bounded in E.
3.28
Journal of Applied Mathematics
11
iii Finally, we prove the case ∞. Here the subcritical condition 1.2 is assumer
as usual, but to make use of Lemma 2.4, f6 is required in this case. Set
tm √
2 c
,
um E
w m t m um √
2 cum
.
um E
3.29
√
Then wm E 2 c and {wm } is bounded in E. Hence, up to a subsequence, we may assume
that: there exists w ∈ E such that 3.8 also holds in this case. If um E → ∞, we claim that
wx ≡
/ 0.
3.30
In fact, if wx ≡ 0 in Ω, then 3.29 and 3.8 imply that
Fx, wm dx −→ 0,
Iwm 4c om 1.
Ω
3.31
√
However, applying Lemma 2.4 with t 2 c/um E , we have
Iwm ≤
1 t2
Ium −→ c,
2m
m −→ ∞,
3.32
which contradicts 3.31, thus 3.30 holds.
Ω \ {x ∈ Ω : wx 0}.
On the other hand, similar to case i, 3.13 holds. Let Ω
Then |Ω| > 0 by 3.30. From assumptions f3 and f4 , pm x ≥ 0 and pm x → ∞ as
where pm x is defined by 3.5. Hence, from 3.8 and 3.13, we have
m → ∞ in Ω,
4c lim infwm 2E lim inf
m→∞
m→∞
≥ lim inf
m→∞
≥
Ω
Ω
pm x|wm |2 dx
pm x|wm |2 dx
3.33
lim inf pm x|wm |2 dx ∞,
m→∞
Ω
which is a contradiction, thus um E ∞, that is, up to a subsequence, {um } is bounded in
E.
Proof of Theorem 1.1. The proof of this theorem is divided in two steps.
Step 1. There exists ρ > 0, α > 0 such that Iu > 0 in Bρ 0 and Iu|∂Bρ ≥ α.
In fact, in each case, assumptions f1 –f3 imply that for any ε > 0, there exists Cε > 0
such that, for all u ∈ R, there holds
fx, u ≤ ε|u| Cε |u|q−1 ,
|Fx, u| ≤ ε|u|2 Cε |u|q ,
3.34
where q is the same as that in 1.2, from which, it is easy to see that there exists ρ > 0, α > 0
such that Iu > 0 in Bρ 0 and Iu|∂Bρ ≥ α.
12
Journal of Applied Mathematics
Step 2. By the Symmetric Mountain Pass Lemma 2.3, to prove Theorem 1.1, it suffices to prove
that for any k ≥ 1, there exists a k-dimensional subspace Ek of E and Rk > 0 such that
Iu ≤ 0,
∀u ∈ Ek \ BRk .
3.35
First, we prove 3.35 in the case ∈ Λk , ∞. Since > Λk , there is ε > 0 such that − ε > Λk .
By the definition of Λk , there exists a k-dimensional subspace Ek of E such that, for the above
ε > 0, there holds
Ψu
ε
ε
≤ Λk < l − ,
2
2
u∈Ek \{0} Φu
3.36
1
Φu
>
.
Ψu
−
ε/2
u∈Ek \{0}
3.37
sup
that is,
sup
By assumption f3 , we have
lim
|u| → ∞
Fx, u
|u|
2
.
2
3.38
Then, for the above ε > 0, there exists M > 0 large enough such that
Fx, u
|u|2
>
ε
1
−
,
2
4
∀|u| > M.
3.39
Therefore, if u ∈ Ek with uE R, by 3.39 and 3.37, we obtain
1
Iu R2 −
2
≤
1 2
R −
2
Ω
Fx, udx
|u|>M
Fx, udx − CM, Ω
ε
1 2 1
R −
−
|u|2 dx − CM, Ω
2
2
4 Ω
2 ε
R2
|u|
1− −
dx − CM, Ω
2
4 Ω R
≤
≤
R2
− ε/4
1−
− CM, Ω
2
− ε/2
< 0,
if R ≥ Rk and Rk > 0 large enough.
3.40
Journal of Applied Mathematics
13
If ∞, similar to 3.37, for any k ≥ 1, there exists Ek ⊂ E such that
1
Φu
>
,
Λk 1/2
u∈Ek \{0} Ψu
sup
3.41
similar to 3.39, from assumption f3 with ∞ it follows that there exists Mk > 0 such
that
2Fx, u
|u|2
> Λk 1,
∀|u| > Mk .
3.42
Then, if u ∈ Ek with uE R, we have
Iu ≤
Λk 1
R2
1−
− CMk , k, Ω < 0,
2
Λk 1/2
3.43
if R ≥ Rk and Rk > 0 large enough. This completes the proof of Theorem 1.1.
Acknowledgment
The authors would like to thank the referees for carefully reading this paper and making
valuable comments and suggestions.
References
1
A. C. Lazer and P. J. McKenna, “Large-amplitude periodic oscillations in suspension bridges: some
new connections with nonlinear analysis,” SIAM Review, vol. 32, no. 4, pp. 537–578, 1990.
2
A. C. Lazer and P. J. McKenna, “Global bifurcation and a theorem of Tarantello,” Journal of Mathematical Analysis and Applications, vol. 181, no. 3, pp. 648–655, 1994.
3
P. J. McKenna and W. Reichel, “Radial solutions of singular nonlinear biharmonic equations and
applications to conformal geometry,” Electronic Journal of Differential Equations, vol. 37, pp. 1–13, 2003.
4
G. Tarantello, “A note on a semilinear elliptic problem,” Differential and Integral Equations, vol. 5, no.
3, pp. 561–565, 1992.
5
A. M. Micheletti and A. Pistoia, “Multiplicity results for a fourth-order semilinear elliptic problem,”
Nonlinear Analysis, vol. 31, no. 7, pp. 895–908, 1998.
6
A. M. Micheletti and A. Pistoia, “Nontrivial solutions for some fourth order semilinear elliptic problems,” Nonlinear Analysis, vol. 34, no. 4, pp. 509–523, 1998.
7
G. Xu and J. Zhang, “Existence results for some fourth-order nonlinear elliptic problems of local
superlinearity and sublinearity,” Journal of Mathematical Analysis and Applications, vol. 281, no. 2, pp.
633–640, 2003.
8
X. Liu and Y. Huang, “On sign-changing solution for a fourth-order asymptotically linear elliptic
problem,” Nonlinear Analysis, vol. 72, no. 5, pp. 2271–2276, 2010.
9
Y. An and R. Liu, “Existence of nontrivial solutions of an asymptotically linear fourth-order elliptic
equation,” Nonlinear Analysis, vol. 68, no. 11, pp. 3325–3331, 2008.
10
Y. Yang and J. Zhang, “Existence of solutions for some fourth-order nonlinear elliptic problems,”
Journal of Mathematical Analysis and Applications, vol. 351, no. 1, pp. 128–137, 2009.
11
J. Zhang and S. Li, “Multiple nontrivial solutions for some fourth-order semilinear elliptic problems,”
Nonlinear Analysis, vol. 60, no. 2, pp. 221–230, 2005.
14
Journal of Applied Mathematics
12
Z. Jihui, “Existence results for some fourth-order nonlinear elliptic problems,” Nonlinear Analysis, vol.
45, pp. 29–36, 2001.
13
J. Zhou and X. Wu, “Sign-changing solutions for some fourth-order nonlinear elliptic problems,” Journal of Mathematical Analysis and Applications, vol. 342, no. 1, pp. 542–558, 2008.
14
J. Zhang and Z. Wei, “Multiple solutions for a class of biharmonic equations with a nonlinearity concave at the origin,” Journal of Mathematical Analysis and Applications, vol. 383, no. 2, pp. 291–306, 2011.
15
Y. Liu and Z. Wang, “Biharmonic equations with asymptotically linear nonlinearities,” Acta Mathematica Scientia. Series B, vol. 27, no. 3, pp. 549–560, 2007.
16
D. Lü, “Multiple solutions for a class of biharmonic elliptic systems with Sobolev critical exponent,”
Nonlinear Analysis, vol. 74, no. 17, pp. 6371–6382, 2011.
17
Y. Yin and X. Wu, “High energy solutions and nontrivial solutions for fourth-order elliptic equations,”
Journal of Mathematical Analysis and Applications, vol. 375, no. 2, pp. 699–705, 2011.
18
P. Mironescu and V. D. Rădulescu, “The study of a bifurcation problem associated to an asymptotically linear function,” Nonlinear Analysis, vol. 26, no. 4, pp. 857–875, 1996.
19
P. Pucci and V. Rădulescu, “The impact of the mountain pass theory in nonlinear analysis: a mathematical survey,” Bollettino della Unione Matematica Italiana. Serie 9, vol. 3, no. 3, pp. 543–582, 2010.
20
V. D. Rădulescu, Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity,
Analytic, and Variational Methods, vol. 6 of Contemporary Mathematics and Its Applications, Hindawi Publishing Corporation, New York, NY, USA, 2008.
21
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,
vol. 65 of CBMS Regional Conference Series in Mathematics, Conference Board of the Mathematical
Sciences, Washington, DC, USA, 1986.
22
D. G. Costa and C. A. Magalhães, “Existence results for perturbations of the p-Laplacian,” Nonlinear
Analysis, vol. 24, no. 3, pp. 409–418, 1995.
23
J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical
Sciences, Springer, New York, NY, USA, 1989.
24
P. Mironescu and V. D. Rădulescu, “A multiplicity theorem for locally Lipschitz periodic functionals,”
Journal of Mathematical Analysis and Applications, vol. 195, no. 3, pp. 621–637, 1995.
25
A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical point theory and applications,” vol. 14, pp. 349–381, 1973.
26
G. Li and H.-S. Zhou, “Multiple solutions to p-Laplacian problems with asymptotic nonlinearity as
up−1 at infinity,” Journal of the London Mathematical Society. Second Series, vol. 65, no. 1, pp. 123–138,
2002.
27
G. Li and H.-S. Zhou, “Asymptotically linear Dirichlet problem for the p-Laplacian,” Nonlinear Analysis, vol. 43, pp. 1043–1055, 2001.
28
C. A. Stuart and H. S. Zhou, “Applying the mountain pass theorem to an asymptotically linear elliptic
equation on RN ,” Communications in Partial Differential Equations, vol. 24, no. 9-10, pp. 1731–1758, 1999.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014