# Existence of Multiple Solutions for a Quasilinear Biharmonic Equation

Authors: Wen-Wu Pan; Cheng-En Yu
Journal: International Scholarly Research Notices
 (2014)

---

## Abstract

Using three critical points theorems, we prove the existence of at least three solutions for a quasilinear biharmonic equation.


---

## Body

## 1. Introduction

In this paper, we show the existence of at least three weak solutions for the Navier boundary value problem(1)
Δ
2
u
-
div
⁡
(
∇
u
)
=
λ
f
(
x
,
u
)
+
g
(
u
)
in
Ω
,
u
=
Δ
u
=
0
on
⁡
∂
Ω
,
where Ω
⊂
R
N
(
4
>
N
≥
1
) is a nonempty bounded open set with a sufficient smooth boundary ∂
Ω, λ
>
0, f
:
Ω
×
R
→
R is an L
1-Carathéodory function, and g
:
R
→
R is a Lipschitz continuous function with Lipschitz constant L
>
0; that is,
(2)
|
g
(
t
1
)
-
g
(
t
2
)
|
≤
L
|
t
1
-
t
2
|
for every t
1
,
t
2
∈
R and g
(
0
)
=
0.Motivated by the fact that such problems are used to describe a large class of physical phenomena, many authors looked for existence and multiplicity of solutions for forth-order nonlinear equations. For an overview on this subject, we cite the papers [1–23]. For instance, when N
=
1, in [22], Liu and Li, using Ricceri’s three critical points theorem [24], established the existence of at least three weak solutions for the following problem:
(3)
u
i
v
+
α
u
′′
+
β
u
=
λ
f
(
x
,
u
)
x
∈
(
a
,
b
)
,
u
(
a
)
=
u
(
b
)
=
u
′′
(
a
)
=
u
′′
(
b
)
=
0
,
where α, β are real constants and λ is a positive parameter and f
:
[
a
,
b
]
×
R
→
R is a L
2-Carathéodory function. Later, some authors generalized this type of equation (see [1–5, 10, 12, 15, 19]). When N
>
1, Liu and Su [21] also used Ricceri’s three critical points theorem [24] to established the existence of at least three weak solutions for the following problem:
(4)
Δ
(
|
Δ
u
|
p
-
2
Δ
u
)
-
div
⁡
(
|
∇
u
|
p
-
2
∇
u
)
=
λ
f
(
x
,
u
)
in
Ω
,
u
=
Δ
u
=
0
on
⁡
∂
Ω
,
where Ω
⊂
R
N
(
N
≥
1
) is a nonempty bounded open set with a sufficient smooth boundary ∂
Ω, p
>
max
⁡
⁡
{
1
,
N
/
2
}, λ
>
0, and f
:
Ω
×
R
→
R is an L
1-Carathéodory function. After that some authors used different critical point theorems to get one nontrivial, at least three, and infinitely many solutions (see [6–8, 16, 18]). Elliptic systems were also considered by [9, 11, 13, 14, 17, 20, 23].The goal of the present paper is to establish some new criteria for (1) to have at least three weak solutions (Theorems 4 and 5). Our analysis is mainly based on recent critical point theorems that are contained in Theorems 2 and 3. In fact, employing rather different three critical points theorems, under different assumptions on the nonlinear term f, we obtain the exact collections of λ for which (1) admits at least three weak solutions in the space W
2,2
(
Ω
)
∩
W
0
1,2
(
Ω
).A special case of our main results is the following theorem.Theorem 1.
Letg
:
R
→
R be a Lipschitz continuous function with the Lipschitz constant L
>
0 and g
(
0
)
=
0 such that L
<
1
/
K
2
m
(
Ω
), where K is defined by (17). Let f
:
R
→
R be a continuous function and put F
(
t
)
=
∫
0
t
‍
f
(
ξ
)
d
ξ for each t
∈
R. Assume that F
(
d
)
>
0 for some d
>
0 and F
(
ξ
)
≥
0 in [
0
,
d
] and
(5)
lim
⁡
⁡
inf
⁡
ξ
→
0
F
(
ξ
)
ξ
2
=
0
,
lim
⁡
⁡
sup
⁡
|
ξ
|
→
+
∞
F
(
ξ
)
ξ
2
=
0
.
Then, there is λ
*
>
0 such that for each λ
>
λ
* the problem
(6)
Δ
2
u
-
div
⁡
(
∇
u
)
=
λ
f
(
u
)
+
g
(
u
)
i
n
Ω
,
u
=
Δ
u
=
0
o
n
∂
Ω
admits at least three weak solutions.


## 2. Preliminaries

First we here recall for the reader’s convenience our main tools to prove the results. The first result has been obtained in [25] and the second one in [26].Theorem 2 (see [25, Theorem 3.1]).
LetX be a separable and reflexive real Banach space, Φ
:
X
→
R a nonnegative continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X
*, and Ψ
:
X
→
R a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Assume that there exists x
0
∈
X such that Φ
(
x
0
)
=
Ψ
(
x
0
)
=
0 and that
(7)
lim
⁡
∥
x
∥
→
+
∞
(
Φ
(
x
)
-
λ
Ψ
(
x
)
)
=
+
∞
∀
λ
∈
[
0
,
+
∞

[
.
Further, assume that there are r
>
0, x
1
∈
X, such that r
<
Φ
(
x
1
) and
(8)
sup
⁡
x
∈
Φ
-
1
(
]
-
∞
,
r
[
)
¯
w
Ψ
(
x
)
<
r
r
+
Φ
(
x
1
)
Ψ
(
x
1
)
;
here Φ
-
1
(
]

-

∞
,
r
[
)
¯
w denotes the closure of Φ
-
1
(
]
-

∞
,
r
[
) in the weak topology. Then, for each
(9)
λ
∈
Λ
1
∶
=
Φ
(
x
1
)
sup
⁡
x
∈
Φ
-
1
(
]
-
∞
,
r
[
)
¯
w
]
Φ
(
x
1
)
Ψ
(
x
1
)
-
sup
⁡
x
∈
Φ
-
1
(
]
-
∞
,
r
[
)
¯
w
Ψ
(
x
)
,
r
sup
⁡
x
∈
Φ
-
1
(
]
-
∞
,
r
[
)
¯
w
Ψ
(
x
)
[
r
sup
⁡
x
∈
Φ
-
1
(
]
-
∞
,
r
[
)
¯
w
,
the equation
(10)
Φ
′
(
u
)
-
λ
Ψ
′
(
u
)
=
0
has at least three solutions in X and, moreover, for each h
>
1, there exists an open interval
(11)
Λ
2
⊆
[
0
,
h
r
r
(
Ψ
(
x
1
)
/
Φ
(
x
1
)
)
-
sup
⁡
x
∈
Φ
-
1
(
-
∞
,
r
)
¯
w
Ψ
(
x
)
]
and a positive real number σ such that, for each λ
∈
Λ
2, (10) has at least three solutions in X whose norms are less than σ.Theorem 3 (see [26, Theorem 3.6]).
LetX be a reflexive real Banach space; let Φ
:
X
→
R be a sequentially weakly lower semicontinuous, coercive, and continuously Gâteaux differentiable whose Gâteaux derivative admits a continuous inverse on X
*, and let Ψ
:
X
→
R be a sequentially weakly upper semicontinuous and continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Assume that there exist r
∈
R and u
1
∈
X with 0
<
r
<
Φ
(
u
1
), such that (A1)
sup
⁡
u
∈
Φ
-
1
(
]
-
∞
,
r
]
)
Ψ
(
u
)
<
r
(
Ψ
(
u
1
)
/
Φ
(
u
1
)
);
(A2)
for eachλ
∈
Λ
r
∶
=
]
Φ
(
u
1
)
/
Ψ
(
u
1
)
,
r
/
sup
⁡
u
∈
Φ
-
1
(
]
-
∞
,
r
]
)
Ψ
(
u
)
[ the functional Φ
-
λ
Ψ is coercive.Then, for each λ
∈
Λ
r, the functional Φ
-
λ
Ψ has at least three distinct critical points in X.Letf
:
Ω
×
R
→
R be an L
1-Carathéodory function and let g
:
R
→
R be a Lipschitz continuous function with the Lipschitz constant L
>
0; that is,
(12)
|
g
(
t
1
)
-
g
(
t
2
)
|
≤
L
|
t
1
-
t
2
|
for every t
1
,
t
2
∈
R, and g
(
0
)
=
0.Put(13)
F
(
x
,
t
)
∶
=
∫
0
t
‍
f
(
x
,
ξ
)
d
ξ
,
G
(
t
)
∶
=
-
∫
0
t
‍
g
(
ξ
)
d
ξ
for all x
∈
Ω and t
∈
R. Denote
(14)
X
∶
=
W
2,2
(
Ω
)
∩
W
0
1,2
(
Ω
)
;
the usual norm in X is defined by
(15)
∥
u
∥
∶
=
(
∫
Ω
‍
|
Δ
u
(
x
)
|
2
+
|
∇
u
(
x
)
|
2
d
x
)
1
/
2
.
Note that X is a separable and reflexive real Banach space.We say that a functionu
∈
X is aweak solution of problem (1) if
(16)
∫
Ω
‍
Δ
u
(
x
)
Δ
v
(
x
)
d
x
+
∫
Ω
‍
∇
u
(
x
)
·
∇
v
(
x
)
d
x
=
λ
∫
Ω
‍
f
(
x
,
u
(
x
)
)
v
(
x
)
d
x
-
∫
Ω
‍
g
(
u
(
x
)
)
v
(
x
)
d
x
=
0
,
for all v
∈
X.It is well known that(
X
,
∥
·
∥
) is embedded in (
C
0
(
Ω
)
,
∥
·
∥
∞
) and
(17)
∥
u
∥
∞
≤
K
∥
u
∥
for all u
∈
X. Since N
<
4, one has K
<
+
∞.Suppose that the Lipschitz constantL
>
0 of the function g satisfies L
<
1
/
K
2
m
(
Ω
). For other basic notations and definitions, we refer the reader to [27–29].


## 3. Main Results

Our main results are the following theorems.Theorem 4.
Assume that there exist a functionw
∈
X, a positive function a
∈
L
1, and two positive constants r and γ with γ
<
p such that(A1)
∥
w
∥
2
>
2
r
/
(
1
-
K
2
L
m
(
Ω
)
);
(A2)
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
<
r
(
∫
Ω
‍
F
(
x
,
w
(
x
)
)
d
x
/
(
r
+
(
(
1
+
K
2
L
m
(
Ω
)
)
/
2
)
∥
w
∥
2
)
);
(A3)
F
(
x
,
t
)
≤
a
(
x
)
(
1
+
|
t
|
γ
) for a.e. x
∈
Ω and for all t
∈
R.Then, for each λ in
(18)
Λ
1
:
=
(
1
)
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
]
(
(
(
1
+
K
2
L
m
(
Ω
)
)
/
2
)
∥
w
∥
2
)
×
(
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
∫
Ω
‍
F
(
x
,
w
(
x
)
)
d
x
-
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
)
-
1
,
r
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
[
r
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
,
problem (1) admits at least three weak solutions in X and, moreover, for each h
>
1, there exists an open interval
(19)
Λ
2
⊆
[
-
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
)
-
1
0
,
(
h
r
)
(
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
2
r
(
1
+
K
2
L
m
(
Ω
)
)
∥
w
∥
2
∫
Ω
‍
F
(
x
,
w
(
x
)
)
d
x
-
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
)
-
1
]
and a positive real number σ such that, for each λ
∈
Λ
2, the problem (1) admits at least three weak solutions in X whose norms are less than σ.Theorem 5.
Assume that there exists a functionw
∈
X and a positive constant r such that (B1)
∥
w
∥
2
>
2
r
/
(
1
-
K
2
L
m
(
Ω
)
);
(B2)
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
/
r
<
(
2
/
(
1
+
K
2
L
m
(
Ω
)
)
)
(
∫
Ω
‍
F
(
x
,
w
(
x
)
)
d
x
/
∥
w
∥
2
);
(B3)
2
K
2
/
(
1
-
K
2
L
m
(
Ω
)
)
lim
⁡
⁡
sup
⁡
|
t
|
→
+
∞
F
(
x
,
t
)
/
t
2
<
(
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
)
/
r.Then, for each
(20)
λ
∈
r
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
]
1
+
K
2
L
m
(
Ω
)
2
∥
w
∥
2
∫
Ω
‍
F
(
x
,
w
(
x
)
)
d
x
,
r
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
[
r
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
,
problem (1) admits at least three weak solutions.Let us give particular consequences of Theorems4 and 5 for a fixed test function w. Now, fix x
0
∈
Ω and pick γ with γ
>
0 such that B
(
x
0
,
γ
)
⊂
Ω where B
(
x
0
,
γ
) denotes the ball with center at x
0 and radius of γ. Put
(21)
Q
=
∫
B
(
x
0
,
γ
)
∖
B
(
x
0
,
γ
/
2
)
‍
|
12
γ
3
|
x
-
x
0
|
l
-
24
γ
2
l
+
9
γ
l
|
x
-
x
0
|
|
2
d
x
,
R
=
π
N
/
2
Γ
(
N
/
2
)
∫
(
γ
/
2
)
2
γ
2
‍
|
12
(
N
+
1
)
γ
3
t
+
9
(
N
-
1
)
γ
1
t
-
24
N
γ
2
|
2
×
t
N
/
2
-
1
d
t
,
(22)
θ
=
K
(
R
+
Q
)
1
/
2
,wherel
=
(
∑
i
=
1
N
‍
x
i
2
)
1
/
2, |
x
-
x
0
|
=
∑
i
=
1
N
‍
(
x
i
-
x
i
0
)
2, and m
(
Ω
) denotes the volume of Ω.Corollary 6.
Assume that there exist a positive functiona
∈
L
1 and three positive constants c, d, and γ with c
<
θ
d and γ
<
2 such that Assumption (A3) in Theorem 4 holds. Furthermore, suppose that (A4)
F
(
x
,
t
)
≥
0 for a.e. x
∈
Ω
∖
B
(
x
0
,
γ
/
2
) and all t
∈
[
0
,
d
];
(A5)
∫
Ω
‍
sup
⁡
t
∈
[
-
c
,
c
]
F
(
x
,
t
)
d
x
<
(
1
-
K
2
L
m
(
Ω
)
)
c
2
(
∫
B
(
x
0
,
γ
/
2
)
‍
F
(
x
,
d
)
d
x
/
(
(
1
-
K
2
L
m
(
Ω
)
)
c
2
+
(
1
+
K
2
L
m
(
Ω
)
)
(
θ
d
)
2
)
).Then, for each λ in
(23)
Λ
1
′
∶
=
1
-
K
2
L
m
(
Ω
)
2
c
2
2
K
2
∫
Ω
‍
sup
⁡
t
∈
[
-
c
,
c
]
F
(
x
,
t
)
d
x
]
1
+
K
2
L
m
(
Ω
)
2
×
(
θ
d
)
2
(
-
∫
Ω
‍
sup
⁡
t
∈
[
-
c
,
c
]
F
(
x
,
t
)
d
x
)
2
K
2
(
∫
B
(
x
0
,
γ
/
2
)
‍
F
(
x
,
d
)
d
x
-
∫
Ω
‍
sup
⁡
t
∈
[
-
c
,
c
]
F
(
x
,
t
)
d
x
)
)
-
1
,
1
-
K
2
L
m
(
Ω
)
2
c
2
2
K
2
∫
Ω
‍
sup
⁡
t
∈
[
-
c
,
c
]
F
(
x
,
t
)
d
x
[
1
-
K
2
L
m
(
Ω
)
2
c
2
2
K
2
∫
Ω
‍
sup
⁡
t
∈
[
-
c
,
c
]
F
(
x
,
t
)
d
x
,
problem (1) admits at least three weak solutions in X and, moreover, for each h
>
1, there exist an open interval
(24)
Λ
2
′
⊆
[
0
,
1
-
K
2
L
m
(
Ω
)
∫
Ω
‍
sup
⁡
t
∈
[
-
c
,
c
]
2
×
(
h
/
2
)
(
c
/
K
)
2
(
c
/
θ
d
)
2
∫
B
(
x
0
,
γ
/
2
)
‍
F
(
x
,
d
)
d
x

-

∫
Ω
‍
sup
⁡
t
∈
[
-
c
,
c
]
F
(
x
,
t
)
d
x

]
and a positive real number σ such that, for each λ
∈
Λ
2
′, problem (1) admits at least three weak solutions in X whose norms are less than σ.Proof.
We claim that all the assumptions of Theorem4 are fulfilled with w given by
(25)
w
(
x
)
=
{
0
for
x
∈
Ω
∖
B
(
x
0
,
γ
)
,
d
(
4
γ
3
|
x
-
x
0
|
3
-
12
γ
2
|
x
-
x
0
|
2
a
a
a
+
9
γ
|
x
-
x
0
|
-
1
)
for
x
∈
B
(
x
0
,
γ
)
∖
B
(
x
0
,
γ
/
2
)
,
d
for
x
∈
B
(
x
0
,
γ
/
2
)
,
and r
∶
=
(
(
1
-
K
2
L
m
(
Ω
)
)
/
2
)
(
c
/
K
)
2. It is easy to verify that w
∈
W
2,2
(
Ω
)
∩
W
0
1,2
(
Ω
), and, in particular, one has
(26)
∥
w
∥
2
=
(
R
+
Q
)
d
2
,
and consequently from (22) we see that
(27)
∥
w
∥
=
θ
d
K
.
Thus, (A1) holds. Since 0
≤
w
(
x
)
≤
d for each x
∈
Ω, the condition (A4) ensures that
(28)
∫
Ω
∖
B
(
x
0
,
γ
)
‍
F
(
x
,
w
(
x
)
)
d
x
+
∫
B
(
x
0
,
γ
)
∖
B
(
x
0
,
γ
/
2
)
‍
F
(
x
,
w
(
x
)
)
d
x
≥
0
,
so, from (A5),
(29)
∫
Ω
‍
sup
⁡
t
∈
[
-
c
,
c
]
F
(
x
,
t
)
d
x
<
(
1
-
K
2
L
m
(
Ω
)
)
c
2
×
∫
B
(
x
0
,
γ
/
2
)
‍
F
(
x
,
d
)
d
x
(
1
-
K
2
L
m
(
Ω
)
)
c
2
+
(
1
+
K
2
L
m
(
Ω
)
)
(
θ
d
)
2
=
(
1
-
K
2
L
m
(
Ω
)
)
c
2
1
2
K
2
×
(
(
∫
B
(
x
0
,
γ
/
2
)
‍
F
(
x
,
d
)
d
x
)
×
(
1
2
K
2
(
1
-
K
2
L
m
(
Ω
)
)
c
2
+
1
2
K
2
(
1
+
K
2
L
m
(
Ω
)
)
(
θ
d
)
2
)
-
1
)
=
r
∫
Ω
‍
F
(
x
,
w
(
x
)
)
d
x
r
+
(
(
1
+
K
2
L
m
(
Ω
)
)
/
2
)
∥
w
∥
2
,
and thus (A2) holds. Next notice that
(30)
(
(
1
+
K
2
L
m
(
Ω
)
)
/
2
)
∥
w
∥
2
∫
Ω
‍
F
(
x
,
w
(
x
)
)
d
x
-
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
≤
(
1
+
K
2
L
m
(
Ω
)
(
θ
d
)
2
)
/
2
K
2
∫
B
(
x
0
,
γ
/
2
)
‍
F
(
x
,
d
)
d
x
-
∫
Ω
‍
sup
⁡
t
∈
[
-
c
,
c
]
F
(
x
,
t
)
d
x
,
r
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
=
(
1
-
K
2
L
m
(
Ω
)
)
c
2
2
K
2
∫
Ω
‍
sup
⁡
t
∈
[
-
c
,
c
]
F
(
x
,
t
)
d
x
.
In addition note that
(31)
(
1
+
K
2
L
m
(
Ω
)
(
θ
d
)
2
)
/
2
K
2
∫
B
(
x
0
,
γ
/
2
)
‍
F
(
x
,
d
)
d
x
-
∫
Ω
‍
sup
⁡
t
∈
[
-
c
,
c
]
F
(
x
,
t
)
d
x
<
(
1
+
K
2
L
m
(
Ω
)
d
(
θ
)
2
2
K
2
)
×
(
(
(
1
-
K
2
L
m
(
Ω
)
)
c
2
(
1
-
K
2
L
m
(
Ω
)
)
c
2
+
(
1
+
K
2
L
m
(
Ω
)
)
(
θ
d
)
2
-
1
)
×
∫
Ω
‍
sup
⁡
t
∈
[
-
c
,
c
]
F
(
x
,
t
)
d
x
(
(
1
-
K
2
L
m
(
Ω
)
)
c
2
(
1
-
K
2
L
m
(
Ω
)
)
c
2
+
(
1
+
K
2
L
m
(
Ω
)
)
(
θ
d
)
2
-
1
)
)
-
1
=
(
1
-
K
2
L
m
(
Ω
)
)
c
2
2
K
2
∫
Ω
‍
sup
⁡
t
∈
[
-
c
,
c
]
F
(
x
,
t
)
d
x
.
Finally note that
(32)
(
h
r
)
(
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
2
r
(
1
+
K
2
L
m
(
Ω
)
)
∥
w
∥
2
∫
Ω
‍
F
(
x
,
w
(
x
)
)
d
x
-
∫
Ω
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
)
-
1
≤
1
-
K
2
L
m
(
Ω
)
2
×
(
h
2
(
c
K
)
2
)
(
∫
Ω
‍
sup
⁡
t
∈
[
-
c
,
c
]
(
c
θ
d
)
2
∫
B
(
x
0
,
γ
/
2
)
‍
F
(
x
,
d
)
d
x
-
∫
Ω
‍
sup
⁡
t
∈
[
-
c
,
c
]
F
(
x
,
t
)
d
x
)
-
1
,
and, taking into account that Λ
1
′
⊆
Λ
1 and Λ
2
⊆
Λ
2
′, we have the desired conclusion directly from Theorem 4.Corollary 7.
Assume that there exist two positive constantsc and d with c
<
θ
d such that the assumption (A4) in Corollary 6 holds. Furthermore, suppose that (B4)
∫
Ω
‍
sup
⁡
t
∈
[
-
c
,
c
]
F
(
x
,
t
)
d
x
/
(
1
-
K
2
L
m
(
Ω
)
)
c
2
<
∫
B
(
x
0
,
γ
/
2
)
‍
F
(
x
,
d
)
d
x
/
(
1
+
K
2
L
m
(
Ω
)
)
(
θ
d
)
2;
(B5)
limsup
⁡
|
t
|
→
+
∞
F
(
x
,
t
)
/
t
2
<
∫
Ω
‍
sup
⁡
t
∈
[
-
c
,
c
]
F
(
x
,
t
)
d
x
/
c
2.Then, for each
(33)
λ
∈
(
θ
d
)
2
∫
B
(
x
0
,
γ
/
2
)
‍
F
(
x
,
d
)
d
x
]
1
+
K
2
L
m
(
Ω
)
2
1
2
K
2
(
θ
d
)
2
∫
B
(
x
0
,
γ
/
2
)
‍
F
(
x
,
d
)
d
x
,
1
-
K
2
L
m
(
Ω
)
2
c
2
2
K
2
∫
Ω
‍
sup
⁡
t
∈
[
-
c
,
c
]
F
(
x
,
t
)
d
x
[
(
θ
d
)
2
∫
B
(
x
0
,
γ
/
2
)
‍
F
(
x
,
d
)
d
x
,
problem (1) admits at least three weak solutions.Proof.
All the assumptions of Theorem5 are fulfilled by choosing w as given in (25) and r
∶
=
(
(
1
-
K
2
L
m
(
Ω
)
)
/
2
)
(
c
/
K
)
2 and bearing in mind that
(34)
∥
w
∥
=
θ
d
K
and recalling
(35)
∫
Ω
∖
B
(
x
0
,
γ
)
‍
F
(
x
,
w
(
x
)
)
d
x
+
∫
B
(
x
0
,
γ
)
∖
B
(
x
0
,
γ
/
2
)
‍
F
(
x
,
w
(
x
)
)
d
x
≥
0
.
Hence, by applying Theorem 5, we have the conclusion.Proof of Theorem1.
Fixλ
>
λ
*
∶
=
m
(
Ω
)
(
1
+
K
2
L
m
(
Ω
)
)
(
θ
d
)
2
/
m
(
B
(
x
0
,
γ
)
)
F
(
d
) for some d
>
0. Since
(36)
lim
⁡
⁡
inf
⁡
ξ
→
0
F
(
ξ
)
ξ
2
=
0
,
there is {
c
m
}
m
∈
N
⊆
]
0
,
+
∞
[ such that lim
⁡
m
→
+
∞
c
m
=
0 and
(37)
lim
⁡
m
→
+
∞
sup
⁡
|
ξ
|
≤
c
m
F
(
ξ
)
c
m
=
0
.
In fact, one has
(38)
lim
⁡
m
→
+
∞
sup
⁡
|
ξ
|
≤
c
m
F
(
ξ
)
c
m
=
lim
⁡
m
→
+
∞
F
(
ξ
c
m
)
ξ
c
m
2
·
ξ
c
m
2
c
m
=
0
,
where F
(
ξ
c
m
)
=
sup
⁡
|
ξ
|
≤
c
m
F
(
ξ
). Hence, there is c
¯
>
0 such that
(39)
sup
⁡
|
ξ
|
≤
c
¯
F
(
ξ
)
c
¯
2
<
min
⁡
{
m
(
B
(
x
0
,
γ
)
)
(
1
-
K
p
L
m
(
Ω
)
)
F
(
d
)
m
(
Ω
)
(
1
+
K
2
L
m
(
Ω
)
)
(
θ
d
)
2
,
(
1
-
K
2
L
m
(
Ω
)
)
λ
}
and c
¯
<
θ
d. From Corollary 7 we have the desired conclusion.


## 4. Proofs

Proof of Theorem4.
Our aim is to apply Theorem2 to our problem. To this end, for each u
∈
X, we let the functionals Φ
,
Ψ
:
X
→
R be defined by
(40)
Φ
(
u
)
∶
=
1
2
∥
u
∥
2
+
∫
Ω
‍
G
(
u
(
x
)
)
d
x
,
Ψ
(
u
)
∶
=
∫
Ω
‍
F
(
x
,
u
(
x
)
)
d
x
,
and put
(41)
I
λ
(
u
)
∶
=
Φ
(
u
)
-
λ
Ψ
(
u
)
∀
u
∈
X
.
The functionals Φ and Ψ satisfy the regularity assumptions of Theorem 2. Indeed, by standard arguments, we have that Φ is Gâteaux differentiable and sequentially weakly lower semicontinuous and its Gâteaux derivative at the point u
∈
X is the functional Φ
′
(
u
)
∈
X
*, given by
(42)
Φ
′
(
u
)
(
v
)
=
∫
Ω
‍
Δ
u
(
x
)
Δ
v
(
x
)
d
x
+
∫
Ω
‍
∇
u
(
x
)
·
∇
v
(
x
)
d
x
-
∫
Ω
‍
g
(
u
(
x
)
)
v
(
x
)
d
x
,
for every v
∈
X. Furthermore, the differential Φ
′
:
X
→
X
* is a Lipschitzian operator. Indeed, for any u
,
v
∈
X, there holds
(43)
∥
Φ
′
(
u
)
-
Φ
′
(
v
)
∥
X
*
=
sup
⁡
∥
w
∥
≤
1
|
(
Φ
′
(
u
)
-
Φ
′
(
v
)
,
w
)
|
≤
sup
⁡
∥
w
∥
≤
1
|
(
u
-
v
,
w
)
|
+
sup
⁡
∥
w
∥
≤
1
∫
Ω
‍
|
g
(
u
(
x
)
)
-
g
(
v
(
x
)
)
|
|
w
(
x
)
|
d
x
≤
sup
⁡
∥
w
∥
≤
1
∥
u
-
v
∥
∥
w
∥
+
sup
⁡
∥
w
∥
≤
1
(
∫
Ω
‍
|
g
(
u
(
x
)
)
-
g
(
v
(
x
)
)
|
2
)
1
/
2
×
(
∫
Ω
‍
|
w
(
x
)
|
2
)
1
/
2
.
Recalling that g is Lipschitz continuous and the embedding X
↪
L
2
(
Ω
) is compact, the claim is true. In particular, we derive that Φ is continuously differentiable. The inequality (17) yields for any u
,
v
∈
X the estimate
(44)
(
Φ
′
(
u
)
-
Φ
′
(
v
)
,
u
-
v
)
=
(
u
-
v
,
u
-
v
)
-
∫
Ω
‍
(
g
(
u
(
x
)
)
-
g
(
v
(
x
)
)
)
×
(
u
(
x
)
-
v
(
x
)
)
d
x
≥
(
1
-
K
2
L
m
(
Ω
)
)
∥
u
-
v
∥
2
.
By the assumption L
<
1
/
K
2
m
(
Ω
), it turns out that Φ
′ is a strongly monotone operator. So, by applying Minty-Browder theorem [29, Theorem 26
.
A],    Φ
′
:
X
→
X
* admits a Lipschitz continuous inverse. On the other hand, the fact that X is compactly embedded into C
0
(
Ω
) implies that the functional Ψ is well defined, continuously Gâteaux differentiable, and with compact derivative, whose Gateaux derivative at the point u
∈
X is given by
(45)
Ψ
′
(
u
)
(
v
)
=
∫
Ω
‍
f
(
x
,
u
(
x
)
)
v
(
x
)
d
x
,
for every v
∈
X. Note that the weak solutions of (1) are exactly the critical points of I
λ. Also, since g is Lipschitz continuous and satisfies g
(
0
)
=
0, we have from (17) that
(46)
1
-
K
2
L
m
(
Ω
)
2
∥
u
∥
2
≤
Φ
(
u
)
≤
1
+
K
2
L
m
(
Ω
)
2
∥
u
∥
2
,
for all u
∈
X, and so Φ is coercive.
Furthermore from (A3) for any fixedλ
∈
[
0
,
+
∞
[, using (46), taking (17) into account, we have
(47)
Φ
(
u
)
-
λ
Ψ
(
u
)
=
1
2
∥
u
∥
2
+
∫
Ω
‍
G
(
u
(
x
)
)
d
x
-
λ
∫
Ω
‍
F
(
x
,
u
(
x
)
)
d
x
≥
1
-
K
2
L
m
(
Ω
)
2
∥
u
∥
2
-
λ
∫
Ω
‍
a
(
x
)
(
1
+
|
u
(
x
)
|
γ
)
d
x
≥
1
-
K
2
L
m
(
Ω
)
2
∥
u
∥
2
-
λ
∥
a
∥
L
1
(
Ω
)
(
m
(
Ω
)
+
K
γ
∥
u
∥
γ
)
,
and so
(48)
lim
⁡
∥
u
∥
→
+
∞
(
Φ
(
u
)
-
λ
Ψ
(
u
)
)
=
+
∞
.
Also according to (A1) we achieve Φ
(
w
)
>
r. From the definition of Φ and by using (46) we have
(49)
Φ
-
1
(
]
-

∞
,
r
[
)
=
{
u
∈
X
:
Φ
(
u
)
<
r
}
⊆
{
u
∈
X
:
∥
u
∥
<
2
r
1
-
L
K
2
m
(
Ω
)
}
⊆
{
u
∈
X
:
|
u
(
x
)
|
<
K
2
r
1
-
L
K
2
m
(
Ω
)
∀
x
∈
Ω
}
.
So, we obtain
(50)
sup
⁡
u
∈
Φ
-
1
(
]
-
∞
,
r
[
)
¯
w
Ψ
(
u
)
≤
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
.
Therefore, from (A2) and (46), we have
(51)
sup
⁡
u
∈
Φ
-
1
(
]
-
∞
,
r
[
)
¯
w
Ψ
(
u
)
≤
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
<
r
r
+
(
(
1
+
K
2
L
m
(
Ω
)
)
/
2
)
∥
w
∥
2
∫
Ω
‍
F
(
x
,
w
(
x
)
)
d
x
<
r
r
+
Φ
(
w
)
Ψ
(
w
)
.
Now, we can apply Theorem 2. Note that, for each x
∈
Ω,
(52)
Φ
(
w
)
Ψ
(
w
)
-
sup
⁡
u
∈
Φ
-
1
(
]
-
∞
,
r
[
)
¯
w
Ψ
(
u
)
≤
(
1
+
K
2
L
m
(
Ω
)
2
∥
w
∥
2
)
×
(
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
∫
Ω
‍
F
(
x
,
w
(
x
)
)
d
x
-
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
)
-
1
,
r
sup
⁡
u
∈
Φ
-
1
(
]
-
∞
,
r
[
)
¯
w
Ψ
(
u
)
≥
r
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
.
Note also that (A2) implies
(53)
(
(
1
+
K
2
L
m
(
Ω
)
)
/
2
)
∥
w
∥
2
∫
Ω
‍
F
(
x
,
w
(
x
)
)
d
x
-
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
<
(
1
+
K
2
L
m
(
Ω
)
2
∥
w
∥
2
)
×
(
sup
⁡
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
(
r
+
(
(
1
+
K
2
L
m
(
Ω
)
)
/
2
)
∥
w
∥
2
r
-
1
)
×
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
)
-
1
=
r
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
.
Also,
(54)
h
r
r
(
Ψ
(
w
)
/
Φ
(
w
)
)
-
sup
⁡
u
∈
Φ
-
1
(
-
∞
,
r
)
¯
w
Ψ
(
u
)
≤
(
h
r
)
(
∫
Ω
sup
⁡
2
r
/
1
-
L
K
2
m
(
Ω
)
2
r
(
1
+
K
2
L
m
(
Ω
)
)
∥
w
∥
2
∫
Ω
‍
F
(
x
,
w
(
x
)
)
d
x
-
∫
Ω
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
)
-
1
=
ρ
.
From (A2) it follows that
(55)
2
r
(
1
+
K
2
L
m
(
Ω
)
)
∥
w
∥
2
∫
Ω
‍
F
(
x
,
w
(
x
)
)
d
x
-
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
>
(
2
r
(
1
+
K
2
L
m
(
Ω
)
)
∥
w
∥
2
-
r
r
+
(
(
1
+
K
2
L
m
(
Ω
)
)
/
2
)
∥
w
∥
2
)
×
∫
Ω
‍
F
(
x
,
w
(
x
)
)
d
x
≥
(
2
r
(
1
+
K
2
L
m
(
Ω
)
)
∥
w
∥
2
-
2
r
(
1
+
K
2
L
m
(
Ω
)
)
∥
w
∥
2
)
×
∫
Ω
‍
F
(
x
,
w
(
x
)
)
d
x
=
0
,
since ∫
Ω
‍
F
(
x
,
w
(
x
)
)
d
x
≥
0 (note F
(
x
,
0
)
=
0 so ∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
≥
0 and now apply (A2)). Now with x
0
=
0 and x
1
=
w from Theorem 2 (note Ψ
(
0
)
=
0) it follows that, for each λ
∈
Λ
1, the problem (1) admits at least three weak solutions and there exist an open interval Λ
2
⊆
[
0
,
ρ
] and a real positive number σ such that, for each λ
∈
Λ
2, the problem (1) admits at least three weak solutions whose norms in X are less than σ. Thus, the conclusion is achieved.Proof of Theorem5.
To apply Theorem3 to our problem, we take the functionals Φ
,
Ψ
:
X
→
R as given in the proof of Theorem 4. Let us prove that the functionals Φ and Ψ satisfy the conditions required in Theorem 3. The regularity assumptions on Φ and Ψ, as requested in Theorem 3, hold. According to (B1) we deduce Φ
(
w
)
>
r. From the definition of Φ we have
(56)
Φ
-
1
(
]
-
∞
,
r
[
)
⊆
{
u
∈
X
:
|
u
(
x
)
|
<
K
2
r
1
-
L
K
2
m
(
Ω
)
∀
x
∈
Ω
}
,
and it follows that
(57)
sup
⁡
u
∈
Φ
-
1
(
]
-
∞
,
r
[
)
Ψ
(
u
)
≤
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
.
Therefore, due to assumption (B2), we have
(58)
sup
⁡
u
∈
Φ
-
1
(
]
-
∞
,
r
[
)
Ψ
(
u
)
r
≤
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
r
<
2
1
+
K
2
L
m
(
Ω
)
∫
Ω
‍
F
(
x
,
w
(
x
)
)
d
x
∥
w
∥
p
≤
Ψ
(
w
)
Φ
(
w
)
.
Furthermore, from (B3) there exist two constants η
,
ϑ
∈
R with
(59)
η
<
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
r
such that
(60)
2
K
2
1
-
K
2
L
m
(
Ω
)
F
(
x
,
t
)
≤
η
t
2
+
ϑ
for all x
∈
Ω and all t
∈
R. Fix u
∈
X. Then
(61)
F
(
x
,
u
(
x
)
)
≤
1
-
K
2
L
m
(
Ω
)
2
K
2
(
η
|
u
(
x
)
|
2
+
ϑ
)
for all x
∈
Ω. Now, to prove the coercivity of the functional Φ
-
λ
Ψ, first we assume that η
>
0. So, for any fixed
(62)
λ
∈
r
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
]
1
+
K
2
L
m
(
Ω
)
2
∥
w
∥
2
∫
Ω
‍
F
(
x
,
w
(
x
)
)
d
x
,
r
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
[
r
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
,
using (61), we have
(63)
Φ
(
u
)
-
λ
Ψ
(
u
)
=
1
2
∥
u
∥
2
+
∫
Ω
‍
G
(
u
(
x
)
)
d
x
-
λ
∫
Ω
‍
F
(
x
,
u
(
x
)
)
d
x
≥
1
-
K
2
L
m
(
Ω
)
2
∥
u
∥
2
-
λ
1
-
K
2
L
m
(
Ω
)
2
K
2
×
(
η
∫
Ω
‍
|
u
(
x
)
|
2
d
x
+
ϑ
)
≥
1
-
K
2
L
m
(
Ω
)
2
×
(
1
-
η
r
∫
Ω
‍
sup
⁡
|
t
|
≤
K
2
r
/
(
1
-
L
K
2
m
(
Ω
)
)
F
(
x
,
t
)
d
x
)
×
∥
u
∥
2
-
1
-
K
2
L
m
(
Ω
)
2
ϑ
,
and thus
(64)
lim
⁡
∥
u
∥
→
+
∞
(
Φ
(
u
)
-
λ
Ψ
(
u
)
)
=
+
∞
.
On the other hand, if η
≤
0, clearly we obtain lim
⁡
∥
u
∥
→
+
∞
(
Φ
(
u
)
-
λ
Ψ
(
u
)
)
=
+
∞. Both cases lead to the coercivity of functional Φ
-
λ
Ψ.
So, assumptions (A1) and (A2) in Theorem3 are satisfied. Hence, by using Theorem 3, the problem (1) admits at least three distinct weak solutions in X.



---

*Source: 370494-2014-10-29.xml*