# Infinitely Many Weak Solutions of thep-Laplacian Equation with Nonlinear Boundary Conditions

Authors: Feng-Yun Lu; Gui-Qian Deng
Journal: The Scientific World Journal
 (2014)

---

## Abstract

We study the followingp-Laplacian equation with nonlinear boundary conditions: -
Δ
p
u
+
μ
(
x
)
|
u
|
p
-
2
u
=
f
(
x
,
u
)
+
g
(
x
,
u
)
,
x
∈
Ω
,
|
∇
u
|
p
-
2
∂
u
/
∂
n
=
η
|
u
|
p
-
2
u
and x
∈
∂
Ω
, where Ω is a bounded domain in ℝ
N with smooth boundary ∂
Ω. We prove that the equation has infinitely many weak solutions by using the variant fountain theorem due to Zou (2001) and f
,
g do not need to satisfy the (
P
.
S
) or (
P
.
S
*
) condition.


---

## Body

## 1. Introduction

In this paper, we study the followingp-Laplacian equation:
(1)
-
Δ
p
u
+
μ
(
x
)
|
u
|
p
-
2
u
=
f
(
x
,
u
)
+
g
(
x
,
u
)
,
x
∈
Ω
,
|
∇
u
|
p
-
2
∂
u
∂
n
=
η
|
u
|
p
-
2
u
,
x
∈
∂
Ω
,
whereΩis a bounded domain inℝ
Nwith smooth boundary∂
Ωand∂
/
∂
nis the outer normal derivative,-
Δ
p
u
=
div
(
|
∇
u
|
p
-
2
∇
u
)is thep-Laplacian with1
<
p
<
N,ηis real parameter, and
(2)
μ
(
x
)
∈
L
∞
(
Ω
)
satisfying
ess
inf
x
∈
Ω
¯
μ
(
x
)
>
0
.
The perturbation functionsf
,
gsatisfy the following conditions:(F1)
f
,
g
∈
C
(
Ω
¯
×
ℝ
,
ℝ
) are odd inu;
(F2)
there existσ
,
δ
∈
(
1
,
p
), c
1
>
0, c
2
>
0, c
3
>
0 such that
(3)
c
1
|
u
|
σ
≤
f
(
x
,
u
)
u
≤
c
2
|
u
|
σ
+
c
3
|
u
|
δ
,
hhhhhhh
for
a
.
e
.
x
∈
Ω
and
u
∈
ℝ
.
(F3)
There existsp
<
q
<
p
* (where p
*
=
p
N
/
N
-
p) such that |
g
(
x
,
u
)
|
≤
c
(
1
+
|
u
|
q
) for a.e.x
∈
Ωandu
∈
ℝ. Moreover, lim
u
→
0
g
(
x
,
u
)
/
|
u
|
p
-
1
=
0 uniformly forx
∈
Ω.
(F4)
Assume that one of the following conditions hold:
(1)
lim
|
u
|
→
∞
g
(
x
,
u
)
/
|
u
|
p
-
2
u
=
0 uniformly forx
∈
Ω;
(2)
lim
|
u
|
→
∞
g
(
x
,
u
)
/
|
u
|
p
-
2
u
=
-
∞ uniformly forx
∈
Ω; furthermore, f
(
x
,
u
)
/
|
u
|
p
-
2
u and g
(
x
,
u
)
/
|
u
|
p
-
2
u are decreasing inuforuis large enough;
(3)
lim
|
u
|
→
∞
g
(
x
,
u
)
/
|
u
|
p
-
2
u
=
∞ uniformly forx
∈
Ω; g
(
x
,
u
)
/
|
u
|
p
-
2
u is increasing inuforuis large enough; moreover, there exists α
>
max
{
σ
,
δ
} such that
(4)
liminf
|
u
|
→
∞
g
(
x
,
u
)
u
-
p
G
(
x
,
u
)
|
u
|
α
≥
c
>
0
uniformly
for
x
∈
Ω
,whereG
(
x
,
u
)
=
∫
0
u
‍
g
(
x
,
t
)
d
t.Remark 1.
The above conditions were given in Zou [1] for the semilinear casep
=
2.Remark 2.
A simple example which satisfies (F1)–(F4) is(5)
f
(
x
,
u
)
+
g
(
x
,
u
)
=
μ
|
u
|
r
-
2
u
+
γ
|
u
|
s
-
2
u
,
where1
<
r
<
p
<
s
<
p
*.Equation (1) is posed in the framework of the Sobolev space
(6)
W
1
,
p
(
Ω
)
=
{
u
∈
L
p
(
Ω
)
:
∫
Ω
‍
|
∇
u
|
p
d
x
<
∞
}
,
with the norm
(7)
∥
u
∥
=
(
∫
Ω
‍
(
|
∇
u
|
p
+
μ
(
x
)
|
u
|
p
)
d
x
)
1
/
p
.The corresponding energy functional of (1) is defined by
(8)
Φ
(
u
)
=
1
p
∫
Ω
‍
(
|
∇
u
|
p
+
μ
(
x
)
|
u
|
p
)
d
x
-
∫
Ω
‍
F
(
x
,
u
)
d
x
-
∫
Ω
‍
G
(
x
,
u
)
d
x
-
η
p
∫
∂
Ω
‍
|
u
|
p
d
s
,
foru
∈
W
1
,
p
(
Ω
), whereF
(
x
,
u
)
=
∫
0
u
‍
f
(
x
,
t
)
d
tandd
sis the measure on the boundary. It is easy to see that Φ
∈
C
1
(
W
1
,
p
(
Ω
)
,
ℝ
) and
(9)
〈
Φ
′
(
u
)
,
v
〉
=
∫
Ω
‍
(
|
∇
u
|
p
-
2
∇
u
∇
v
+
μ
(
x
)
|
u
|
p
-
2
u
v
)
d
x
-
∫
Ω
‍
f
(
x
,
u
)
v
d
x
-
∫
Ω
‍
g
(
x
,
u
)
v
d
x
-
η
∫
∂
Ω
‍
|
u
|
p
-
2
u
v
d
s
,
for allu
,
v
∈
W
1
,
p
(
Ω
). It is well-known that the weak solution of (1) corresponds to the critical point of the energy functional Φ onW
1
,
p
(
Ω
).Remark 3.
Under condition (2), it is easy to check that norm (7) is equivalent to the usual one, that is, the norm defined in (7) with μ
(
x
)
≡
1.In [2], the author researched (1) (
η
=
0
) and obtained the existence of infinitely many weak solutions. Moreover, the existence of three solutions for (1) (
η
=
0
,
p
>
N
) was researched in [3] by using a three-critical-point theorem due to Ricceri [4]. Also, some authors researched and obtained the existence of infinitely many weak solution without requiring any symmetric conditions and also with discontinuous nonlinearities; see [5, 6]. Recently, this equation was studied by J.-H. Zhao and P.-H. Zhao [7] via Bartsch’s dual fountain theorem in [8] and obtained the existence of infinitely many weak solutions for (1) under the case of Remark 2. They obtained the following theorem.Theorem A.
Letf
(
x
,
t
)
+
g
(
x
,
t
)
=
μ
|
u
|
r
-
2
u
+
γ
|
u
|
s
-
2
u, where1
<
r
<
p
<
s
<
p
*. Then there exists a constantΛ
>
0such that, for anyη
<
Λ, (1)
for anyγ
>
,
μ
∈
ℝ, (1) has a sequence of solutionsu
ksuch thatΦ
(
u
k
)
→
∞ask
→
∞;
(2)
for anyμ
>
0
,
γ
∈
ℝ, (1) has a sequence of solutionsv
ksuch thatΦ
(
v
k
)
→
0
-ask
→
∞.The main ingredient for the proof of the above theorem is a dual fountain theorem in [8]. It should be noted that the(
P
.
S
)or(
P
.
S
*
)condition and its variants play an important role in this theorem and its application. While the variant fountain theorem in Zou [1] does not need not the(
P
.
S
)or(
P
.
S
*
)condition, we obtain the following generalized result by using Zou’s theorem.Theorem 4.
Assume that (F1)–(F4) hold; then there exists a constantΛ
>
0such that, for anyη
<
Λ, (1) has infinitely many weak solutions{
u
k
}satisfying
(10)
Φ
(
u
k
)
⟶
0
-
a
s
k
⟶
∞
.This paper is organized as follows. In Section2, we recall some preliminary theorems and lemmas. In Section 3, we give the proof of Theorem 4.


## 2. Preliminaries

In what follows, we make use of the following notations:E(orW
1
,
p
(
Ω
)) denotes Banach space with the norm∥
·
∥;E
*denotes the conjugate space forE;L
p
(
Ω
)denotes Lebesgue space with the norm |
·
|
p; 〈
·
,
·
〉 is the dual pairing of the spacesE
* andE; we denote by→(resp., ⇀) the strong (resp., weak) convergence; c
,
c
1
,
c
2
,
… denote (possibly different) positive constants.For completeness, we first recall the variant fountain theorem in Zou [1]. LetEbe a Banach space with norm ∥
·
∥ and E
=
⊕
j
∈
N
X
j
¯ with dim
X
j
<
∞ for anyj
∈
N. SetY
k
=
⊕
j
=
0
k
X
j, Z
k
=
⊕
j
=
k
∞
X
j
¯.Theorem 5 (see [1, Theorem 2.2]).
TheC
1-functional Φ
λ
:
E
→
ℝ defined by Φ
λ
(
u
)
=
A
(
u
)
-
λ
B
(
u
),λ
∈
[
1,2
], satisfies(A1)
Φ
λ maps bounded sets to bounded sets uniformly forλ
∈
[
1,2
]; furthermore, Φ
λ
(
-
u
)
=
Φ
λ
(
u
) for all (
λ
,
u
)
∈
[
1,2
]
×
E.
(A2)
B
(
u
)
≥
0 for allu
∈
E; B
(
u
)
→
∞ as ∥
u
∥
→
∞on any finite dimensional subspace ofE.
(A3)
There existsρ
k
>
r
k
>
0 such that
(11)
a
k
(
λ
)
:
=
inf
u
∈
Z
k
,
∥
u
∥
=
ρ
k
Φ
λ
(
u
)
≥
0
>
b
k
(
λ
)
:
=
max
u
∈
Y
k
,
∥
u
∥
=
r
k
Φ
λ
(
u
)
;for allλ
∈
[
1,2
],
(12)
d
k
(
λ
)
:
=
inf
u
∈
Z
k
,
∥
u
∥
≤
ρ
k
Φ
λ
(
u
)
⟶
0
a
s
k
⟶
∞
u
n
i
f
o
r
m
l
y
f
o
r
λ
∈
[
1,2
]
.Then there existλ
n
→
1, u
(
λ
n
)
∈
Y
n, such that
(13)
Φ
′
λ
n
∣
Y
n
(
u
(
λ
n
)
)
=
0
,
Φ
λ
n
(
u
(
λ
n
)
)
⟶
c
k
∈
[
d
k
(
2
)
,
b
k
(
1
)
]
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhi
as
n
⟶
∞
.Particularly, if{
u
(
λ
n
)
} has a convergent subsequence for everyk, thenΦ
1has infinitely many nontrivial critical points {
u
k
}
∈
E
∖
{
0
} satisfying Φ
1
(
u
k
)
→
0
- as k
→
∞.Remark 6.
Obviously, the sequence{
u
(
λ
n
)
}is a(
P
.
S
*
)sequence.For our working spaceE
=
W
1
,
p
(
Ω
), E is a reflexive and separable Banach space; then there aree
j
∈
Eand e
j
*
∈
E
* such that
(14)
E
=
span
{
e
j
:
j
=
1,2
,
…
}
¯
,
E
*
=
span
{
e
j
*
:
j
=
1,2
,
…
}
¯
,
〈
e
j
*
,
e
j
〉
=
{
1
,
i
=
j
,
0
,
i
≠
j
.We writeX
j
:
=
span
{
e
j
}; thenY
k
,
Z
kcan be defined as that in the beginning of Theorem 5. ConsiderΦ
λ
:
E
→
ℝ defined by
(15)
Φ
λ
(
u
)
:
=
1
p
∥
u
∥
p
-
∫
Ω
‍
G
(
x
,
u
)
d
x
-
η
p
∫
∂
Ω
‍
|
u
|
p
d
s
-
λ
∫
Ω
‍
F
(
x
,
u
)
d
x
:
=
A
(
u
)
-
λ
B
(
u
)
,
λ
∈
[
1,2
]
.ThenB
(
u
)
≥
0for allu
∈
E;B
(
u
)
→
∞as∥
u
∥
→
∞on any finite dimensional subspace ofE;Φ
λ
(
-
u
)
=
Φ
λ
(
u
)for allλ
∈
[
1,2
]
,
u
∈
E
.We need the following lemmas.Lemma 7 (see [7, Lemma 3.5]).
If1
≤
q
<
p
*, then one has
(16)
β
k
:
=
sup
u
∈
Z
k
,
∥
u
∥
=
1
|
u
|
q
⟶
0
a
s
k
⟶
∞
.


## 3. Proof of Theorem4

First, we check the condition of Theorem5.Lemma 8.
Assume (F1)–(F3); then (A1)–(A3) hold.Proof.
(A1) and (A2) are obvious. Letn
>
k
>
2; we assume that σ
≤
δand define
(17)
β
k
(
σ
)
:
=
sup
u
∈
Z
k
,
∥
u
∥
=
1
|
u
|
σ
,
β
k
(
δ
)
:
=
sup
u
∈
Z
k
,
∥
u
∥
=
1
|
u
|
δ
.
Observe that(18)
|
u
|
σ
≤
β
k
(
σ
)
∥
u
∥
,
|
u
|
δ
≤
β
k
(
δ
)
∥
u
∥
,
for any u
∈
Z
k. Note that q
<
p
*; there exists a constant c
>
0 such that
(19)
|
u
|
q
≤
c
∥
u
∥
.
By theSobolevtrace imbedding inequality, we have
(20)
|
u
|
L
p
(
∂
Ω
)
p
≤
K
∥
u
∥
p
.
Then we takeΛ
*
=
1
/
4
Ksuch that, for allη
<
Λ
*,
(21)
η
p
|
u
|
L
p
(
∂
Ω
)
p
≤
1
4
p
∥
u
∥
p
.
By (F3), for anyɛ
>
0, there existsc
ɛsuch that
(22)
|
G
(
x
,
u
)
|
≤
ɛ
|
u
|
p
+
c
ɛ
|
u
|
q
.
Then, by (F1)–(F3) and (18)–(21), we obtain
(23)
Φ
λ
(
u
)
=
1
p
∥
u
∥
p
-
∫
Ω
‍
G
(
x
,
u
)
d
x
-
η
p
∫
∂
Ω
‍
|
u
|
p
d
s
-
λ
∫
Ω
‍
F
(
x
,
u
)
d
x
≥
3
4
p
∥
u
∥
p
-
ɛ
|
u
|
p
p
-
c
ɛ
|
u
|
q
q
-
c
|
u
|
σ
σ
-
c
|
u
|
δ
δ
≥
3
4
p
∥
u
∥
p
-
ɛ
c
∥
u
∥
p
-
c
∥
u
∥
q
-
c
β
k
(
σ
)
σ
∥
u
∥
σ
-
c
β
k
(
δ
)
δ
∥
u
∥
δ
.
Note that p
<
q; we may choose ɛ
>
0 and R
>
0 sufficiently small that
(24)
1
4
p
∥
u
∥
p
-
ɛ
c
∥
u
∥
p
-
c
∥
u
∥
q
≥
0
holds true for anyu
∈
W
1
,
p
(
Ω
)with∥
u
∥
≤
R. So we have
(25)
Φ
λ
(
u
)
≥
1
2
p
∥
u
∥
p
-
c
β
k
(
σ
)
σ
∥
u
∥
σ
-
c
β
k
(
δ
)
δ
∥
u
∥
δ
,
for any u
∈
Z
kwith ∥
u
∥
≤
R. Choosing
(26)
ρ
k
:
=
(
4
p
c
β
k
(
σ
)
σ
+
4
p
c
β
k
(
δ
)
δ
)
1
/
(
p
-
δ
)
,
by Lemma 7, forβ
k
(
σ
)
→
0,β
k
(
δ
)
→
0ask
→
∞, it follows thatρ
k
→
0ask
→
∞, so there existsk
0such that ρ
k
≤
R when k
≥
k
0. Thus, for k
≥
k
0, u
∈
Z
k, and ∥
u
∥
=
ρ
k, we have Φ
λ
(
u
)
≥
ρ
k
p
/
4
p
>
0; then a
k
(
λ
)
≥
0 for all λ
∈
[
1,2
].
On the other hand, ifu
∈
Y
kwith ∥
u
∥ being small enough, since all the norms are equivalent on the finite dimensional space andσ
<
p, thenb
k
(
λ
)
<
0for allλ
∈
[
1,2
].
Furthermore, ifu
∈
Z
k with ∥
u
∥
≤
ρ
k, k
≥
k
0, we see that
(27)
Φ
λ
(
u
)
≥
-
c
β
k
(
σ
)
σ
ρ
k
σ
-
c
β
k
(
δ
)
δ
ρ
k
δ
⟶
0
as
k
⟶
∞
.
Therefore, d
k
(
λ
)
→
0 as k
→
∞. Thus, (A3) holds.By Theorem5, we have the following lemma.Lemma 9.
There existλ
n
→
1 and u
(
λ
n
)
∈
Y
n such that
(28)
Φ
′
λ
n
|
Y
n
(
u
(
λ
n
)
)
=
0
,
Φ
λ
n
(
u
(
λ
n
)
)
⟶
c
k
∈
[
d
k
(
2
)
,
b
k
(
1
)
]
a
s
n
⟶
∞
.In order to complete our proof of Theorem4, by a standard argument (see the proof of Lemma 3.4 in Zhao [7]), we only need to show that {
u
(
λ
n
)
} is bounded.Lemma 10.
{
u
(
λ
n
)
} is bounded inW
1
,
p
(
Ω
).Proof.
SinceΦ
λ
n
′
|
Y
n
(
u
(
λ
n
)
)
=
0, then
(29)
1
-
η
∫
∂
Ω
‍
|
u
(
λ
n
)
|
p
∥
u
(
λ
n
)
∥
p
d
s
=
∫
Ω
‍
λ
n
f
(
x
,
u
(
λ
n
)
u
(
λ
n
)
)
+
g
(
x
,
u
(
λ
n
)
)
u
(
λ
n
)
∥
u
(
λ
n
)
∥
p
d
x
.
We can choose0
<
Λ
<
Λ
* and if η
<
Λ such that 1
-
η
K
>
0. If, up to a subsequence, ∥
u
(
λ
n
)
∥
→
∞ as n
→
∞, then, by (F2),
(30)
1
+
|
η
|
K
≥
∫
Ω
‍
g
(
x
,
u
(
λ
n
)
)
u
(
λ
n
)
∥
u
(
λ
n
)
∥
p
d
x
≥
1
2
(
1
-
η
K
)
,
for n is large enough. Obviously, it is a condition if (F4)(1) holds.
Otherwise, we setw
n
=
u
(
λ
n
)
/
∥
u
(
λ
n
)
∥; then, up to a subsequence,
(31)
w
n
⇀
w
in
E
,
w
n
⟶
w
in
L
t
(
Ω
)
for
1
≤
t
<
p
*
,
w
n
(
x
)
⟶
w
(
x
)
a
.
e
.
x
∈
Ω
.
If w
≠
0 in E and lim
|
u
|
→
∞
g
(
x
,
u
)
/
|
u
|
p
-
2
u
=
-
∞ in (F4)(2), then, for n is large enough, by Fatou’s Lemma, we have that
(32)
-
1
2
(
1
-
η
K
)
≥
∫
Ω
‍
-
g
(
x
,
u
(
λ
n
)
)
u
(
λ
n
)
|
u
(
λ
n
)
|
p
|
w
n
|
p
d
x
≥
c
+
∫
{
x
∈
Ω
:
w
≠
0
,
|
u
(
λ
n
)
|
≥
c
}
-
g
(
x
,
u
(
λ
n
)
)
u
(
λ
n
)
|
u
(
λ
n
)
|
p
|
w
n
|
p
d
x
⟶
∞
;
this is a contradiction. It is similar if lim
|
u
|
→
∞
g
(
x
,
u
)
/
|
u
|
p
-
2
u
=
∞ in (F4)(3). Thus, w
=
0.
Lett
n
∈
[
0,1
] such that
(33)
Φ
λ
n
(
t
n
u
(
λ
n
)
)
:
=
max
t
∈
[
0,1
]
Φ
λ
n
(
t
u
(
λ
n
)
)
.
For anyc
>
0large enough, and w
¯
n
:
=
(
2
p
c
)
1
/
p
w
n, for n is large enough, we have that
(34)
Φ
λ
n
(
t
n
u
(
λ
n
)
)
≥
Φ
λ
n
(
w
¯
n
)
=
2
c
-
∫
Ω
‍
G
(
x
,
w
¯
n
)
d
x
-
η
p
∫
∂
Ω
‍
|
w
¯
n
|
p
d
s
-
λ
n
∫
Ω
‍
F
(
x
,
w
¯
n
)
d
x
≥
c
,
which implies that lim
n
→
∞
Φ
λ
n
(
t
n
u
(
λ
n
)
)
→
∞. Obviously,
(35)
〈
Φ
λ
n
′
(
t
n
u
(
λ
n
)
)
,
t
n
u
(
λ
n
)
〉
=
0
.
It follows that(36)
∞

=

lim
n
→
∞
(
Φ
λ
n
(
t
n
u
(
λ
n
)
)

-

1
p
〈
Φ
′
λ
n
(
t
n
u
(
λ
n
)
)
,
t
n
u
(
λ
n
)
〉
)
≤
lim
n
→
∞
λ
n
∫
Ω
‍
(
1
p
f
(
x
,
t
n
u
(
λ
n
)
)
t
n
u
(
λ
n
)
-
F
(
x
,
t
n
u
(
λ
n
)
)
1
p
)
d
x
+
∫
Ω
‍
(
1
p
g
(
x
,
t
n
u
(
λ
n
)
)
t
n
u
(
λ
n
)
-
G
(
x
,
t
n
u
(
λ
n
)
)
)
d
x
.
If (F4)(2) holds, we have that(
1
/
p
)
f
(
x
,
u
)
u
-
F
(
x
,
u
) and (
1
/
p
)
g
(
x
,
u
)
u
-
G
(
x
,
u
) are decreasing inufor u is large enough. Therefore,
(37)
1
p
f
(
x
,
s
u
)
s
u
-
F
(
x
,
s
u
)
+
1
p
g
(
x
,
s
u
)
s
u
-
G
(
x
,
s
u
)
≤
c
for all s
>
0 and u
∈
ℝ; it is a contradiction.
If (F4)(3) holds, then we have that(38)
∞
≤
c
∫
Ω
‍
|
u
(
λ
n
)
|
σ
d
x
+
∫
Ω
‍
(
1
p
g
(
x
,
u
(
λ
n
)
)
u
(
λ
n
)
-
G
(
x
,
u
(
λ
n
)
)
)
d
x
,
which implies
(39)
∫
Ω
‍
(
1
p
g
(
x
,
u
(
λ
n
)
)
u
(
λ
n
)
-
G
(
x
,
u
(
λ
n
)
)
)
d
x
⟶
∞
.
On the other hand, by the property ofu
(
λ
n
), for n is large enough, since α
>
max
{
δ
,
σ
}, we have that
(40)
b
k
(
1
)
≥
λ
n
∫
Ω
‍
(
1
p
f
(
x
,
u
(
λ
n
)
)
u
(
λ
n
)
-
F
(
x
,
u
(
λ
n
)
)
)
d
x
+
∫
Ω
‍
(
1
p
g
(
x
,
u
(
λ
n
)
)
u
(
λ
n
)
-
G
(
x
,
u
(
λ
n
)
)
)
d
x
≥
1
2
∫
Ω
‍
(
1
p
g
(
x
,
u
(
λ
n
)
)
u
(
λ
n
)
-
G
(
x
,
u
(
λ
n
)
)
)
d
x
+
1
2
c
∫
Ω
‍
|
u
(
λ
n
)
|
α
d
x
-
1
2
c
∫
Ω
‍
|
u
(
λ
n
)
|
δ
d
x
-
1
2
c
∫
Ω
‍
|
u
(
λ
n
)
|
σ
d
x
≥
c
∫
Ω
‍
(
1
p
g
(
x
,
u
(
λ
n
)
)
u
(
λ
n
)
-
G
(
x
,
u
(
λ
n
)
)
)
d
x
-
c
;
this implies that ∫
Ω
‍
(
(
1
/
p
)
g
(
x
,
u
(
λ
n
)
)
u
(
λ
n
)
-
G
(
x
,
u
(
λ
n
)
)
)
d
x is bounded, which contradicts (39).
By the above arguments, we have that{
u
(
λ
n
)
} is bounded.Remark 11.
In fact, our result still holds if we consider a weaker condition than (F4)(2); that is, there isc
>
0 such that
(41)
H
(
x
,
t
)
≤
H
(
x
,
s
)
+
c
,
H
¯
(
x
,
t
)
≤
H
¯
(
x
,
s
)
+
c
for all 0
<
s
<
t or t
<
s
<
0,∀
x
∈
Ω, where H
(
x
,
t
)
=
(
1
/
p
)
f
(
x
,
t
)
t
-
F
(
x
,
t
) and H
¯
(
x
,
t
)
=
(
1
/
p
)
g
(
x
,
t
)
t
-
G
(
x
,
t
).


## Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.



---

*Source: 194310-2014-01-14.xml*