# Nontrivial Solutions for Dirichlet Boundary Value Systems with the(

Authors: Shang-Kun Wang; Wen-Wu Pan
Journal: Journal of Mathematics
 (2014)

---

## Abstract

Using critical point theory due to Bonanno (2012), we prove the existence of at least one nontrivial solution for Dirichlet boundary value systems with the(
p
1
,
…
,
p
n
)-Laplacian.


---

## Body

## 1. Introduction

In this note, we prove the existence of at least one nontrivial solution for Dirichlet boundary value systems with the(
p
1
,
…
,
p
n
)-Laplacian as follows:
(1)
Δ
p
1
u
1
+
λ
F
u
1
x
,
u
1
,
…
,
u
n
=
0
,
in
Ω
,
⋮
Δ
p
n
u
n
+
λ
F
u
n
x
,
u
1
,
…
,
u
n
=
0
,
in
Ω
,
u
1
=
⋯
=
u
n
=
0
,
on
⁡
∂
Ω
,
for 1
≤
i
≤
n, where n
≥
1 is an integer, where Δ
p
i
u
i
=
div
⁡
(
|
∇
u
i
|
p
-
2
∇
u
i
) denotes the p
i-Laplacian differential operator and p
i
>
1 for i
=
1
,
…
,
n, Ω
⊂
R
N
(
N
≥
1
), is a nonempty bounded open set with smooth boundary ∂
Ω, λ
>
0
,
F
:
Ω
×
R
n
→
R is a function such that F
(
·
,
t
1
,
…
,
t
n
) is measurable in Ω for all (
t
1
,
…
,
t
n
)
∈
R
n, F
(
x
,
·
,
…
,
·
) is C
1 in R
n for every x
∈
Ω and for every ϱ
>
0,
(2)
sup
⁡
(
t
1
,
…
,
t
n
)
≤
ϱ
∑
i
=
1
n
F
t
i
(
x
,
t
1
,
…
,
t
n
)
∈
L
1
(
Ω
)
,
and F
u
i denotes the partial derivative of F with respect to u
i for 1
≤
i
≤
n.Due to importance of second-order Dirichlet and Neumann problems in describing a large class of physic phenomena, many authors have studied the existence and multiplicity of solutions for such a problem; we refer the reader to [1–13] and references therein. Some authors also study the system case; see [14–20].In [6], the authors, employing a three-critical-point theorem due to Bonanno [3], determined an exact open interval of the parameter λ for which system (1) in the case n
=
1 admits non nontrivial weak solution.The aim of this paper is to prove the existence of at least one nontrivial weak solution for (1) for appropriate values of the parameter λ belonging to a precise real interval, which extend the results in [7]. For basic notation and definitions and also for a thorough account on the subject, we refer the reader to [21].


## 2. Preliminaries and Basic Notation

First, we recall for the reader’s convenience [22, Theorem 2.5] as given in [23, Theorem 2.1] (see also [3, Proposition 2.1]) which is our main tool to transfer the question of existence of at least one weak solution of (1) to the existence of a critical point of the Euler functional as follows.For a given nonempty setX and two functionals Φ
,
Ψ
:
X
→
R, we define the following two functions as follows:
(3)
β
r
1
,
r
2
=
inf
⁡
v
∈
Φ
-
1
(
]
r
1
,
r
2
[
)
sup
⁡
u
∈
Φ
-
1
(
]
r
1
,
r
2
[
)
Ψ
u
-
Ψ
v
r
2
-
Φ
v
,
ρ
(
r
1
,
r
2
)
=
sup
⁡
v
∈
Φ
-
1
(
]
r
1
,
r
2
[
)
Ψ
v
-
sup
⁡
u
∈
Φ
-
1
(
]
-
∞
,
r
1
[
)
Ψ
u
Φ
v
-
r
1
,
for all r
1
,
r
2
∈
R, r
1
<
r
2.Theorem 1 ([3, Theorem 5.1]).
LetX be a reflexive real Banach space and let Φ
:
X
→
R be a sequentially weakly lower semicontinuous, coercive, and continuously Gâteaux differentiable functional whose Gâteaux derivative admits a continuous inverse on X
* and let Ψ
:
X
→
R be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Put I
λ
=
Φ
-
λ
Ψ and assume that there are r
1
,
r
2
∈
R, r
1
<
r
2, such that
(4)
β
(
r
1
,
r
2
)
<
ρ
(
r
1
,
r
2
)
.
Then, for each λ
∈
]
1
/
ρ
(
r
1
,
r
2
)
,
1
/
β
(
r
1
,
r
2
)
[, there is u
0
,
λ
∈
Φ
-
1
(
]
r
1
,
r
2
[
) such that I
λ
(
u
0
,
λ
)
≤
I
λ
(
u
)
∀
u
∈
Φ
-
1
(
]
r
1
,
r
2
[
) and I
λ
′
(
u
0
,
λ
)
=
0.Let us introduce notation that will be used later. LetY
∶
=
W
0
1
,
p
i
(
Ω
) be the Sobolev space with the usual norm given by ∥
u
i
∥
=
∫
Ω
‍
|
∇
u
i
|
p
i
d
x
1
/
p
i, where i
=
1
,
…
,
n. Let X
∶
=
W
0
1
,
p
1
Ω
×
⋯
×
W
0
1
,
p
n
(
Ω
) with the norm
(5)
u
=
∑
i
=
1
n
∫
Ω
‍
∇
u
i
p
i
d
x
1
/
p
i
=
∑
i
=
1
n
u
i
,
where u
=
(
u
1
,
u
2
,
…
,
u
n
)
∈
X. Let
(6)
c
∶
=
max
⁡
sup
⁡
u
i
∈
W
0
1
,
p
i
(
Ω
)
∖
{
0
}
max
⁡
x
∈
Ω
¯
⁡
u
i
p
i
u
i
p
i
,
1
≤
i
≤
n
.
Since p
i
>
N for 1
≤
i
≤
n, one has c
<
+
∞. In addition, it is known [24, formula (6b)] that
(7)
sup
⁡
u
i
∈
W
0
1
,
p
i
(
Ω
)
∖
{
0
}
max
⁡
x
∈
Ω
¯
⁡
u
i
u
i
≤
N
-
1
/
p
i
π
Γ
1
+
N
2
1
/
N
p
i
-
1
p
i
-
N
1
-
1
/
p
i
×
m
Ω
1
/
N
-
1
/
p
i
for 1
≤
i
≤
n, where Γ denotes the Gamma function and m
(
Ω
) is the Lebesgue measure of the set Ω and equality occurs when Ω is a ball.Now, fixx
0
∈
Ω and pick r
1
,
r
2 with 0
<
r
1
<
r
2 such that S
(
x
0
,
r
1
)
⊂
S
(
x
0
,
r
1
)
⊆
Ω, where S
(
x
,
r
) stands for the open ball in R
N of radius r centered at x. Set
(8)
k
i
∶
=
k
i
N
,
p
i
,
r
1
,
r
2
=
1
r
2
-
r
1
r
2
N
-
r
1
N
c
π
N
/
2
Γ
1
+
N
/
2
1
/
p
i
,
1
≤
i
≤
n
.We say thatu
=
(
u
1
,
…
,
u
n
) is a weak solution to (1) if u
=
(
u
1
,
…
,
u
n
)
∈
X and
(9)
∫
Ω
‍
∑
i
=
1
n
‍
∇
u
i
p
i
-
2
∇
u
i
∇
v
i
d
x
=
λ
∫
Ω
‍
∑
i
=
1
n
‍
F
u
i
(
x
,
u
1
,
…
,
u
n
)
v
i
d
x
for every v
=
(
v
1
,
…
,
v
n
)
∈
X. For γ
>
0, we denote the set
(10)
K
(
γ
)
=
t
1
,
…
,
t
n
∣
∑
i
=
1
n
‍
t
i
p
i
p
i
≤
γ
∏
i
=
1
n
p
i
.Now, we prove the following lemma to get the main result.Lemma 2.
LetT
:
X
→
X
* be the operator defined by
(11)
T
u
1
,
…
,
u
n
h
1
,
…
,
h
n
=
∫
Ω
‍
∑
i
=
1
n
‍
∇
u
i
x
p
i
-
2
∇
u
i
(
x
)
·
∇
h
i
(
x
)
d
x
for every (
u
1
,
…
,
u
n
), (
h
1
,
…
,
h
n
)
∈
X. Then, T admits a continuous inverse on X
*.Proof.
Since(12)
T
u
1
,
…
,
u
n
u
1
,
…
,
u
n
=
∫
Ω
‍
∑
i
=
1
n
‍
∇
u
i
x
p
i
d
x
=
∑
i
=
1
n
‍
u
i
p
i
p
i
,
T is coercive. For any p
>
1, by (2.2) in [25], there exists C
p
>
0 such that
(13)
〈
x
p
x
-
y
p
y
,
x
-
y
〉
≥
C
p
x
-
y
p
,
if
p
≥
2
,
C
p
x
-
y
2
x
+
y
2
-
p
,
if
1
<
p
<
2
for any x
,
y
∈
R
n. Now, for any u
=
(
u
1
,
…
,
u
n
)
∈
X and v
=
(
v
1
,
…
,
v
n
)
∈
X, we have
(14)
〈
T
(
u
)
-
T
(
v
)
,
u
-
v
〉
=
∑
i
=
1
n
‍
∫
Ω
‍
∇
u
i
x
p
i
-
2
∇
u
i
x
-
∇
v
i
x
p
i
-
2
∇
v
i
x
kkkkkkkkk
×
∇
u
i
x
-
∇
v
i
x
d
x
.
Then, from (13), we see that, for u
,
v
∈
X with u
≠
v,
(15)
T
u
-
T
v
,
u
-
v
=
∑
i
∈
I
1
‍
C
p
i
∫
Ω
‍
∇
u
i
x
-
∇
v
i
x
p
i
d
x
+
∑
i
∈
I
2
‍
C
p
i
∫
Ω
‍
C
p
∇
u
i
x
-
∇
v
i
x
2
∇
u
i
x
+
∇
v
i
x
2
-
p
i
d
x
,
where I
1
=
{
i
∈
{
1
,
…
,
n
}
:
p
i
≥
2
} and I
2
=
{
i
∈
{
1
,
…
,
n
}
:
1
<
p
i
<
2
}. Thus, T is strictly monotone. Clearly, T is also demicontinuous; so, by [26, Theorem 26.A(d)], the inverse operator T
-
1 of T exists.
Now, we show thatT satisfies property as follows:(
S
2
)
if(
u
1
n
,
…
,
u
n
n
)
⇀
(
u
1
,
…
,
u
n
) and T
(
u
1
n
,
…
,
u
n
n
)
→
T
(
u
1
,
…
,
u
n
), then (
u
1
n
,
…
,
u
n
n
)
→
(
u
1
,
…
,
u
n
). Let us take a sequence (
u
1
n
,
…
,
u
n
n
)
∈
X such that (
u
1
n
,
…
,
u
n
n
)
⇀
(
u
1
,
…
,
u
n
) in X and
(16)
T
(
u
1
n
,
…
,
u
n
n
)
⟶
T
u
1
,
…
,
u
n
in
X
*
.
Then,
(17)
T
u
1
n
,
…
,
u
n
n
,
u
1
n
,
…
,
u
n
n
⟶
T
(
u
1
,
…
,
u
n
)
,
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
K
K
k
k
k
(
u
1
,
…
,
u
n
)
.
So,
(18)
u
1
n
p
1
p
1
+
⋯
+
u
n
n
p
n
p
n
⟶
u
1
p
1
p
1
+
⋯
+
u
n
p
n
p
n
.
According to the uniform convexity of X, (
u
1
n
,
…
,
u
n
n
)
→
(
u
1
,
…
,
u
n
) in X.
It suffices then to show the continuity ofT
-
1. Let (
(
f
1
n
)
n
,
…
,
(
f
n
n
)
n
) be a sequence of X
* such that f
i
n
→
f
i in (
W
0
1
,
p
i
(
Ω
)
)
*. Let (
u
1
n
,
…
,
u
n
n
) and (
u
1
,
…
,
u
n
) in X be such that
(19)
T
p
i
-
1
f
i
n
=
u
i
n
,
T
p
i
-
1
(
f
i
)
=
u
i
.
By the coercivity of T, one deduces that the sequence (
u
1
n
,
…
,
u
n
n
) is bounded in the reflexive space X. For a subsequence, we have (
u
1
n
,
…
,
u
n
n
)
⇀
(
u
1
^
,
…
,
u
n
^
) in X, which implies
(20)
lim
⁡
n
→
+
∞
T
(
u
1
n
,
…
,
u
n
n
)
-
T
(
u
1
,
…
,
u
n
)
,
(
u
1
,
…
,
u
n
)
k
k
k
k
k
k
-
(
u
1
^
,
…
,
u
n
^
)
T
(
u
1
n
,
…
,
u
n
n
)
-
T
(
u
1
,
…
,
u
n
)
,
(
u
1
,
…
,
u
n
)
=
lim
⁡
n
→
+
∞
(
f
1
n
,
…
,
f
n
n
)
-
(
f
1
,
…
,
f
n
)
,
(
u
1
,
…
,
u
n
)
kkkkkkkkkkk
(
f
1
n
,
…
,
f
n
n
)
-
(
f
1
,
…
,
f
n
)
,
(
u
1
,
…
,
u
n
)
-
(
u
1
^
,
…
,
u
n
^
)
=
0
.
It follows from the property of (
S
2
) and the continuity of T that (
u
1
n
,
…
,
u
n
n
)
→
(
u
1
^
,
…
,
u
n
^
) in X and T
(
u
1
,
…
,
u
n
)
→
T
(
u
1
^
,
…
,
u
n
^
)
=
T
(
u
1
,
…
,
u
n
) in X
*.
Moreover, sinceT is an injection, we conclude that (
u
1
,
…
,
u
n
)
=
(
u
1
^
,
…
,
u
n
^
).


## 3. Results

For a given nonnegative constantθ and a positive constant τ with ∑
i
=
1
n
(
τ
k
i
)
p
i
/
p
i
≠
θ
/
∏
i
=
1
n
‍
p
i, put
(21)
a
τ
θ
≔
∫
Ω
sup
⁡
t
1
,
…
,
t
n
∈
K
θ
⁡
F
x
,
t
1
,
…
,
t
n
d
x
-
∫
S
x
0
,
r
1
‍
F
x
,
τ
,
…
,
τ
d
x
k
k
k
k
-
∫
S
x
0
,
r
1
‍
F
x
,
τ
,
…
,
τ
d
x
×
θ
∏
i
=
1
n
p
i
-
∑
i
=
1
n
‍
τ
k
i
p
i
p
i
-
1
,
where K
θ
=
{
(
t
1
,
…
,
t
n
)
∈
R
n
:
∑
i
=
1
n
|
t
i
|
p
i
/
p
i
≤
θ
/
∏
i
=
1
n
‍
p
i
} (see (10)).We formulate our main result as follows.Theorem 3.
Assume that there exist a nonnegative constantθ
1 and two positive constants θ
2 and τ with ∑
i
=
1
n
(
(
τ
k
i
)
p
i
/
p
i
)
>
θ
1
/
∏
i
=
1
n
p
i and ∑
i
=
1
n
‍
(
(
τ
k
i
)
p
i
/
p
i
)
<
θ
2
/
∏
i
=
1
n
‍
p
i such that (A1)
F
(
x
,
t
1
,
…
,
t
n
)
≥
0 for each (
x
,
t
1
,
…
,
t
n
)
∈
(
Ω
¯
∖
S
(
x
0
,
r
1
)
)
×
[
0
,
τ
]
×
⋯
×
[
0
,
τ
];
(A2)
a
τ
(
θ
2
)
<
a
τ
(
θ
1
). Then, for each λ
∈
]
1
/
a
τ
(
θ
1
)
,
1
/
a
τ
(
θ
2
)
[, system (1) admits at least one nontrivial weak solution u
0
=
(
u
01
,
…
,
u
0
n
)
∈
X such that
(22)
θ
1
c
∏
i
=
1
n
‍
p
i
<
∑
i
=
1
n
‍
u
0
i
p
i
p
i
<
θ
2
c
∏
i
=
1
n
p
i
.Proof.
To apply Theorem1 to our problem, arguing as in [7], we introduce the functionals Φ
,
Ψ
:
X
→
R for each u
=
(
u
1
,
…
,
u
n
)
∈
X as follows:
(23)
Φ
u
=
∑
i
=
1
n
u
i
P
i
P
i
,
Ψ
(
u
)
=
∫
Ω
‍
F
(
x
,
u
1
(
x
)
,
…
,
u
n
(
x
)
)
d
x
.
It is well known that Φ and Ψ are well defined and continuously differentiable functionals whose derivatives at the point u
=
(
u
1
,
…
,
u
n
)
∈
X are the functionals Φ
′
(
u
)
,
Ψ
′
(
u
)
∈
X
*, given by
(24)
〈
Φ
′
(
u
)
,
v
〉
=
∫
Ω
‍
∑
i
=
1
n
‍
∇
u
i
p
i
-
2
∇
u
i
∇
v
i
d
x
,
Ψ
′
(
u
)
(
v
)
=
∫
Ω
‍
∑
i
=
1
n
‍
F
u
i
(
x
,
u
1
(
x
)
,
…
,
u
n
(
x
)
)
v
i
(
x
)
d
x
for every v
=
(
v
1
,
…
,
v
n
)
∈
X, respectively. Moreover, Φ is sequentially weakly lower semicontinuous, Φ
′ admits a continuous inverse on X
*, and Ψ is sequentially weakly upper semicontinuous (see Lemma 2). 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
n
)
∈
X, let (
u
1
m
,
…
,
u
n
m
)
→
(
u
1
,
…
,
u
n
) weakly in X as m
→
+
∞; then, we have that (
u
1
m
,
…
,
u
n
m
) converges uniformly to (
u
1
,
…
,
u
n
) on Ω as m
→
+
∞ (see [26]). Since F
(
x
,
·
,
…
,
·
) is C
1 in R
n for every x
∈
Ω, the derivatives of F are continuous in R
n for every x
∈
Ω; so, for 1
≤
i
≤
n, F
u
i
(
x
,
u
1
m
,
…
,
u
n
m
)
→
F
u
i
(
x
,
u
1
,
…
,
u
n
) strongly as m
→
+
∞ which follows Ψ
′
(
u
1
m
,
…
,
u
n
m
)
→
Ψ
′
(
u
1
,
…
,
u
n
) strongly as m
→
+
∞. Thus, we proved that Ψ
′ is strongly continuous on X, which implies that Ψ
′ is a compact operator by Proposition 26.2 of [26]. Set w
(
x
)
=
(
w
1
(
x
)
,
…
,
w
n
(
x
)
) such that, for 1
≤
i
≤
n,
(25)
w
i
x
=
0
,
x
∈
Ω
¯
∖
S
x
0
,
r
2
,
τ
r
2
-
r
1
r
2
-
∑
j
=
1
N
‍
x
j
-
x
j
0
2
,
x
∈
S
x
0
,
r
2
∖
S
x
0
,
r
1
,
τ
,
x
∈
s
x
0
,
r
1
,
r
1
=
θ
1
/
c
∏
i
=
1
n
‍
p
i, and r
2
=
θ
2
/
c
∏
i
=
1
n
‍
p
i. It is easy to see that w
(
x
)
=
(
w
1
(
x
)
,
…
,
w
n
(
x
)
)
∈
X and, in particular, one has, for 1
≤
i
≤
n,
(26)
w
i
p
i
=
r
2
N
-
r
1
N
π
N
/
2
Γ
(
1
+
N
/
2
)
τ
r
2
-
r
1
p
i
=
k
i
τ
p
i
c
,
so, from the definition of Φ, we have
(27)
Φ
(
w
)
=
∑
i
=
1
n
‍
k
i
τ
p
i
c
.
From the conditions ∑
i
=
1
n
(
(
τ
k
i
)
p
i
/
p
i
)
>
θ
1
/
∏
i
=
1
n
‍
p
i and ∑
i
=
1
n
(
(
τ
k
i
)
p
i
/
p
i
)
<
θ
2
/
∏
i
=
1
n
‍
p
i, we obtain
(28)
r
1
<
Φ
(
w
)
<
r
2
.
Moreover, from (6), one has
(29)
sup
⁡
x
∈
Ω
∑
i
=
1
n
‍
u
i
x
p
i
≤
c
∑
i
=
1
n
‍
u
i
p
i
for each u
=
(
u
1
,
…
,
u
n
)
∈
X; so, from the definition of Φ, we observe that
(30)
Φ
-
1
-
∞
,
r
2
=
u
1
,
…
,
u
n
∈
X
;
Φ
u
1
,
…
,
u
n
<
r
2
=
u
1
,
…
,
u
n
∈
X
;
∑
i
=
1
n
‍
u
i
p
i
p
i
<
r
2
⊆
(
u
1
,
…
,
u
n
)
∈
X
;
∑
i
=
1
n
‍
u
i
x
p
i
p
i
<
θ
2
∏
i
=
1
n
‍
p
i
∀
x
∈
Ω
,
from which it follows
(31)
sup
⁡
u
1
,
…
,
u
n
∈
Φ
-
1
(
]
-
∞
,
r
2
[
)
Ψ
u
=
sup
⁡
u
1
,
…
,
u
n
∈
Φ
-
1
(
]
-
∞
,
r
2
[
)
∫
Ω
‍
F
x
,
u
1
x
,
…
,
u
n
x
d
x
≤
∫
Ω
‍
sup
⁡
(
t
1
,
…
,
t
n
)
∈
K
(
θ
2
)
F
(
x
,
t
1
,
…
,
t
n
)
d
x
.
Since, for 1
≤
i
≤
n, 0
≤
w
i
(
x
)
≤
τ for each x
∈
Ω, condition (A1) ensures that
(32)
∫
Ω
¯
∖
S
x
0
,
r
2
‍
F
x
,
ω
1
,
…
,
ω
n
d
x
+
∫
S
(
x
0
,
r
2
)
∖
S
(
x
0
,
r
1
)
‍
F
(
x
,
ω
1
,
…
,
ω
n
)
d
x
≥
0
.
So, one has
(33)
β
r
1
,
r
2
≤
sup
⁡
u
∈
Φ
-
1
(
]
-
∞
,
r
2
[
)
Ψ
(
u
)
-
Ψ
(
w
)
r
2
-
Φ
(
w
)
≤
∫
Ω
‍
sup
⁡
(
t
1
,
…
,
t
n
)
∈
K
(
θ
2
)
F
(
x
,
t
1
,
…
,
t
n
)
d
x
-
Ψ
(
w
)
r
2
-
Φ
(
w
)
≤
a
τ
(
θ
2
)
.
On the other hand, by similar reasoning as before, one has
(34)
ρ
r
1
,
r
2
≥
Ψ
(
w
)
-
sup
⁡
u
∈
Φ
-
1
(
]
-
∞
,
r
1
[
)
Ψ
(
u
)
Φ
(
w
)
-
r
1
≥
Ψ
(
w
)
-
∫
Ω
‍
sup
⁡
(
t
1
,
…
,
t
n
)
∈
K
(
θ
1
)
F
(
x
,
t
1
,
…
,
t
n
)
d
x
Φ
(
w
)
-
r
1
≥
a
τ
(
θ
1
)
.
Hence, from Assumption (A2), one has β
(
r
1
,
r
2
)
<
ρ
(
r
1
,
r
2
). Therefore, from Theorem 1, taking into account that the weak solutions of the system (1) are exactly the solutions of the equation Φ
′
(
u
)
-
λ
Ψ
′
(
u
)
=
0, we have the conclusion.Now, we point out the following consequence of Theorem3.Theorem 4.
Suppose that there exist two positive constantsθ and τ with ∑
i
=
1
n
(
(
τ
k
i
)
p
i
/
p
i
)
<
θ
/
∏
i
=
1
n
‍
p
i such that (A3)
m
(
Ω
)
∑
i
=
1
n
(
(
τ
k
i
)
p
i
/
p
i
)
sup
⁡
(
x
,
t
1
,
…
,
t
n
)
∈
Ω
¯
×
K
(
θ
)
F
(
x
,
t
1
,
…
,
t
n
)
<
(
θ
/
∏
i
=
1
n
p
i
)
∫
S
(
x
0
,
r
1
)
‍
F
(
x
,
τ
,
…
,
τ
)
d
x;
(A4)
F
(
x
,
0
,
…
,
0
)
=
0 for every x
∈
Ω. Then, for each
(35)
λ
∈
∑
i
=
1
n
‍
k
i
τ
p
i
/
p
i
m
Ω
sup
⁡
x
,
t
1
,
…
,
t
n
∈
Ω
¯
×
K
θ
F
x
,
t
1
,
…
,
t
n
∑
i
=
1
n
k
i
τ
p
i
/
p
i
m
Ω
sup
⁡
x
,
t
1
,
…
,
t
n
∈
Ω
¯
×
K
θ
F
x
,
t
1
,
…
,
t
n
,
k
k
k
k
k
k
θ
/
∏
i
=
1
n
p
i
∫
S
x
0
,
r
1
‍
F
x
,
τ
,
…
,
τ
d
x
∑
i
=
1
n
‍
k
i
τ
p
i
/
p
i
m
Ω
sup
⁡
x
,
t
1
,
…
,
t
n
∈
Ω
¯
×
K
θ
F
x
,
t
1
,
…
,
t
n
,
system (1) admits at least one nontrivial weak solution u
0
=
(
u
01
,
…
,
u
0
n
)
∈
X such that ∑
i
=
1
n
‍
(
∥
u
0
i
∥
p
i
/
p
i
)
<
θ
/
c
∏
i
=
1
n
‍
p
i.Proof.
The conclusion follows from Theorem3, by taking θ
1
=
0 and θ
2
=
θ. Indeed, owing to our assumptions, one has
(36)
a
τ
θ
2
<
1
-
∏
i
=
1
n
‍
p
i
∑
i
=
1
n
k
i
τ
p
i
/
p
i
θ
×
∫
Ω
‍
sup
⁡
t
1
,
…
,
t
n
∈
K
θ
F
x
,
t
1
,
…
,
t
n
d
x
×
θ
∏
i
=
1
n
p
i
-
∑
i
=
1
n
‍
τ
k
i
p
i
p
i
-
1
=
1
-
∏
i
=
1
n
‍
p
i
∑
i
=
1
n
k
i
τ
p
i
/
p
i
θ
×
∫
Ω
‍
sup
⁡
t
1
,
…
,
t
n
∈
K
θ
F
x
,
t
1
,
…
,
t
n
d
x
×
θ
∏
i
=
1
n
p
i
-
∑
i
=
1
n
‍
τ
k
i
p
i
p
i
-
1
=
∫
Ω
‍
sup
⁡
t
1
,
…
,
t
n
∈
K
θ
F
x
,
t
1
,
…
,
t
n
d
x
×
θ
∏
i
=
1
n
p
i
-
1
.
On the other hand, taking Assumption (A4) into account, one has
(37)
∫
S
(
x
0
,
r
1
)
‍
F
(
x
,
τ
,
…
,
τ
)
d
x
∑
i
=
1
n
k
i
τ
p
i
/
c
=
a
τ
(
θ
1
)
.
Moreover, since
(38)
sup
⁡
x
∈
Ω
∑
i
=
1
n
‍
u
i
x
p
i
p
i
≤
c
∑
i
=
1
n
‍
u
i
p
i
p
i
for each u
=
(
u
1
,
…
,
u
n
)
∈
X, an easy computation ensures that ∑
i
=
1
n
(
∥
u
i
∥
∞
<
θ
) whenever Φ
(
u
)
<
r
2. Now, owing to Assumption (A3), it is sufficient to invoke Theorem 3 for concluding the proof.Letf be a continuous function in Ω
¯ and let g
i, for 1
≤
i
≤
n, be a function in C
1 and
(39)
F
x
,
u
1
x
,
…
,
u
n
x
=
f
(
x
)
∏
i
=
1
n
‍
g
i
(
u
i
(
x
)
)
kkkkkkkkl
for
each
(
x
,
u
1
,
…
,
u
n
)
∈
Ω
¯
×
R
n
.Theorem 5.
Assume that Assumption (A1) in Theorem3 holds. Suppose that there exist two positive constants θ and τ with ∑
i
=
1
n
‍
(
(
τ
k
i
)
p
i
/
p
i
)
<
θ
/
∏
i
=
1
n
‍
p
i such that (B1)
f
(
x
)
≥
0 for each x
∈
(
Ω
¯
∖
S
(
x
0
,
r
1
)
) and g
i
(
t
i
)
≥
0 for t
i
∈
[
0
,
τ
]
,
1
≤
i
≤
n;
(B2)
m
Ω
∑
i
=
1
n
(
τ
k
i
)
p
i
/
p
i
max
⁡
x
,
t
1
,
…
,
t
n
∈
Ω
¯
×
K
f
x
∏
i
=
1
n
g
i
(
t
i
x
)
<
θ
∏
i
=
1
n
‍
(
g
i
(
τ
)
/
p
i
)
∫
S
(
x
0
,
r
1
)
‍
f
(
x
)
d
x;
(B3)
g
i
(
0
)
=
0 for every x
∈
Ω for 1
≤
i
≤
n. Then, for each
(40)
λ
∈
∑
i
=
1
n
‍
k
i
τ
p
i
/
p
i
m
Ω
max
⁡
x
,
t
1
,
…
,
t
n
∈
Ω
¯
×
K
f
x
∏
i
=
1
n
‍
g
i
t
i
x
∑
i
=
1
n
k
i
τ
p
i
/
p
i
m
Ω
max
⁡
x
,
t
1
,
…
,
t
n
∈
Ω
¯
×
K
f
x
∏
i
=
1
n
g
i
t
i
x
,
k
k
k
k
k
k
k
θ
/
∏
i
=
1
n
p
i
∏
i
=
1
n
g
i
τ
∫
S
x
0
,
r
1
‍
f
x
d
x
∑
i
=
1
n
‍
k
i
τ
p
i
/
p
i
m
Ω
max
⁡
x
,
t
1
,
…
,
t
n
∈
Ω
¯
×
K
f
x
∏
i
=
1
n
‍
g
i
t
i
x
,
system
(41)
Δ
p
1
u
1
+
λ
f
(
x
)
g
1
′
(
u
1
)
∏
i
=
1
,
i
≠
1
n
‍
g
i
(
u
i
)
=
0
,
in
Ω
,
⋮
Δ
p
n
u
n
+
λ
f
(
x
)
g
n
′
(
u
n
)
∏
i
=
1
,
i
≠
n
n
‍
g
i
(
u
i
)
=
0
,
in
Ω
,
u
1
=
⋯
=
u
n
=
0
,
o
n
∂
Ω
admits at least one nontrivial weak solution u
0
=
(
u
01
,
…
,
u
0
n
)
∈
X such that ∑
i
=
1
n
‍
(
∥
u
0
i
∥
p
i
/
p
i
)
<
θ
/
c
∏
i
=
1
n
‍
p
i.The following result gives the existence of at least one nontrivial weak solution inX to the problem (1) in the autonomous case. For a given nonnegative constant θ and a positive constant τ with ∑
i
=
1
n
‍
(
(
τ
k
i
)
p
i
/
p
i
)
≠
θ
/
∏
i
=
1
n
‍
p
i, put
(42)
b
τ
θ
∶
=
m
Ω
sup
⁡
t
1
,
…
,
t
n
∈
K
θ
H
t
1
,
…
,
t
n
k
k
k
k
k
k
-
r
1
N
π
N
/
2
Γ
1
+
N
/
2
H
τ
,
…
,
τ
m
Ω
sup
⁡
t
1
,
…
,
t
n
∈
K
θ
H
t
1
,
…
,
t
n
×
θ
∏
i
=
1
n
p
i
-
∑
i
=
1
n
‍
τ
k
i
p
i
p
i
-
1
,
where K
θ
=
{
(
t
1
,
…
,
t
n
)
∈
R
n
:
∑
i
=
1
n
‍
(
|
t
i
|
p
i
/
p
i
)
≤
θ
/
∏
i
=
1
n
‍
p
i
} (see (10)). We have the following result as a direct consequence of Theorem 5.Theorem 6.
LetH
:
R
n
→
R be a C
1 function and assume that there exist a nonnegative constant θ
1 and two positive constants θ
2 and τ with ∑
i
=
1
n
‍
(
(
τ
k
i
)
p
i
/
p
i
)
>
θ
1
/
∏
i
=
1
n
‍
p
i and ∑
i
=
1
n
‍
(
(
τ
k
i
)
p
i
/
p
i
)
<
θ
2
/
∏
i
=
1
n
‍
p
i such that (C1)
H
(
t
1
,
…
,
t
n
)
≥
0 for each (
t
1
,
…
,
t
n
)
∈
[
0
,
τ
]
×
⋯
×
[
0
,
τ
];
(C2)
m
(
Ω
)
∑
i
=
1
n
‍
(
(
τ
k
i
)
p
i
/
p
i
)
max
⁡
t
1
,
…
,
t
n
∈
K
H
(
t
1
,
…
,
t
n
)
<
(
θ
r
1
N
π
N
/
2
/
Γ
(
1
+
N
/
2
)
∏
i
=
1
n
‍
p
i
)
H
(
τ
,
…
,
τ
). Then, for each λ
∈
1
/
b
τ
(
θ
1
)
,
1
/
b
τ
(
θ
2
), the problem
(43)
Δ
p
1
u
1
+
λ
H
u
1
(
u
1
,
…
,
u
n
)
=
0
,
i
n
Ω
,
⋮
Δ
p
n
u
n
+
λ
H
u
n
(
u
1
,
…
,
u
n
)
=
0
,
i
n
Ω
,
u
1
=
⋯
=
u
n
=
0
,
o
n
∂
Ω
admits at least one nontrivial weak solution u
0
∈
X such that θ
1
/
c
∏
i
=
1
n
‍
p
i
<
∑
i
=
1
n
‍
(
∥
u
0
i
∥
p
i
/
p
i
)
<
θ
2
/
c
∏
i
=
1
n
‍
p
i.Proof.
Note that∫
S
x
0
,
r
1
‍
H
(
τ
,
…
,
τ
)
d
x
=
(
r
1
N
π
N
/
2
/
Γ
(
1
+
N
/
2
)
)
H
(
τ
,
…
,
τ
), which together with (B2) implies
(44)
m
Ω
∑
i
=
1
n
‍
τ
k
i
p
i
p
i
max
⁡
t
1
,
…
,
t
n
∈
K
H
t
1
,
…
,
t
n
<
θ
∏
i
=
1
n
p
i
∫
S
(
x
0
,
r
1
)
‍
H
(
τ
,
…
,
τ
)
d
x
.
Therefore, we have the conclusion by Theorem 5.As an example, we point out the following special case of our main result.Theorem 7.
Leth
:
Ω
→
R be a positive and essentially bounded function and let g
:
R
→
R be a nonnegative continuous function such that lim
⁡
t
→
0
+
(
g
(
t
)
/
t
p
-
1
)
=
+
∞. Then, for each λ
∈
]
0
,
(
1
/
p
c
∫
Ω
‍
h
x
d
x
)
sup
⁡
θ
>
0
(
θ
p
/
∫
0
θ
‍
g
ξ
d
ξ
)
[, the problem
(45)
Δ
p
u
+
λ
h
x
g
u
=
0
,
x
∈
Ω
,
u
=
0
,
x
∈
∂
Ω
admits at least one nontrivial weak solution in Y.Proof.
For fixedλ as in the conclusion, there exists positive constant θ such that
(46)
λ
<
1
p
c
∫
Ω
‍
h
(
x
)
d
x
sup
⁡
θ
>
0
θ
p
∫
0
θ
‍
g
(
ξ
)
d
ξ
.
Moreover, lim
⁡
t
→
0
+
(
g
(
t
)
/
t
p
-
1
)
=
+
∞ implies lim
⁡
t
→
0
+
(
∫
0
t
‍
g
(
ξ
)
d
ξ
/
t
p
)
=
+
∞. Therefore, a positive constant τ satisfying (
k
τ
)
p
<
θ can be chosen such that
(47)
k
p
λ
p
c
1
∫
B
(
x
0
,
r
1
)
‍
h
(
x
)
d
x
<
∫
0
τ
‍
g
(
ξ
)
d
ξ
τ
p
.
Hence, arguing as in the proof of Theorem 4, the conclusion follows from Theorem 6 with θ
1
=
0, θ
2
=
θ, and f
(
t
)
=
g
(
t
) for every t
∈
R.We remark that ifh
(
x
)
=
1 for every x
∈
Ω, then Theorem 7 gives [7, Theorem 1.1].Remark 8.
For fixedρ, put λ
ρ
∶
=
(
1
/
p
c
∫
Ω
‍
h
x
d
x
)
sup
⁡
θ
>
0
(
θ
p
/
∫
0
θ
‍
g
(
ξ
)
d
ξ
). The result of Theorem 7 for every λ
∈
]
0
,
λ
ρ
[ holds with |
u
0
(
x
)
|
<
ρ for all x
∈
Ω, where u
0 is the ensured nontrivial weak solution in Y (see [7, Corollary 3.1]).We close this paper by presenting the following example to illustrate our results.Example 9.
Consider the problem(48)
-
Δ
3
u
=
λ
1
+
e
-
u
+
u
+
2
3
-
u
+
,
x
∈
Ω
,
u
=
0
,
x
∈
∂
Ω
,
where u
+
=
max
⁡
{
u
,
0
}, N
=
2, and Ω
=
{
(
x
,
y
)
:
x
2
+
y
2
≤
4
}. Let(49)
g
(
t
)
=
1
+
e
-
t
+
t
+
2
3
-
t
+
for all t
∈
R, where t
+
=
max
⁡
{
t
,
0
}. It is clear that lim
⁡
t
→
0
+
(
g
(
t
)
/
t
)
=
+
∞. Note that p
=
3 and N
=
2; we have c
=
2
/
π
3, and so k
=
54
π
2
6. Hence, taking Remark 8 into account, since sup
⁡
θ
∈
]
0,1
[
(
θ
2
/
∫
0
θ
‍
g
ξ
d
ξ
)
=
sup
⁡
θ
∈
]
0,1
[
(
θ
2
/
(
θ
+
e
-
θ
θ
3
)
)
=
e
/
(
1
+
e
), by applying Theorem 7, for every λ
∈
]
0
,
(
π
3
/
3
2
3
)
(
e
/
(
1
+
e
)
)
[, the problem (48) has at least one nontrivial classical solution u
0
∈
Y such that ∥
u
0
∥
∞
<
1.



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*Source: 505290-2014-11-19.xml*