# Positive Solutions of a Kind of Equations Related to the Laplacian andp-Laplacian

Authors: Fangfang Zhang; Zhanping Liang
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

Positive solutions of a kind of equations related to the Laplacian andp-Laplacian on a bounded domain in R
N with N
⩾
1 are studied by using variational method. A sufficient condition of the existence of positive solutions is characterized by the eigenvalues of linear and another nonlinear eigenvalue problems.


---

## Body

## 1. Introduction

In this paper, we study the equation(1)
-
Δ
u
-
Δ
p
u
=
f
(
x
,
u
)
,
in
Ω
,
u
=
0
,
on
⁡
∂
Ω
,
where Δ
p
u
=
div
⁡
(
|
∇
u
|
p
-
2
∇
u
)
,
p
>
2, Ω is a smooth bounded domain in R
N for N
⩾
1, and f satisfies the following conditions: (f1)
f
∈
C
(
Ω
¯
×
R
,
R
)
,
f
(
x
,
t
)
⩾
0 for any x
∈
Ω
¯
,
t
>
0 and f
(
x
,
t
)
=
0 for any x
∈
Ω
¯
,
t
⩽
0;
(f2)
forf
0
,
f
∞
<
∞, the limits
(2)
lim
⁡
t
→
0
+
f
(
x
,
t
)
t
=
f
0
,
lim
⁡
t
→
∞
f
(
x
,
t
)
t
p
-
1
=
f
∞
exist uniformly forx
∈
Ω
¯.The asymptotic behaviors off near zero and infinity lead us to define
(3)
λ
1
=
inf
⁡
{
∫
Ω
‍
|
∇
u
|
2
:
u
∈
H
0
1
(
Ω
)
,
∫
Ω
‍
|
u
|
2
=
1
}
,
μ
1
=
inf
⁡
{
∫
Ω
‍
|
∇
u
|
p
:
u
∈
W
0
1
,
p
(
Ω
)
,
∫
Ω
‍
|
u
|
p
=
1
}
,
where H
0
1
(
Ω
) and W
0
1
,
p
(
Ω
) are the usual Sobolev spaces defined as the completion of C
0
∞
(
Ω
) with respect to the norms ∥
u
∥
2
=
(
∫
Ω
‍
|
∇
u
|
2
)
1
/
2 and ∥
u
∥
p
=
(
∫
Ω
‍
|
∇
u
|
p
)
1
/
p, respectively. Then it is well known that μ
1 is the first eigenvalue of the nonlinear eigenvalue problem
(4)
-
Δ
p
ϕ
=
μ
|
ϕ
|
p
-
2
ϕ
,
in
Ω
,
ϕ
=
0
,
on
⁡
∂
Ω
.
Moreover μ
1 is a simple eigenvalue of (4), the associated eigenfunction ϕ
1 can be chosen as positive in Ω, and any eigenfunction corresponding to an eigenvalue larger than μ
1 must change sign. The reader is referred to [1–3] for details.By a solutionu of (1), we mean that u solves (1) in the weak sense; that is, u satisfies
(5)
∫
Ω
‍
∇
u
·
∇
v
+
∫
Ω
‍
|
∇
u
|
p
-
2
∇
u
·
∇
v
=
∫
Ω
‍
f
(
x
,
u
)
v
,
i
v
∈
W
0
1
,
p
(
Ω
)
.
Moreover, by a positive solution u of (1), we mean that u is a weak solution of (1), u
≠
0 and u
(
x
)
⩾
0 for x
∈
Ω.Our main result is the following theorem.Theorem 1.
Suppose thatf satisfies (
f
1
) and (
f
2
) with f
0
<
λ
1, f
∞
>
μ
1. Then (1) has a positive solution.Assume thatf
=
g
+
h. Equation (1) can be viewed as combination of the following equations:
(6)
-
Δ
u
=
g
(
x
,
u
)
,
in
Ω
,
u
=
0
,
on
⁡
∂
Ω
,
-
Δ
p
u
=
h
(
x
,
u
)
,
in
Ω
,
u
=
0
,
on
⁡
∂
Ω
.
In the last decade or so, there was an extensive effort in studying the existence of solutions of (6); see [4–8].Before concluding this section, we recall a theorem from [9], which will be used to prove our main theorem in this paper.Theorem 2.
Let(
E
,
∥
·
∥
) be a Banach space and I
⊂
R
+ an interval. Consider the family of C
1 functionals on E,
(7)
J
ν
(
u
)
=
S
(
u
)
-
ν
T
(
u
)
,
ν
∈
I
,
with J
ν
(
0
)
=
0, ν
∈
I
,
T nonnegative and either S
(
u
)
→
∞ or T
(
u
)
→
∞ as ∥
u
∥
→
∞. For any ν
∈
I, we set
(8)
Γ
ν
=
{
γ
∈
C
(
[
0,1
]
,
E
)
:
γ
(
0
)
=
0
,
J
ν
(
γ
(
1
)
)
<
0
}
.
If for every ν
∈
I the set Γ
ν is nonempty and
(9)
c
ν
=
inf
⁡
γ
∈
Γ
ν
max
⁡
t
∈
[
0,1
]
J
ν
(
γ
(
t
)
)
>
0
,
then for almost every ν
∈
I there exists a sequence {
u
n
ν
}
⊂
E such that (i)
{
u
n
ν
} is bounded;
(ii)
J
ν
(
u
n
ν
)
→
c
ν as n
→
∞;
(iii)
J
ν
′
(
u
n
ν
)
→
0 in the dual E
* as n
→
∞.Throughout this paper, we denote byX the Sobolev space W
0
1
,
p
(
Ω
) with the norm ∥
u
∥
X
=
∥
u
∥
p, by X
* the duality space of X, by ⇀  the weak convergence in X, and by 〈
·
,
·
〉 the duality pairing between X
* and X. The letters C
1
,
C
2
,
… will denote various positive constants whose exact values are not essential to the analysis of the problem. Let P
=
{
u
∈
X
:
u
(
x
)
⩾
0, a.e. x
∈
Ω
} and p
*
=
N
p
/
(
N
-
p
) if p
<
N or ∞ if p
⩾
N.


## 2. Proof of Theorem1

In this section, we always assume(
f
1
) and (
f
2
) hold with f
0
<
λ
1 and f
∞
>
μ
1. Hence, there exist ε
1
>
0
,
C
ε
1
>
0 and q
∈
(
p
,
p
*
) such that
(10)
F
(
x
,
t
)
⩾
1
p
(
μ
1
+
ε
1
)
t
p
-
C
ε
1
,
x
∈
Ω
¯
,
t
⩾
0
,
(11)
F
(
x
,
t
)
⩽
1
2
(
1
-
ε
1
)
λ
1
t
2
+
C
ε
1
t
q
,
x
∈
Ω
¯
,
t
∈
R
,
where F
(
x
,
t
)
=
∫
0
t
‍
f
(
x
,
s
)
d
s. In the following, we utilize Theorem 2 to complete the proof of Theorem 1. In the setting of Theorem 2 we have E
=
X
,
I
=
[
δ
,
1
] with μ
1
/
(
μ
1
+
ε
1
)
<
δ
<
1, and
(12)
S
(
u
)
=
1
2
∥
u
∥
2
2
+
1
p
∥
u
∥
X
p
,
T
(
u
)
=
∫
Ω
‍
F
(
x
,
u
)
,
J
ν
(
u
)
=
1
2
∥
u
∥
2
2
+
1
p
∥
u
∥
X
p
-
ν
∫
Ω
‍
F
(
x
,
u
)
,
u
∈
X
,
ν
∈
I
.
It is easy to verify that
(13)
〈
J
ν
′
(
u
)
,
v
〉
=
∫
Ω
‍
∇
u
·
∇
v
+
∫
Ω
‍
|
∇
u
|
p
-
2
∇
u
·
∇
v
-
ν
∫
Ω
‍
f
(
x
,
u
)
v
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
k
l
h
h
h
l
h
h
u
,
v
∈
X
,
ν
∈
I
.
Firstly, we show that J
ν satisfies the conditions of Theorem 2 by proving several lemmas.Lemma 3.
Γ
ν
≠
∅ for any ν
∈
I.Proof.
Letϕ
1
>
0 be a μ
1-eigenfunction. For t
>
0, we have by (10) that
(14)
J
ν
(
t
ϕ
1
)
=
1
2
t
2
∥
ϕ
1
∥
2
2
+
1
p
t
p
∥
ϕ
1
∥
X
p
-
ν
∫
Ω
‍
F
(
x
,
t
ϕ
1
)
⩽
1
2
t
2
∥
ϕ
1
∥
2
2
+
1
p
μ
1
t
p
∫
Ω
‍
|
ϕ
1
|
p
-
1
p
(
μ
1
+
ε
1
)
δ
t
p
∫
Ω
‍
|
ϕ
1
|
p
+
C
1
=
1
2
t
2
∥
ϕ
1
∥
2
2
-
1
p
C
2
t
p
∫
Ω
‍
|
ϕ
1
|
p
+
C
1
,
where C
2
=
(
μ
1
+
ε
1
)
δ
-
μ
1. Noting that C
2
>
0, we can choose t
0
>
0 large enough so that J
ν
(
t
0
ϕ
1
)
<
0, where t
0 is independent of ν
∈
I. The proof is completed.Lemma 4.
There exists a constantc
>
0 such that c
v
⩾
c for any ν
∈
I.Proof.
For anyu
∈
X, it follows from (11) that
(15)
J
ν
(
u
)
=
1
2
∥
u
∥
2
2
+
1
p
∥
u
∥
X
p
-
ν
∫
Ω
‍
F
(
x
,
u
)
⩾
1
2
∥
u
∥
2
2
+
1
p
∥
u
∥
X
p
-
1
2
(
1
-
ε
1
)
λ
1
∫
Ω
‍
|
u
|
2
-
C
ε
1
∫
Ω
‍
|
u
|
q
⩾
1
2
∥
u
∥
2
2
+
1
p
∥
u
∥
X
p
-
1
2
(
1
-
ε
1
)
∥
u
∥
2
2
-
C
ε
1
∫
Ω
‍
|
u
|
q
⩾
1
p
∥
u
∥
X
p
-
C
ε
1
∫
Ω
‍
|
u
|
q
.
By Sobolev’s embedding theorem, we conclude that there exist ρ
>
0 and c
>
0 such that J
ν
(
u
)
>
0 for ∥
u
∥
∈
(
0
,
ρ
] and
(16)
J
ν
(
u
)
⩾
c
,
∥
u
∥
X
=
ρ
.
Fix ν
∈
I and γ
∈
Γ
ν. By the definition of Γ
ν, we have that ∥
γ
(
1
)
∥
>
ρ. Hence, there exists t
γ
∈
(
0,1
) such that ∥
γ
(
t
γ
)
∥
=
ρ. So
(17)
c
ν
=
inf
⁡
γ
∈
Γ
ν
max
⁡
t
∈
[
0,1
]
J
ν
(
γ
(
t
)
)
⩾
inf
⁡
γ
∈
Γ
ν
J
ν
(
γ
(
t
γ
)
)
⩾
c
.
The proof is completed.Lemma 5.
For anyν
∈
I, if {
u
n
} is bounded and J
ν
′
(
u
n
)
→
0 in X
* as n
→
∞, then {
u
n
} admits a convergent subsequence.Proof.
Givenν
∈
I, assume that {
u
n
} is bounded, J
ν
′
(
u
n
)
→
0 in X
* as n
→
∞. By extracting a subsequence, we may suppose that there exists u
∈
X such that as n
→
∞
(18)
u
n
⇀
u
in
X
,
u
n
⟶
u
in
L
s
(
Ω
)
,
s
∈
[
1
,
p
*
)
.
It follows from (
f
1
) and (
f
2
) that there exist C
1
,
C
2
>
0 such that
(19)
f
(
x
,
t
)
⩽
C
1
|
t
|
+
C
2
|
t
|
p
-
1
,
x
∈
Ω
¯
,
t
∈
R
.
Hence, by Hölder’s inequality and Sobolev’s embedding theorem, we have
(20)
|
∫
Ω
‍
f
(
x
,
u
n
)
(
u
n
-
u
)
|
⩽
C
1
∫
Ω
‍
|
u
n
|
|
u
n
-
u
|
+
C
2
∫
Ω
‍
|
u
n
|
p
-
1
|
u
n
-
u
|
⩽
C
1
(
∫
Ω
‍
|
u
n
|
2
)
1
/
2
(
∫
Ω
‍
|
u
n
-
u
|
2
)
1
/
2
+
C
2
(
∫
Ω
‍
|
u
n
|
p
)
(
p
-
1
)
/
p
(
∫
Ω
‍
|
u
n
-
u
|
p
)
1
/
p
⩽
C
3
(
∫
Ω
‍
|
u
n
-
u
|
2
)
1
/
2
+
C
4
(
∫
Ω
‍
|
u
n
-
u
|
p
)
1
/
p
⟶
0
,
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
.
n
⟶
∞
.
Similarly, we have
(21)
|
∫
Ω
‍
f
(
x
,
u
)
(
u
n
-
u
)
|
⟶
0
,
n
⟶
∞
.
Noting that
(22)
〈
J
ν
′
(
u
n
)
-
J
ν
′
(
u
)
,
u
n
-
u
〉
=
〈
J
ν
′
(
u
n
)
,
u
n
-
u
〉
-
〈
J
ν
′
(
u
)
,
u
n
-
u
〉
=
∫
Ω
‍
∇
u
n
·
∇
(
u
n
-
u
)
+
∫
Ω
‍
|
∇
u
n
|
p
-
2
∇
u
n
·
∇
(
u
n
-
u
)
-
ν
∫
Ω
‍
f
(
x
,
u
n
)
(
u
n
-
u
)
-
∫
Ω
‍
∇
u
·
∇
(
u
n
-
u
)
-
∫
Ω
‍
|
∇
u
|
p
-
2
∇
u
·
∇
(
u
n
-
u
)
+
ν
∫
Ω
‍
f
(
x
,
u
)
(
u
n
-
u
)
=
∫
Ω
‍
|
∇
(
u
n
-
u
)
|
2
+
∫
Ω
‍
(
|
∇
u
n
|
p
-
2
∇
u
n
-
|
∇
u
|
p
-
2
∇
u
)
·
∇
(
u
n
-
u
)
-
ν
∫
Ω
‍
f
(
x
,
u
n
)
(
u
n
-
u
)
+
ν
∫
Ω
‍
f
(
x
,
u
)
(
u
n
-
u
)
,
and the inequality deduced from an inequality in Appendix of [3],
(23)
∫
Ω
‍
(
|
∇
u
n
|
p
-
2
∇
u
n
-
|
∇
u
|
p
-
2
∇
u
)
·
∇
(
u
n
-
u
)
⩾
2
p
(
2
p
-
1
-
1
)
∫
Ω
‍
|
∇
(
u
n
-
u
)
|
p
,
it follows from (20) and (21) that
(24)
2
p
(
2
p
-
1
-
1
)
∫
Ω
‍
|
∇
(
u
n
-
u
)
|
p
⩽
〈
J
ν
′
(
u
n
)
-
J
ν
′
(
u
)
,
u
n
-
u
〉
+
ν
∫
Ω
‍
f
(
x
,
u
n
)
(
u
n
-
u
)
-
ν
∫
Ω
‍
f
(
x
,
u
)
(
u
n
-
u
)
⟶
0
,
n
⟶
∞
,
where we have used the fact that
(25)
〈
J
ν
′
(
u
n
)
-
J
ν
′
(
u
)
,
u
n
-
u
〉
⟶
0
,
n
⟶
∞
.
Hence u
n
→
u in X. The proof is completed.Lemma 6.
There exists a sequence{
ν
n
}
⊂
I with ν
n
→
1
- as n
→
∞ and {
u
ν
n
}
⊂
X such that J
ν
n
(
u
ν
n
)
=
c
ν
n, J
ν
n
′
(
u
ν
n
)
=
0.Proof.
We only need to show that for almost everyν
∈
I there exists u
ν
∈
X such that J
ν
(
u
ν
)
=
c
ν and J
ν
′
(
u
ν
)
=
0. By Theorem 2, for almost each ν
∈
I, there exists a bounded sequence {
u
n
ν
}
⊂
X such that
(26)
J
ν
(
u
n
ν
)
⟶
c
ν
,
J
ν
′
(
u
n
ν
)
⟶
0
,
n
⟶
∞
.
By Lemma 5, we may assume that u
n
ν
→
u
ν in X as n
→
∞. Then the continuity of J
ν and J
ν
′ implies that J
ν
(
u
ν
)
=
c
ν and J
ν
′
(
u
ν
)
=
0. The proof is completed.Define〈
L
u
,
v
〉
=
∫
Ω
‍
f
(
x
,
u
)
v
,
〈
K
u
,
v
〉
=
∫
Ω
‍
|
u
|
p
-
2
u
v, u
,
v
∈
X. Then we have the following.Lemma 7.
Supposing(
f
1
) and (
f
2
) hold, then
(27)
lim
⁡
∥
u
∥
X
→
∞
,
u
∈
P
L
u
-
f
∞
K
u
∥
u
∥
X
p
-
1
=
0
.Proof.
By(
f
1
), (
f
2
), for every ε
>
0, there is a constant C
ε
>
0 such that
(28)
|
f
(
x
,
t
)
-
f
∞
t
p
-
1
|
⩽
C
ε
+
ε
t
p
-
1
,
x
∈
Ω
¯
,
t
⩾
0
.
For u
∈
P
∖
{
0
}, letting w
=
u
/
∥
u
∥
X, by Hölder’s inequality and Sobolev’s embedding theorem, we have
(29)
sup
⁡
∥
v
∥
X
⩽
1
|
〈
L
u
-
f
∞
K
u
∥
u
∥
X
p
-
1
,
v
〉
|
⩽
sup
⁡
∥
v
∥
X
⩽
1
∫
Ω
‍
|
f
(
x
,
u
)
-
f
∞
u
p
-
1
|
∥
u
∥
X
p
-
1
|
v
|
⩽
sup
⁡
∥
v
∥
X
⩽
1
∫
Ω
‍
C
ε
∥
u
∥
X
-
(
p
-
1
)
|
v
|
+
ε
w
p
-
1
|
v
|
⩽
C
6
∥
u
∥
X
-
(
p
-
1
)
+
ε
C
5
,
where C
5 is independent of ε. The proof is completed.Proof of Theorem1.
By Lemma6, there exists a sequence {
ν
n
}
⊂
I with ν
n
→
1
- as n
→
∞ and {
u
ν
n
}
⊂
X such that
(30)
J
ν
n
(
u
ν
n
)
=
c
ν
n
,
J
ν
n
′
(
u
ν
n
)
=
0
.
By Lemma 4 and (30), we have c
ν
n
⩾
c
>
0 and 〈
J
ν
n
′
(
u
ν
n
)
,
u
ν
n
-
〉
=
0. Hence u
ν
n
∈
P
∖
{
0
}. In the following, we first claim that {
u
ν
n
} is bounded in X. Suppose by contradiction that lim
⁡
n
→
∞
∥
u
ν
n
∥
X
=
∞. Let w
n
=
u
ν
n
/
∥
u
ν
n
∥
X. Hence, we have, for v
∈
X,
(31)
1
∥
u
ν
n
∥
X
p
-
2
∫
Ω
‍
∇
w
n
∇
v
+
∫
Ω
‍
|
∇
w
n
|
p
-
2
∇
w
n
∇
v
=
ν
n
f
∞
∫
Ω
‍
w
n
p
-
1
v
+
ν
n
∫
Ω
‍
f
(
x
,
u
ν
n
)
-
f
∞
u
ν
n
p
-
1
∥
u
ν
n
∥
X
p
-
1
v
.
Since {
w
n
} is bounded in X, we may assume that w
n
⇀
w
0
∈
P
⊂
X
,
w
n
→
w
0 in L
p
(
Ω
) and w
n
(
x
)
→
w
0
(
x
) a.e. on Ω as n
→
∞. Letting v
=
w
n
-
w
0 in (31) and n
→
∞, we get
(32)
lim
⁡
n
→
∞
∫
Ω
‍
|
∇
w
n
|
p
-
2
∇
w
n
·
∇
(
w
n
-
w
0
)
=
0
.
It follows from [5, Theorem 10] that w
n
→
w
0 in X as n
→
∞. Passing to limit n
→
∞ in (31), we obtain by Lemma 7 that
(33)
∫
Ω
‍
|
∇
w
0
|
p
-
2
∇
w
0
·
∇
v
=
f
∞
∫
Ω
‍
w
0
p
-
1
v
,
v
∈
X
.
From (33) and the fact that ∥
w
0
∥
X
=
1, we know that f
∞
=
μ
1, which contradicts the assumption f
∞
>
μ
1. Since ν
n
→
1
-, we can show that
(34)
J
1
′
(
u
ν
n
)
⟶
0
in
X
*
,
n
⟶
∞
.
In fact, for any v
∈
X, it follows from (19), Hölder’s inequality, and Sobolev’s embedding theorem that
(35)
|
∫
Ω
‍
f
(
x
,
u
ν
n
)
v
|
⩽
C
1
∫
Ω
‍
|
u
ν
n
|
|
v
|
+
C
2
∫
Ω
‍
|
u
ν
n
|
p
-
1
|
v
|
⩽
C
7
∥
v
∥
X
.
Furthermore, (30) implies that
(36)
〈
J
1
′
(
u
ν
n
)
,
v
〉
+
(
1
-
ν
n
)
∫
Ω
‍
f
(
x
,
u
ν
n
)
v
=
〈
J
ν
n
′
(
u
ν
n
)
,
v
〉
=
0
,
llllllllllllllllllllll
.
.
v
∈
X
.
Hence, J
1
′
(
u
ν
n
)
→
0 in X
* as n
→
∞. By Lemma 5, {
u
ν
n
} has a convergent subsequence. Without loss of generality, we may assume that u
ν
n
→
u as n
→
∞. According to Lemma 4, (30) and
(37)
|
∫
Ω
‍
F
(
x
,
u
ν
n
)
|
⩽
C
8
,
we have
(38)
J
1
(
u
)
=
lim
⁡
n
→
∞
J
1
(
u
ν
n
)
=
lim
⁡
n
→
∞
J
ν
n
(
u
ν
n
)
⩾
c
>
0
,
J
1
′
(
u
)
=
lim
⁡
n
→
∞
J
1
′
(
u
ν
n
)
=
0
.
The standard process shows that u is a positive solution to (1). The proof is completed.



---

*Source: 364010-2014-10-14.xml*