# Multiple Solutions for Nonlinear Navier Boundary Systems Involving(

Authors: Qing Miao
Journal: Discrete Dynamics in Nature and Society
 (2016)

---

## Abstract

We improve some results on the existence and multiplicity of solutions for the(
p
1
(
x
)
,
…
,
p
n
(
x
)
)-biharmonic system. Our main results are new. Our approach is based on general variational principle and the theory of the variable exponent Sobolev spaces.


---

## Body

## 1. Introduction

In this paper, we consider the existence of solutions for the following system:(1)
Δ
Δ
u
i
p
i
x
-
2
Δ
u
i
=
λ
F
u
i
x
,
u
1
,
u
2
,
…
,
u
n
i
n
Ω
,
u
i
=
Δ
u
i
=
0
o
n
∂
Ω
,for 1
≤
i
≤
n, where Ω
⊂
R
N
(
N
≥
2
) is a bounded domain with smooth boundary ∂
Ω. λ is a positive parameter and F
:
Ω
×
R
n
→
R is a function such that the mapping (
t
1
,
t
2
,
…
,
t
n
)
→
F
(
x
,
t
1
,
t
2
,
…
,
t
n
) is in C
1 in R
n for all x
∈
Ω, F
t
i denotes the partial derivative of F with respect to t
i, and F
t
i is continuous in Ω
×
R
n, for i
=
1,2
,
…
,
n. p
i
(
x
)
∈
C
(
Ω
¯
)
(
i
=
1,2
,
…
,
n
) with N
/
2
<
p
i
-
≔
i
n
f
x
∈
Ω
¯
p
i
(
x
)
≤
p
i
+
≔
s
u
p
x
∈
Ω
¯
p
i
(
x
)
<
+
∞.In recent years, many authors considered the existence and multiplicity of solutions for some fourth order problems [1–10]. In [4], based on critical point theory, the existence of infinitely many solutions has been established for a class of nonlinear elliptic equations involving the p-biharmonic operator and under Navier boundary value conditions. The p
(
x
)-Laplacian operator is more complicated nonlinearities than p-Laplacian; it is inhomogeneous and usually it does not have the so-called first eigenvalue, since the infimum of its principle eigenvalue is zero. In [11], based on variational methods, the authors established the existence of an unbounded sequence of weak solutions for a class of differential equations with p
(
x
)-Laplacian. In [12], when the nonlinearity f has the subcritical growth and via variational methods [13], the authors obtained the existence of at least one, two, or three weak solutions for a class of differential equations with p
(
x
)-Laplacian whenever the parameter λ belongs to a precise positive interval. Recently, the p
(
x
)-biharmonic problems have attracted more and more attention; we refer the reader to [11, 14–21]. In [16], El Amrouss and Ourraoui studied the p
(
x
)-biharmonic equation with Navier and Neumann boundary condition; the technical approach is based on Ricceri’s variational principle and local mountain pass theorem, without Palais-Smale condition. In [20], the authors established the existence of at least three solutions for elliptic systems involving the (
p
x
,
q
x
)-biharmonic operator. In [15], Allaoui et al. considered the existence of infinitely many solutions for the (
p
(
x
)
,
q
(
x
)
)-biharmonic problem by a general Ricceri’s variational principle. However, there are rare results on (
p
1
x
,
…
,
p
n
(
x
)
)-biharmonic problem.Inspired by the aforementioned papers, our objective is to prove the existence and multiplicity solutions for problem (1); we study problem (1) by using the results as follows.Theorem A (see [13, 22]).
LetX be a reflexive real Banach space; Φ
:
X
→
R is a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X
∗; Ψ
:
X
→
R is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that (2)
Φ
0
=
Ψ
0
=
0
.Assume that there exist r
>
0 and u
¯
∈
X, with r
<
Φ
(
u
¯
), such that(i)
s
u
p
Φ
(
u
)
≤
r
Ψ
(
u
)
/
r
<
Ψ
(
u
¯
)
/
Φ
(
u
¯
);
(ii)
for eachλ
∈
Λ
r
≔
(
Φ
(
u
¯
)
/
Ψ
(
u
¯
)
,
r
/
s
u
p
Φ
(
u
)
≤
r
Ψ
(
u
)
), the functional Φ
-
λ
Ψ is coercive.
Then, for each compact interval[
α
,
β
]
⊆
Λ
r, there exists ρ
>
0 with the following property: for every λ
∈
[
α
,
β
], the equation (3)
Φ
′
u
-
λ
Ψ
′
u
=
0has at least three solutions in X whose norms are less than ρ.Theorem B (see [23]).
LetX be a reflexive real Banach space; Φ
,
Ψ
:
X
→
R are two Gâteaux differentiable functionals such that Φ is sequentially weakly lower semicontinuous and coercive and Ψ is sequentially weakly upper semicontinuous. For every r
>
i
n
f
X
Φ, let one put (4)
φ
r
≔
inf
u
∈
Φ
-
1
-
∞
,
r
⁡
sup
v
∈
Φ
-
1
-
∞
,
r
⁡
Ψ
v
-
Ψ
u
r
-
Φ
u
,
γ
≔
liminf
r
→
+
∞
⁡
φ
r
,
δ
≔
liminf
r
→
inf
X
⁡
Φ
+
⁡
φ
r
.Then, one has the following:(a)
For everyr
>
i
n
f
X
Φ and every λ
∈
(
0
,
1
/
φ
(
r
)
), the restriction of the functional I
λ
=
Φ
-
λ
Ψ to Φ
-
1
(
(
-
∞
,
r
)
) admits a global minimum, which is a critical point (local minimum) of I
λ in X.
(b)
Ifγ
<
+
∞, then, for each λ
∈
(
0
,
1
/
γ
), the following alternative holds: either
(b1)
I
λ possesses a global minimum, or
(b2)
there is a sequence{
u
n
} of critical points (local minima) of I
λ such that l
i
m
n
→
+
∞
Φ
(
u
n
)
=
+
∞.
(c)
Ifδ
<
+
∞, then, for each λ
∈
(
0
,
1
/
δ
), the following alternative holds: either
(c1)
there is a global minimum ofΦ which is a local minimum of I
λ, or
(c2)
there is a sequence of pairwise distinct critical points local minima ofI
λ which weakly converges to a global minimum of Φ.This paper is organized as follows. In Section2, we recall some basic facts about the variable exponent Lebesgue and Sobolev spaces, some important properties of the p
(
x
)-biharmonic operator. In Section 3, we establish the main results.


## 2. Preliminaries

In order to deal with thep
(
x
)-biharmonic problem, we need some theories on spaces L
p
(
x
)
(
Ω
), W
m
,
p
(
x
)
(
Ω
) and introduce some notations used in the following.Denote(5)
C
+
Ω
¯
=
h
∈
C
Ω
¯
;
h
x
>
1
,
∀
x
∈
Ω
¯
,
L
p
x
Ω
=
u
:
u
i
s
a
m
e
a
s
u
r
a
b
l
e
r
e
a
l
-
v
a
l
u
e
d
f
u
n
c
t
i
o
n
,
∫
Ω
u
p
x
d
x
<
∞
.We introduce a norm onL
p
(
x
)
(
Ω
): (6)
u
p
x
=
inf
⁡
λ
>
0
:
∫
Ω
u
x
λ
p
x
d
x
≤
1
.Then (
L
p
(
x
)
(
Ω
)
,
|
·
|
p
(
x
)
) becomes a Banach space; we call it a generalized Lebesgue space.Proposition 1 (see [24]).
The conjugate space ofL
p
(
x
)
(
Ω
) is L
p
0
(
x
)
(
Ω
), where 1
/
p
(
x
)
+
1
/
p
0
(
x
)
=
1. For any u
∈
L
p
(
x
)
(
Ω
) and v
∈
L
p
0
(
x
)
(
Ω
), one has the following Hölder-type inequality: (7)
∫
Ω
u
v
d
x
≤
1
p
-
+
1
p
0
-
u
p
x
v
p
0
x
.
The variable exponent Sobolev spaceW
m
,
p
(
x
)
(
Ω
) is defined by(8)
W
m
,
p
x
Ω
=
u
∈
L
p
x
Ω
∣
D
α
u
∈
L
p
x
Ω
,
α
≤
m
,where α is the multi-index and |
α
| is the order, m is a positive integer, and it can be equipped with the norm (9)
u
m
,
p
x
=
∑
α
≤
m
D
α
u
p
x
.From [24], we know that spaces L
p
(
x
)
(
Ω
) and W
m
,
p
(
x
)
(
Ω
) are separable, reflexive, and uniform convex Banach spaces.
We denote byW
0
m
,
p
(
x
)
(
Ω
) the closure of C
0
∞
(
Ω
) in W
m
,
p
(
x
)
(
Ω
).
LetX
≔
∏
i
=
1
n
‍
(
W
2
,
p
i
(
x
)
(
Ω
)
∩
W
0
1
,
p
i
(
x
)
(
Ω
)
) endow with the norm (10)
u
1
,
u
2
,
…
,
u
n
=
∑
i
=
1
n
u
i
p
i
x
,where (11)
u
i
p
i
x
=
inf
⁡
λ
>
0
:
∫
Ω
Δ
u
i
λ
p
i
x
+
∇
u
i
λ
p
i
x
+
u
i
λ
p
i
x
d
x
.Remark 2.
According to [25], the norm ·
2
·
p
(
x
) is equivalent to the norm |
Δ
·
|
p
(
x
) in the space W
2
,
p
(
x
)
(
Ω
)
∩
W
0
1
,
p
(
x
)
(
Ω
). Consequently, the norms ·
2
,
p
(
x
), |
Δ
·
|
p
(
x
), and ·
p
(
x
) are equivalent.Proposition 3 (see [24]).
Putρ
(
u
)
=
∫
Ω
‍
|
Δ
u
|
p
(
x
)
d
x
,
∀
u
∈
W
2
,
p
(
x
)
(
Ω
)
∩
W
0
1
,
p
(
x
)
(
Ω
); then(1)
u
p
(
x
)
<
1
(
=
1
;
>
1
)
⇔
ρ
(
u
)
<
1
(
=
1
;
>
1
);
(2)
u
p
(
x
)
≥
1
⇒
u
p
(
x
)
p
-
≤
ρ
(
u
)
≤
u
p
(
x
)
p
+;
(3)
u
p
(
x
)
≤
1
⇒
u
p
(
x
)
p
+
≤
ρ
(
u
)
≤
u
p
(
x
)
p
-;
(4)
l
i
m
k
→
+
∞
u
k
p
(
x
)
=
0
⇔
l
i
m
k
→
+
∞
ρ
(
u
k
)
=
0.Proposition 4 (see [20, 26]).
The embeddingW
0
1
,
p
i
(
x
)
∩
W
2
,
p
i
(
x
)
↪
C
(
Ω
¯
) is compact whenever p
i
-
>
N
/
2
,
i
=
1,2
,
…
,
n. So there is a constant C
>
0 such that (12)
C
≔
max
⁡
sup
u
i
∈
W
2
,
p
i
x
Ω
∩
W
0
1
,
p
i
x
Ω
∖
0
⁡
max
x
∈
Ω
¯
⁡
u
i
x
u
i
x
p
i
x
<
+
∞
.


## 3. Main Results

Definition 5.
One says thatu
=
(
u
1
,
u
2
,
…
,
u
n
)
∈
X is a weak solution to the system (1) if u
=
(
u
1
,
u
2
,
…
,
u
n
)
∈
X and (13)
∫
Ω
∑
i
=
1
n
Δ
u
i
x
p
i
x
-
2
Δ
u
i
x
Δ
v
i
x
d
x
-
λ
∫
Ω
∑
i
=
1
n
F
u
i
x
,
u
1
,
u
2
,
…
,
u
n
v
i
x
d
x
=
0
,for every v
=
(
v
1
,
v
2
,
…
,
v
n
)
∈
X.
Letp
_
=
m
i
n
{
p
i
-
;
i
=
1,2
,
…
,
n
}, p
^
=
m
a
x
{
p
i
+
;
i
=
1,2
,
…
,
n
}. For σ
>
0, one denotes the set (14)
Q
σ
=
t
1
,
t
2
,
…
,
t
n
∈
R
n
,
∑
i
=
1
n
t
i
≤
σ
.
Define the functionI
λ
:
X
→
R by (15)
I
λ
u
=
Φ
u
-
λ
Ψ
ufor all u
=
(
u
1
,
u
2
,
…
,
u
n
)
∈
X, where (16)
Φ
u
=
∑
i
=
1
n
∫
Ω
1
p
i
x
Δ
u
i
p
i
x
d
x
,
Ψ
u
=
∫
Ω
F
x
,
u
1
,
…
,
u
n
d
x
.Then the operator Φ
′
:
X
→
X
∗, where X
∗ is the dual space of X, is defined by (17)
Φ
′
u
v
=
∑
i
=
1
n
∫
Ω
Δ
u
i
p
i
x
-
2
Δ
u
i
Δ
v
i
d
x
,for v
=
(
v
1
,
v
2
,
…
,
v
n
)
∈
X.Proposition 6.
Φ
′ is continuous, coercive, and strictly monotone. (
Φ
′
) admits a continuous inverse on X
∗.Proof.
Since(18)
Φ
′
u
u
=
∑
i
=
1
n
∫
Ω
Δ
u
i
p
i
x
d
x
≥
∑
i
=
1
n
min
⁡
u
i
p
i
x
p
i
+
,
u
i
p
i
x
p
i
-
,and p
i
-
>
1, then Φ
′ is coercive.
Using the elementary inequalities(19)
x
p
-
2
x
-
y
p
-
2
y
,
x
-
y
≥
1
2
p
x
-
y
p
,
p
≥
2
,
C
p
x
-
y
2
x
+
y
2
-
p
,
p
<
2
.We deduce that (20)
Φ
′
u
-
Φ
′
v
,
u
-
v
>
0
,which means that Φ
′ is strictly monotone. The inverse operator (
Φ
′
)
-
1 of Φ
′ exists and the continuity of (
Φ
′
)
-
1 can be proved essentially by the same way as the latter part of the proof of [16, Proposition 2.5]; we omit the details.From Proposition6, we see that Φ
∈
C
1
(
X
,
R
). Since X is compactly embedded in C
(
Ω
¯
)
×
⋯
×
C
(
Ω
¯
), we can see that Φ
:
X
→
R are sequentially weakly lower semicontinuous.The functionalΨ
:
X
→
R is Gateaux differentiable functional and (21)
Ψ
′
u
v
=
∫
Ω
∑
i
=
1
n
F
u
i
x
,
u
1
,
u
2
,
…
,
u
n
v
i
x
d
x
,for v
=
(
v
1
,
v
2
,
…
,
v
n
)
∈
X. Ψ is sequentially weakly upper semicontinuous. Furthermore, Ψ
′
:
X
→
X
∗ is a compact operator. Indeed, it is enough to show that Ψ
′ is strongly continuous on X. For this, for fixed (
u
1
,
u
2
,
…
,
u
n
)
∈
X, let (
u
1
k
,
u
2
k
,
…
,
u
n
k
)
→
(
u
1
,
u
2
,
…
,
u
n
) weakly in X as k
→
+
∞. Then we have (
u
1
k
,
u
2
k
,
…
,
u
n
k
) converges uniformly to (
u
1
,
u
2
,
…
,
u
n
) on Ω as k
→
+
∞ [27]. Since F
(
x
,
·
,
…
,
·
) is C
1 in R
n for every x
∈
Ω, so for 1
≤
i
≤
n, F
u
i
(
x
,
u
1
k
,
…
,
u
n
k
)
→
F
u
i
(
x
,
u
1
,
…
,
u
n
) strongly as k
→
+
∞, from which follows Ψ
′
(
x
,
u
1
k
,
…
,
u
n
k
)
→
Ψ
′
(
x
,
u
1
,
…
,
u
n
) strongly as k
→
+
∞. Thus we have that Ψ
′ is strongly continuous on X, which implies that Ψ
′ is a compact operator by [27, Proposition 26.2].Theorem 7.
Assume the following:(A1)
F
(
x
,
0,0
,
…
,
0
)
=
0 for x
∈
Ω.
(A2)
There existα
(
x
)
∈
L
1
(
Ω
) and n positive constants β
i with β
i
<
p
i
- for 1
≤
i
≤
n, such that (22)
0
≤
F
x
,
t
1
,
…
,
t
n
≤
α
x
1
+
∑
i
=
1
n
t
i
β
i
for a.e.x
∈
Ω, (
t
1
,
…
,
t
n
)
∈
R
+
n, where R
+
n
=
(
t
1
,
t
2
,
…
,
t
n
)
∈
R
n, t
i
≥
0, for i
=
1,2
,
…
,
n.
(A3)
There existx
0
∈
Ω, 0
<
R
1
<
R
2, M
i
>
0, and δ
∈
R with δ
≥
R
2
2
-
R
1
2
/
2
N and (
2
δ
N
/
R
2
2
-
R
1
2
)
p
i
-
π
N
/
2
/
Γ
(
1
+
N
/
2
)
(
R
2
N
-
R
1
N
)
>
1 such that (23)
∫
Ω
sup
t
1
≤
b
1
,
…
,
t
n
≤
b
n
⁡
F
x
,
t
1
,
…
,
t
n
d
x
<
min
⁡
1
/
p
i
+
b
i
/
C
p
i
+
:
1
≤
i
≤
n
∑
i
=
1
n
1
/
p
i
-
2
δ
N
/
R
2
2
-
R
1
2
p
i
+
π
N
/
2
/
Γ
1
+
N
/
2
R
2
N
-
R
1
N
∫
B
x
0
,
R
1
F
x
,
δ
,
…
,
δ
d
x
,
whereb
i
=
m
i
n
{
C
,
M
i
} for 1
≤
i
≤
n.
Then, setting(24)
Λ
≔
∑
i
=
1
n
1
/
p
i
-
2
δ
N
/
R
2
2
-
R
1
2
p
i
+
π
N
/
2
/
Γ
1
+
N
/
2
R
2
N
-
R
1
N
∫
B
x
0
,
R
1
F
x
,
δ
,
…
,
δ
d
x
,
min
⁡
1
/
p
i
+
b
i
/
C
p
i
+
:
1
≤
i
≤
n
∫
Ω
sup
t
1
≤
b
1
,
…
,
t
n
≤
b
n
⁡
F
x
,
t
1
,
…
,
t
n
d
x
,for each compact interval [
α
,
β
]
⊆
Λ, there exists a positive real number ρ with the following property: for every λ
∈
[
α
,
β
], problem (1) admits at least three weak solutions whose norms are less than ρ.Proof.
To apply Theorem A to our problem, the functionalsΦ
,
Ψ satisfy the conditions of Theorem A. Now, we show that the hypotheses of Theorem A are fulfilled.
Now we setu
0
=
(
0
,
…
,
0
); from (A1), we have Φ
(
u
0
)
=
Ψ
(
u
0
)
=
0. Let x
0
∈
Ω, 0
<
R
1
<
R
2, and take (25)
w
x
=
0
,
x
∈
Ω
∖
B
x
0
,
R
2
,
δ
,
x
∈
B
x
0
,
R
1
,
δ
R
2
2
-
R
1
2
R
2
2
-
∑
i
=
1
N
x
i
-
x
i
0
2
,
x
∈
B
x
0
,
R
2
∖
B
x
0
,
R
1
,
∑
i
=
1
N
∂
2
w
x
∂
x
i
2
=
0
,
x
∈
Ω
∖
B
x
0
,
R
2
∪
B
x
0
,
R
1
,
-
2
δ
N
R
2
2
-
R
1
2
,
x
∈
B
x
0
,
R
2
∖
B
x
0
,
R
1
.Let u
¯
=
(
w
(
x
)
,
…
,
w
(
x
)
), and r
=
min
⁡
{
1
/
p
i
+
(
b
i
/
C
)
p
i
+
:
1
≤
i
≤
n
}. Clearly, u
¯
∈
X, and we have (26)
Φ
u
¯
=
∑
i
=
1
n
∫
Ω
1
p
i
x
Δ
w
x
p
i
x
d
x
≥
∑
i
=
1
n
1
p
i
+
∫
Ω
Δ
w
p
i
x
d
x
≥
∑
i
=
1
n
1
p
i
+
2
δ
N
R
2
2
-
R
1
2
p
i
-
π
N
/
2
Γ
1
+
N
/
2
R
2
N
-
R
1
N
≥
r
.
On the other way, whenΦ
(
u
)
≤
r, we have (27)
∑
i
=
1
n
1
p
i
+
∫
Ω
Δ
u
i
p
i
x
d
x
≤
r
.So, by Proposition 3, we have (28)
1
p
i
+
m
i
n
u
i
p
i
x
p
i
+
,
u
i
p
i
x
p
i
-
≤
r
.We deduce that (29)
u
i
p
i
x
<
max
⁡
r
p
i
+
1
/
p
i
+
,
r
p
i
+
1
/
p
i
-
.For r
=
m
i
n
{
1
/
p
i
+
(
b
i
/
C
)
p
i
+
:
1
≤
i
≤
n
}, we have u
i
p
i
(
x
)
≤
b
i
/
C for 1
≤
i
≤
n.
From (12) we have m
a
x
|
u
i
(
x
)
|
≤
C
u
i
p
i
(
x
); we obtain for all x
∈
Ω, (30)
u
i
x
≤
b
i
,
1
≤
i
≤
n
.
It follows that, for everyu
=
(
u
1
,
u
2
,
…
,
u
n
)
∈
X, (31)
sup
Φ
u
≤
r
⁡
Ψ
u
=
sup
Φ
u
≤
r
⁡
∫
Ω
F
x
,
u
1
,
…
,
u
n
d
x
≤
∫
Ω
sup
t
1
≤
b
1
,
…
,
t
n
≤
b
n
⁡
F
x
,
t
1
,
…
,
t
n
d
x
.
Since(32)
Φ
u
¯
=
∑
i
=
1
n
∫
Ω
1
p
i
x
Δ
w
x
p
i
x
d
x
≤
∑
i
=
1
n
1
p
i
-
2
δ
N
R
2
2
-
R
1
2
p
i
+
π
N
/
2
Γ
1
+
N
/
2
R
2
N
-
R
1
N
,
Ψ
u
¯
>
∫
B
x
0
,
R
1
F
x
,
δ
,
…
,
δ
d
x
,therefore, from (A3), we have (33)
sup
Φ
u
≤
r
⁡
Ψ
u
r
≤
∫
Ω
sup
t
1
≤
b
1
,
…
,
t
n
≤
b
n
⁡
F
x
,
t
1
,
…
,
t
n
d
x
min
⁡
1
/
p
i
+
b
i
/
C
p
i
+
:
1
≤
i
≤
n
<
Ψ
u
¯
Φ
u
¯
,and the assumption (i) of Theorem A is satisfied.
From Proposition3, we know that if u
i
p
i
(
x
)
<
1, then (34)
1
p
i
+
u
i
p
i
x
p
i
+
≤
∫
Ω
1
p
i
x
Δ
u
i
p
i
x
d
x
≤
1
p
i
-
u
i
p
i
x
p
i
-
,let k
i
>
0, such that k
i
≥
1
/
p
i
+
u
i
p
i
(
x
)
p
i
-
-
1
/
p
i
+
u
i
p
i
(
x
)
p
i
+, and then (35)
∫
Ω
1
p
i
x
Δ
u
i
p
i
x
d
x
≥
1
p
i
+
u
i
p
i
x
p
i
-
-
k
i
.If u
i
p
i
(
x
)
≥
1, then (36)
1
p
i
+
u
i
p
i
x
p
i
-
≤
∫
Ω
1
p
i
x
Δ
u
i
p
i
x
d
x
≤
1
p
i
-
u
i
p
i
x
p
i
+
.From (A2), (12), (35), and (36), we have (37)
Φ
u
-
λ
Ψ
u
=
∑
i
=
1
n
∫
Ω
1
p
i
x
Δ
u
i
p
i
x
d
x
-
λ
∫
Ω
F
x
,
u
1
,
…
,
u
n
d
x
≥
∑
i
=
1
n
1
p
i
+
u
i
p
i
x
p
i
-
-
k
i
-
λ
∫
Ω
α
x
1
+
∑
i
=
1
n
u
i
β
i
d
x
,noting that p
i
-
>
β
i; therefore for λ
≥
0, we see that (38)
lim
u
→
+
∞
⁡
Φ
u
+
λ
Ψ
u
=
∞
,in particular, for every λ
∈
Λ. Then the assumption (ii) of Theorem A holds.
Then all the assumptions of Theorem A are fulfilled. By Theorem A, we know that there exist an open intervalΛ
⊆
[
0
,
∞
) and a positive constant ρ such that, for any λ
∈
Λ, problem (1) has at least three weak solutions whose norms are less than ρ.Remark 8.
Graef et al. [5] studied the problem and established the existence of at least three solutions in the particular case when p
i
(
x
)
=
p
i
(
>
1
).Theorem 9.
Assume the following:(A4)
F
(
x
,
t
1
,
t
2
,
…
,
t
n
)
≥
0, for each (
x
,
t
1
,
t
2
,
…
,
t
n
)
∈
Ω
×
R
+
n.
(A5)
There existx
1
∈
Ω, 0
<
R
3
<
R
4 such that, if one puts (39)
α
≔
liminf
ξ
→
+
∞
⁡
∫
Ω
sup
t
1
,
t
2
,
…
,
t
n
∈
Q
ξ
⁡
F
x
,
t
1
,
t
2
,
…
,
t
n
d
x
ξ
p
_
,
β
≔
limsup
t
1
,
t
2
,
…
,
t
n
→
+
∞
,
…
,
+
∞
⁡
∫
B
x
1
,
R
3
F
x
,
t
1
,
t
2
,
…
,
t
n
d
x
∑
i
=
1
n
t
i
p
i
+
/
p
i
-
,one has (40)
α
<
L
β
,where L
≔
m
i
n
{
L
p
i
+
,
i
=
1,2
,
…
,
n
}, (41)
L
p
i
+
=
Γ
1
+
N
/
2
C
p
_
∑
i
=
1
n
p
i
+
1
/
p
i
-
p
_
π
N
/
2
1
R
4
N
-
R
3
N
R
4
2
-
R
3
2
2
N
p
i
+
.Then, for every (42)
λ
∈
Λ
≔
1
C
p
_
∑
i
=
1
n
p
i
+
1
/
p
i
-
p
_
1
L
β
,
1
αproblem (1) admits an unbounded sequence of weak solutions.Proof.
To apply Theorem B to our problem, the functionalsΦ
,
Ψ satisfy the conditions of Theorem B. Now, let us verify that γ
<
+
∞. Let {
ξ
k
} be a real sequence such that ξ
k
→
+
∞ as k
→
+
∞ and (43)
lim
k
→
+
∞
⁡
∫
Ω
sup
t
1
,
t
2
,
…
,
t
n
∈
Q
ξ
k
⁡
F
x
,
t
1
,
t
2
,
…
,
t
n
d
x
ξ
k
p
_
=
liminf
ξ
→
+
∞
⁡
∫
Ω
sup
t
1
,
t
2
,
…
,
t
n
∈
Q
ξ
⁡
F
x
,
t
1
,
t
2
,
…
,
t
n
d
x
ξ
p
_
=
α
<
∞
.Put r
k
=
ξ
k
p
_
/
C
p
_
∑
i
=
1
n
‍
(
p
i
+
)
1
/
p
i
-
p
_ for all k
∈
N, (44)
Φ
-
1
-
∞
,
r
k
=
u
=
u
1
,
u
2
,
…
,
u
n
∈
X
;
Φ
u
<
r
k
⊆
u
∈
X
;
∑
i
=
1
n
∫
Ω
1
p
i
x
Δ
u
i
p
i
x
d
x
<
r
k
⊆
u
∈
X
;
∑
i
=
1
n
1
p
i
+
∫
Ω
Δ
u
i
p
i
x
d
x
<
r
k
.So, by Proposition 3, we have (45)
1
p
i
+
min
⁡
u
i
p
i
x
p
i
+
,
u
i
p
i
x
p
i
-
<
r
k
.Hence for k large enough (r
k
>
1),(46)
u
i
p
i
x
<
p
i
+
r
k
1
/
p
i
-
.From (12) we have m
a
x
|
u
i
(
x
)
|
≤
C
u
i
p
i
(
x
); we obtain for all x
∈
Ω, (47)
u
i
x
≤
C
p
i
+
r
k
1
/
p
i
-
.Thus ∑
i
=
1
n
‍
|
u
i
(
x
)
|
≤
∑
i
=
1
n
‍
C
p
i
+
r
k
1
/
p
i
-
≤
ξ
k. Then we have (48)
Φ
-
1
-
∞
,
r
k
⊆
u
∈
X
;
∑
i
=
1
n
u
i
x
≤
ξ
k
.Note that Φ
(
0
,
…
,
0
)
=
0, Ψ
(
0
,
…
,
0
)
≥
0; then (49)
φ
r
k
=
inf
u
∈
Φ
-
1
-
∞
,
r
k
⁡
sup
v
∈
Φ
-
1
-
∞
,
r
k
⁡
Ψ
v
-
Ψ
u
r
k
-
Φ
u
≤
sup
v
∈
Φ
-
1
-
∞
,
r
k
⁡
Ψ
v
r
k
≤
C
p
_
∑
i
=
1
n
p
i
+
1
/
p
i
-
p
_
∫
Ω
sup
t
1
,
t
2
,
…
,
t
n
∈
Q
ξ
k
⁡
F
x
,
t
1
,
t
2
,
…
,
t
n
d
x
ξ
k
p
_
.Therefore, from (A5), we have (50)
γ
≤
liminf
k
→
+
∞
⁡
φ
r
k
≤
C
p
_
∑
i
=
1
n
p
i
+
1
/
p
i
-
p
_
lim
k
→
+
∞
⁡
∫
Ω
sup
t
1
,
t
2
,
…
,
t
n
∈
Q
ξ
k
⁡
F
x
,
t
1
,
…
,
t
n
d
x
ξ
k
p
_
<
+
∞
.It is clear that Λ
⊆
(
0
,
1
/
γ
).
For the fixedλ
∈
Λ, the other step is to show that the functional I
λ has no global minimum. Arguing as in [15], since 1
/
λ
<
C
p
_
∑
i
=
1
n
‍
(
p
i
+
)
1
/
p
i
-
p
_
L
β, we can consider n positive real sequences {
η
i
,
k
}
i
=
1
n and θ
>
0 such that ∑
i
=
1
n
‍
η
i
,
k
2
→
+
∞ as k
→
+
∞ and (51)
1
λ
<
θ
<
L
C
p
_
∑
i
=
1
n
p
i
+
1
/
p
i
-
p
_
∫
B
x
1
,
R
3
F
x
,
η
1
,
k
,
…
,
η
n
,
k
d
x
∑
i
=
1
n
η
i
,
k
p
i
+
/
p
i
-
.
Let{
u
k
(
x
)
=
(
u
1
k
,
u
2
k
,
…
,
u
n
k
)
} be a sequence in X defined by (52)
u
i
k
x
=
0
,
x
∈
Ω
∖
B
x
1
,
R
4
,
η
i
,
k
,
x
∈
B
x
1
,
R
3
,
η
i
,
k
R
4
2
-
R
3
2
R
4
2
-
∑
i
=
1
N
x
i
-
x
i
1
2
,
x
∈
B
x
1
,
R
4
∖
B
x
1
,
R
3
,for 1
≤
i
≤
n. For any fixed k
∈
N, it is to see that u
k
∈
X, and (53)
∑
i
=
1
N
∂
2
u
i
k
x
∂
x
i
2
=
0
,
x
∈
Ω
∖
B
x
1
,
R
4
∪
B
x
1
,
R
3
,
-
2
η
i
,
k
N
R
4
2
-
R
3
2
,
x
∈
B
x
1
,
R
4
∖
B
x
1
,
R
3
.Then (54)
Φ
u
k
=
∑
i
=
1
n
∫
Ω
1
p
i
x
Δ
u
i
k
x
p
i
x
d
x
≤
∑
i
=
1
n
1
p
i
-
∫
Ω
Δ
u
i
k
x
p
i
x
d
x
≤
∑
i
=
1
n
1
p
i
-
∫
B
x
1
,
R
4
∖
B
x
1
,
R
3
Δ
u
i
k
x
p
i
x
d
x
≤
∑
i
=
1
n
1
p
i
-
2
η
i
,
k
N
R
4
2
-
R
3
2
p
i
+
·
π
N
/
2
Γ
1
+
N
/
2
R
4
N
-
R
3
N
=
∑
i
=
1
n
1
C
p
_
∑
i
=
1
n
p
i
+
1
/
p
i
-
p
_
η
i
,
k
p
i
+
p
i
-
L
p
i
+
.By (A1), we have (55)
Ψ
u
k
=
∫
Ω
F
x
,
u
1
k
,
…
,
u
n
k
d
x
≥
∫
B
x
1
,
R
3
F
x
,
η
1
,
k
,
…
,
η
n
,
k
d
x
,and combining (51), (54), and (55), we obtain (56)
I
λ
u
k
=
Φ
u
k
-
λ
Ψ
u
k
≤
1
C
p
_
∑
i
=
1
n
p
i
+
1
/
p
i
-
p
_
∑
i
=
1
n
η
i
,
k
p
i
+
p
i
-
L
p
i
+
-
λ
∫
B
x
1
,
R
3
F
x
,
η
1
,
k
,
…
,
η
n
,
k
d
x
≤
1
L
C
p
_
∑
i
=
1
n
p
i
+
1
/
p
i
-
p
_
∑
i
=
1
n
η
i
,
k
p
i
+
p
i
-
-
λ
∫
B
x
1
,
R
3
F
x
,
η
1
,
k
,
…
,
η
n
,
k
d
x
<
1
-
λ
θ
L
C
p
_
∑
i
=
1
n
p
i
+
1
/
p
i
-
p
_
∑
i
=
1
n
η
i
,
k
p
i
+
p
i
-
,for k large enough, so (57)
I
λ
u
k
=
-
∞
.Hence, our claim is proved. Since all assumptions of Theorem B case (b) are satisfied, the functional I
λ admits an unbounded sequence {
u
k
=
(
u
1
k
,
…
,
u
n
k
)
}
⊂
X of critical points. This completes the proof of Theorem 9.Theorem 10.
Assume that (A1), (A4) hold and consider the following:(A6)
There existx
1
∈
Ω, 0
<
R
3
<
R
4 such that, if one puts (58)
α
0
≔
liminf
ξ
→
0
+
⁡
∫
Ω
sup
t
1
,
t
2
,
…
,
t
n
∈
Q
ξ
⁡
F
x
,
t
1
,
t
2
,
…
,
t
n
d
x
ξ
p
^
,
β
0
≔
limsup
t
1
,
…
,
t
n
→
0
+
,
…
,
0
+
⁡
∫
B
x
1
,
R
3
F
x
,
t
1
,
t
2
,
…
,
t
n
d
x
∑
i
=
1
n
t
i
p
i
-
/
p
i
-
,one has (59)
α
0
<
L
1
β
0
,where L
1
≔
m
i
n
{
L
p
i
-
,
i
=
1,2
,
…
,
n
}, (60)
L
p
i
-
=
Γ
1
+
N
/
2
C
∑
i
=
1
n
p
i
+
1
/
p
i
+
p
^
π
N
/
2
1
R
4
N
-
R
3
N
R
4
2
-
R
3
2
2
N
p
i
-
.Then, for every (61)
λ
∈
Λ
≔
1
C
∑
i
=
1
n
p
i
+
1
/
p
i
+
p
^
1
L
1
β
0
,
1
α
0problem (1) admits a sequence of weak solutions which converges to 0.Proof.
From condition (A1), we havem
i
n
X
Φ
=
Φ
(
0
,
…
,
0
)
=
0, Ψ
(
0
,
…
,
0
)
=
0.
Let{
ξ
k
} be a real sequence such that ξ
k
→
0
+ as k
→
+
∞ and (62)
lim
k
→
+
∞
⁡
∫
Ω
sup
t
1
,
t
2
,
…
,
t
n
∈
Q
ξ
k
⁡
F
x
,
t
1
,
t
2
,
…
,
t
n
d
x
ξ
k
p
^
=
α
0
<
∞
.Put r
k
=
ξ
k
p
^
/
C
∑
i
=
1
n
‍
(
p
i
+
)
1
/
p
i
+
p
^ for all k
∈
N. Therefore, from (A6), we have (63)
δ
≤
liminf
k
→
+
∞
⁡
φ
r
k
≤
C
∑
i
=
1
n
p
i
+
1
/
p
i
+
p
^
α
0
<
+
∞
.It is clear that Λ
⊆
(
0
,
1
/
δ
).
For the fixedλ
∈
Λ, the other step is to show that the functional I
λ has not a local minimum at zero. Arguing as in [15], since 1
/
λ
<
C
∑
i
=
1
n
‍
(
p
i
+
)
1
/
p
i
+
p
^
L
1
β
0, we can consider n positive real sequences {
η
i
,
k
}
i
=
1
n and θ
>
0 such that ∑
i
=
1
n
‍
η
i
,
k
2
→
0 as k
→
+
∞ and (64)
1
λ
<
θ
<
L
1
C
∑
i
=
1
n
p
i
+
1
/
p
i
+
p
^
∫
B
x
1
,
R
3
F
x
,
η
1
,
k
,
…
,
η
n
,
k
d
x
∑
i
=
1
n
η
i
,
k
p
i
-
/
p
i
-
.
Let{
u
k
(
x
)
=
(
u
1
k
,
u
2
k
,
…
,
u
n
k
)
} be a sequence in X defined by (52):(65)
Φ
u
k
≤
∑
i
=
1
n
1
p
i
-
∫
Ω
Δ
u
i
k
x
p
i
x
d
x
≤
∑
i
=
1
n
1
p
i
-
∫
B
x
1
,
R
4
∖
B
x
1
,
R
3
Δ
u
i
k
x
p
i
x
d
x
≤
∑
i
=
1
n
1
p
i
-
2
η
i
,
k
N
R
4
2
-
R
3
2
p
i
-
·
π
N
/
2
Γ
1
+
N
/
2
R
4
N
-
R
3
N
=
∑
i
=
1
n
1
C
∑
i
=
1
n
p
i
+
1
/
p
i
+
p
^
η
i
,
k
p
i
-
p
i
-
L
p
i
-
.
Combining (55), (64), and (65), for k large enough, we have (66)
I
λ
u
k
=
Φ
u
k
-
λ
Ψ
u
k
<
1
-
λ
θ
L
1
C
∑
i
=
1
n
p
i
+
1
/
p
i
+
p
^
∑
i
=
1
n
η
i
,
k
p
i
-
p
i
-
<
0
=
I
λ
0
,
…
,
0
.The alternative of Theorem B case (c) ensures the existence of sequence (
u
k
) of pairwise distinct critical points (local minima) of I
λ which weakly converges to 0. This completes the proof of Theorem 10.Example 11.
LetΩ
=
(
(
-
1,1
)
)
3, with p
,
q
,
r being three functions defined on Ω by p
(
x
1
,
x
2
,
x
3
)
=
x
1
2
+
x
2
2
+
x
3
2
+
3, q
(
x
1
,
x
2
,
x
3
)
=
x
1
2
+
x
2
2
+
x
3
2
+
4, and r
(
x
1
,
x
2
,
x
3
)
=
x
1
2
+
x
2
2
+
x
3
2
+
5, and consider the increasing sequence of positive real numbers given by (67)
a
1
=
2
,
a
n
+
1
=
n
!
a
n
3
+
2
n
≥
1
.Define the function F
:
Ω
×
R
3
→
R by (68)
F
x
1
,
x
2
,
x
3
,
t
1
,
t
2
,
t
3
=
a
n
+
1
9
e
1
-
1
/
1
-
∑
i
=
1
3
t
i
-
a
n
+
1
2
+
x
1
2
+
x
2
2
+
x
3
2
,
i
f
x
1
,
x
2
,
x
3
,
t
1
,
t
2
,
t
3
∈
Ω
×
⋃
n
≥
1
S
a
n
+
1
,
a
n
+
1
,
a
n
+
1
,
1
,
0
,
o
t
h
e
r
w
i
s
e
,where S
(
(
a
n
+
1
,
a
n
+
1
,
a
n
+
1
)
,
1
) denotes the open unit ball with center at (
a
n
+
1
,
a
n
+
1
,
a
n
+
1
). It is easy to verify that F is nonnegative function such that F
(
·
,
·
,
·
,
t
1
,
t
2
,
t
3
) is continuous in Ω for all (
t
1
,
t
2
,
t
3
)
∈
R
3. F
(
x
1
,
x
2
,
x
3
,
·
,
·
,
·
) is C
1 in R
3 for every (
x
1
,
x
2
,
x
3
)
∈
Ω. F
(
x
1
,
x
2
,
x
3
,
0,0
,
0
)
=
0 for all (
x
1
,
x
2
,
x
3
)
∈
Ω, for every ρ
>
0: (69)
sup
t
1
,
t
2
,
t
3
<
ρ
⁡
F
t
1
x
1
,
x
2
,
x
3
,
t
1
,
t
2
,
t
3
+
F
t
2
x
1
,
x
2
,
x
3
,
t
1
,
t
2
,
t
3
+
F
t
3
x
1
,
x
2
,
x
3
,
t
1
,
t
2
,
t
3
∈
L
1
Ω
.
The restriction ofF on S
(
(
a
n
+
1
,
a
n
+
1
,
a
n
+
1
)
,
1
) attains its maximum in S
(
(
a
n
+
1
,
a
n
+
1
,
a
n
+
1
)
,
1
) and F
(
x
1
,
x
2
,
x
3
,
a
n
+
1
,
a
n
+
1
,
a
n
+
1
)
=
(
a
n
+
1
)
9
e
x
1
2
+
x
2
2
+
x
3
2.
Hence(70)
limsup
n
→
∞
⁡
F
x
1
,
x
2
,
x
3
,
a
n
+
1
,
a
n
+
1
,
a
n
+
1
a
n
+
1
6
/
3
+
a
n
+
1
7
/
4
+
a
n
+
1
8
/
5
=
+
∞
.Therefore (71)
β
=
limsup
t
1
→
+
∞
,
t
2
→
+
∞
,
t
3
→
+
∞
⁡
∫
B
x
1
,
R
3
F
x
1
,
x
2
,
x
3
,
t
1
,
t
2
,
t
3
d
x
1
d
x
2
d
x
3
t
1
6
/
3
+
t
2
7
/
4
+
t
3
8
/
5
=
B
x
1
,
R
3
limsup
t
1
→
+
∞
,
t
2
→
+
∞
,
t
3
→
+
∞
⁡
F
x
1
,
x
2
,
x
3
,
t
1
,
t
2
,
t
3
t
1
6
/
3
+
t
2
7
/
4
+
t
3
8
/
5
=
+
∞
.Moreover, by choosing ξ
n
=
a
n
+
1
-
1, for every n
∈
N, we have (72)
sup
t
1
+
t
2
+
t
3
≤
a
n
+
1
-
1
⁡
F
x
1
,
x
2
,
x
3
,
t
1
,
t
2
,
t
3
=
a
n
9
e
x
1
2
+
x
2
2
+
x
3
2
,then (73)
lim
n
→
+
∞
⁡
sup
t
1
+
t
2
+
t
3
≤
a
n
+
1
-
1
⁡
F
x
1
,
x
2
,
x
3
,
t
1
,
t
2
,
t
3
a
n
+
1
-
1
3
=
0
,and so (74)
liminf
ξ
→
+
∞
⁡
sup
t
1
+
t
2
+
t
3
≤
ξ
⁡
F
x
1
,
x
2
,
x
3
,
t
1
,
t
2
,
t
3
ξ
3
=
0
.Then, (75)
α
=
liminf
ξ
→
+
∞
⁡
∫
Ω
sup
t
1
+
t
2
+
t
3
≤
ξ
⁡
F
x
1
,
x
2
,
x
3
,
t
1
,
t
2
,
t
3
d
x
1
d
x
2
d
x
3
ξ
3
=
Ω
liminf
ξ
→
+
∞
⁡
sup
t
1
+
t
2
+
t
3
≤
ξ
⁡
F
x
1
,
x
2
,
x
3
,
t
1
,
t
2
,
t
3
ξ
3
=
0
<
L
β
=
+
∞
.Hence, from Theorem 9, for each λ
>
0, the problem (76)
Δ
Δ
u
x
1
2
+
x
2
2
+
x
3
2
+
1
Δ
u
=
λ
F
u
x
1
,
x
2
,
x
3
,
u
,
v
,
w
i
n
Ω
,
Δ
Δ
v
x
1
2
+
x
2
2
+
x
3
2
+
2
Δ
v
=
λ
F
v
x
1
,
x
2
,
x
3
,
u
,
v
,
w
i
n
Ω
,
Δ
Δ
w
x
1
2
+
x
2
2
+
x
3
2
+
3
Δ
w
=
λ
F
w
x
1
,
x
2
,
x
3
,
u
,
v
,
w
i
n
Ω
,
u
=
v
=
w
=
Δ
u
=
Δ
v
=
Δ
w
=
0
o
n
∂
Ω
,admits an unbounded sequence of weak solutions.



---

*Source: 3050417-2016-03-02.xml*