# Existence of Nontrivial Solutions ofp-Laplacian Equation with Sign-Changing Weight Functions

Authors: Ghanmi Abdeljabbar
Journal: ISRN Mathematical Analysis
 (2014)

---

## Abstract

This paper shows the existence and multiplicity of nontrivial solutions of thep-Laplacian problem -
Δ
p
u
=
1
/
σ
(
∂
F
(
x
,
u
)
/
∂
u
)
+
λ
a
(
x
)
|
u
|
q
-
2
u for x
∈
Ω with zero Dirichlet boundary conditions, where Ω is a bounded open set in ℝ
n, 1
<
q
<
p
<
σ
<
p
*
(
p
*
=
n
p
/
(
n
-
p
) if p
<
n, p
*
=
∞ if p
≥
n), λ
∈
ℝ
∖
{
0
}, a is a smooth function which may change sign in Ω
̅,, and F
∈
C
1
(
Ω
̅
×
ℝ
,
ℝ
). The method is based on Nehari results on three submanifolds of the space W
0
1
,
p
(
Ω
).


---

## Body

## 1. Introduction

In this paper, we are concerned with the multiplicity of nontrivial nonnegative solutions of the following elliptic equation:(1)
-
Δ
p
u
=
1
σ
∂
F
(
x
,
u
)
∂
u
+
λ
a
(
x
)
|
u
|
q
-
2
u
in
Ω
,
u
=
0
,
on
⁡
∂
Ω
,
where Ω is a bounded domain of ℝ
n
(
n
≥
3
), 1
<
q
<
p
<
σ
<
p
*
(
p
*
=
n
p
/
(
n
-
p
) if p
<
n, p
*
=
∞ if p
≥
n
), λ
∈
ℝ
∖
{
0
}, F
∈
C
1
(
Ω
¯
×
ℝ
,
ℝ
) is positively homogeneous of degree σ; that is, F
(
x
,
t
u
)
=
t
σ
F
(
x
,
u
) holds for all (
x
,
u
)
∈
Ω
¯
×
ℝ and the sign-changing weight function a satisfies the following condition:(A)a
∈
C
(
Ω
) with ∥
a
∥
∞
=
1,  a
+
≔
max
⁡
(
+
a
,
0
)
≢
0, and  a
-
≔
max
⁡
(
-
a
,
0
)
≢
0.In recent years, several authors have used the Nehari manifold and fibering maps (i.e., maps of the formt
↦
J
λ
(
t
u
), where J
λ is the Euler function associated with the equation) to solve semilinear and quasilinear problems. For instance, we cite papers [1–9] and references therein. More precisely, Brown and Zhang [10] studied the following subcritical semilinear elliptic equation with sign-changing weight function:
(2)
-
Δ
u
=
λ
a
(
x
)
u
+
b
(
x
)
|
u
|
γ
-
2
u
(
x
)
,
in
Ω
,
u
=
0
,
on
⁡
∂
Ω
,
where γ
>
2. Also, the authors in [10] by the same arguments considered the following semilinear elliptic problem:
(3)
-
Δ
u
=
λ
f
(
x
)
|
u
|
q
-
2
u
(
x
)
+
g
(
x
)
|
u
|
p
-
2
u
(
x
)
,
in
Ω
,
u
=
0
,
on
⁡
∂
Ω
,
where 1
<
q
<
2
<
p. Exploiting the relationship between the Nehari manifold and fibering maps, they gave an interesting explanation of the well-known bifurcation result. In fact, the nature of the Nehari manifold changes as the parameter λ crosses the bifurcation value.Inspired by the work of Brown and Zhang [10], Nyamouradi [11] treated the following problem:
(4)
Δ
(
|
Δ
u
|
p
-
2
Δ
u
)
=
1
p
*
f
(
x
,
u
)
+
λ
|
u
|
q
-
2
u
(
x
)
;
in
Ω
,
u
=
Δ
u
=
0
,
on
⁡
∂
Ω
,
where f is positively homogeneous of degree p
*
-
1.In this work, motivated by the above works, we give a very simple variational method to prove the existence of at least two nontrivial solutions of problem (1). In fact, we use the decomposition of the Nehari manifold as λ vary to prove our main result.Before stating our main result, we need the following assumptions:(H1)
F
:
Ω
¯
×
ℝ
→
ℝ is a C
1 function such that
(5)
F
(
x
,
t
u
)
=
t
σ
F
(
x
,
u
)
(
t
>
0
)
∀
x
∈
Ω
¯
,
u
∈
ℝ
.
(H2)
F
(
x
,
0
)
=
(
∂
F
/
∂
u
)
(
x
,
0
)
=
0, F
+
(
x
,
u
)
=
max
⁡
(
+
F
(
x
,
u
)
,
0
)
≢
0, and F
-
(
x
,
u
)
=
max
⁡
(
-
F
(
x
,
u
)
,
0
)
≢
0 for all u
≠
0.We remark that using assumption (H1), for all x
∈
Ω
¯, u
∈
ℝ, we have the so-called Euler identity:
(6)
u
∂
F
(
x
,
u
)
∂
u
=
σ
F
(
x
,
u
)
,
|
F
(
x
,
u
)
|
≤
K
|
u
|
σ
for
some
constant
K
>
0
.
Our main result is the following.Theorem 1.
Under the assumptions (A), (H1), and (H2), there exists λ
0
>
0 such that for all 0
<
|
λ
|
<
λ
0, problem (1) has at least two nontrivial nonnegative solutions.This paper is organized as follows. In Section2, we give some notations and preliminaries and we present some technical lemmas which are crucial in the proof of Theorem 1. Theorem 1 is proved in Section 3.


## 2. Some Notations and Preliminaries

Throughout this paper, we denote byS
l the best Sobolev constant for the operators W
0
1
,
p
(
Ω
)
↪
L
l
(
Ω
), given by
(7)
S
l
=
inf
⁡
u
∈
W
0
1
,
p
(
Ω
)
∖
{
0
}
∫
Ω
|
∇
u
|
p
d
x
(
∫
Ω
|
u
|
l
)
p
/
l
d
x
,
where 1
<
l
≤
p
*. In particular, we have
(8)
∫
Ω
|
u
|
l
d
x
≤
S
l
-
l
/
p
∥
u
∥
l
∀
u
∈
W
0
1
,
p
(
Ω
)
,
with the standard norm
(9)
∥
u
∥
=
(
∫
Ω
|
∇
u
|
p
d
x
)
1
/
p
.
Problem (1) is posed in the framework of the Sobolev space E
=
W
0
1
,
p
(
Ω
). Moreover, a function u in E is said to be a weak solution of problem (1) if
(10)
∫
Ω
|
∇
u
|
p
-
2
∇
u
∇
φ
d
x
-
1
σ
∫
Ω
‍
∂
F
(
x
,
u
)
∂
u
φ
d
x
-
λ
∫
Ω
a
|
u
|
q
-
2
u
φ
d
x
=
0
,
∀
φ
∈
E
.
Thus, by (6) the corresponding energy functional of problem (1) is defined in E by
(11)
J
λ
(
u
)
=
1
p
∥
u
∥
p
-
1
σ
∫
Ω
F
(
x
,
u
)
d
x
-
λ
q
∫
Ω
a
(
x
)
|
u
|
q
d
x
.
In order to verify J
λ
∈
C
1
(
E
,
ℝ
), we need the following lemmas.Lemma 2.
Assume thatF
∈
C
1
(
Ω
¯
×
ℝ
,
ℝ
) is positively homogeneous of degree σ; then ∂
F
/
∂
u
∈
C
(
Ω
¯
×
ℝ
,
ℝ
) is positively homogeneous of degree σ
-
1.Proof.
The proof is the same as that in Chu and Tang [4].
In addition, by Lemma2, we get the existence of positive constant M such that
(12)
|
∂
F
(
x
,
u
)
∂
u
|
≤
M
|
u
|
σ
-
1
,
∀
x
∈
Ω
¯
,
u
∈
ℝ
.Lemma 3 (see [12], Theorem A.2).
Letp
,
r
∈
[
1
,
∞
) and f
∈
C
(
Ω
¯
×
ℝ
,
ℝ
) such that
(13)
|
f
(
x
,
u
)
|
≤
c
(
1
+
|
u
|
p
/
r
)
,
∀
x
∈
Ω
¯
,
∀
u
∈
ℝ
.
Then for every u
∈
L
p
(
Ω
), one has f
(
·
,
u
)
∈
L
r
(
Ω
); moreover the operator A
:
L
p
(
Ω
)
→
L
r
(
Ω
) defined by A
(
u
)
(
x
)
=
f
(
x
,
u
(
x
)
) is continuous.Lemma 4 (See Proposition 1 in [13]).
Suppose that∂
F
(
x
,
u
)
/
∂
u
∈
C
(
Ω
¯
×
ℝ
,
ℝ
) verifies condition (12). Then, the functional J
λ belongs to C
1
(
E
,
ℝ
), and
(14)
〈
J
λ
′
(
u
)
,
u
〉
=
∥
u
∥
p
-
∫
Ω
‍
F
(
x
,
u
)
d
x
-
λ
∫
Ω
‍
a
(
x
)
|
u
|
q
d
x
,
where 〈
·
,
·
〉 denotes the usual duality between E and E
*
:
=
W
-
1
,
p
′
(
Ω
) (the dual space of the sobolev space E).
As the energy functionalJ
λ is not bounded below in E, it is useful to consider the functional on the Nehari manifold:
(15)
N
λ
=
{
u
∈
E
∖
{
0
}
:
〈
J
λ
′
(
u
)
,
u
〉
=
0
}
.
Thus, u
∈
N
λ if and only if
(16)
∥
u
∥
p
-
∫
Ω
‍
F
(
x
,
u
)
d
x
-
λ
∫
Ω
‍
a
(
x
)
|
u
|
q
d
x
=
0
.
Note that N
λ contains every nonzero solution of problem (1). Moreover, one has the following result.Lemma 5.
The energy functionalJ
λ is coercive and bounded below on N
λ.Proof.
Ifu
∈
N
λ, then by (16) and condition (A) we obtain
(17)
J
λ
(
u
)
=
σ
-
p
σ
p
∥
u
∥
p
-
λ
σ
-
q
σ
q
∫
Ω
‍
a
(
x
)
|
u
|
q
d
x
,
≥
σ
-
p
σ
p
∥
u
∥
p
-
|
λ
|
σ
-
q
σ
q
∫
Ω
‍
|
u
|
q
d
x
.
So, it follows from (8) that
(18)
J
λ
(
u
)
≥
σ
-
p
σ
p
∥
u
∥
p
-
|
λ
|
S
q
q
/
p
σ
-
q
σ
q
∥
u
∥
q
.
Thus, J
λ is coercive and bounded below on N
λ.
Define(19)
ϕ
λ
(
u
)
=
〈
J
λ
′
(
u
)
,
u
〉
.
Then, by (16) it is easy to see that for u
∈
N
λ,
(20)
〈
ϕ
λ
′
(
u
)
,
u
〉
=
p
∥
u
∥
p
-
σ
∫
Ω
‍
F
(
x
,
u
)
d
x
-
λ
q
∫
Ω
‍
a
(
x
)
|
u
|
q
d
x
,
(21)
=
λ
(
p
-
q
)
∫
Ω
‍
a
(
x
)
|
u
|
q
d
x
-
(
σ
-
p
)
∫
Ω
‍
F
(
x
,
u
)
d
x
,
(22)
=
λ
(
σ
-
q
)
∫
Ω
‍
a
(
x
)
|
u
|
q
d
x
-
(
σ
-
p
)
∥
u
∥
p
,
(23)
=
(
p
-
q
)
∥
u
∥
p
-
(
σ
-
q
)
∫
Ω
‍
F
(
x
,
u
)
d
x
.
Now, we split N
λ into three parts
(24)
N
λ
+
=
{
u
∈
N
λ
:
〈
ϕ
λ
′
(
u
)
,
u
〉
>
0
}
,
N
λ
0
=
{
u
∈
N
λ
:
〈
ϕ
λ
′
(
u
)
,
u
〉
=
0
}
,
N
λ
-
=
{
u
∈
N
λ
:
〈
ϕ
λ
′
(
u
)
,
u
〉
<
0
}
.Lemma 6.
Assume thatu
0 is a local minimizer for J
λ on N
λ and that u
0
∉
N
λ
0. Then, J
λ
′
(
u
0
)
=
0 in E
-
1 (the dual space of the Sobolev space E).Proof.
Our proof is the same as that in Brown-Zhang [10, Theorem 2.3].Lemma 7.
One has the following:(i)
ifu
∈
N
λ
+, then λ
∫
Ω
‍
a
(
x
)
|
u
|
q
d
x
>
0;
(ii)
ifu
∈
N
λ
0, then λ
∫
Ω
‍
a
(
x
)
|
u
|
q
d
x
>
0 and ∫
Ω
‍
F
(
x
,
u
)
d
x
>
0;
(iii)
ifu
∈
N
λ
-, then ∫
Ω
‍
F
(
x
,
u
)
d
x
>
0.Proof.
The proof is immediate from (21), (22), and (23).
From now on, we denote byλ
0 the constant defined by
(25)
λ
0
=
q
(
σ
-
p
)
p
(
σ
-
q
)
S
q
q
/
p
(
p
-
q
K
(
σ
-
q
)
S
σ
σ
/
p
)
(
p
-
q
)
(
σ
-
p
)
,
then we have the following.Lemma 8.
If0
<
|
λ
|
<
λ
0, then N
λ
0
=
⌀.Proof.
Suppose otherwise, that0
<
|
λ
|
<
λ
0 such that N
λ
0
≠
⌀. Then for u
∈
N
λ
0, we have
(26)
0
=
〈
ϕ
λ
′
(
u
)
,
u
〉
=
λ
(
σ
-
q
)
∫
Ω
‍
a
(
x
)
|
u
|
q
d
x
-
(
σ
-
p
)
∥
u
∥
p
(27)
=
(
p
-
q
)
∥
u
∥
p
-
(
σ
-
q
)
∫
Ω
‍
F
(
x
,
u
)
d
x
.
From the Hölder inequality, (6) and (8), it follows that
(28)
∫
Ω
F
(
x
,
u
)
d
x
≤
∫
Ω
|
F
(
x
,
u
)
|
d
x
≤
K
∫
Ω
|
u
|
σ
d
x
≤
K
S
σ
-
σ
/
p
∥
u
∥
σ
.
Hence, it follows from (27) that
(29)
∥
u
∥
p
=
σ
-
q
p
-
q
∫
Ω
F
(
x
,
u
)
d
x
≤
σ
-
q
p
-
q
K
S
σ
-
σ
/
p
∥
u
∥
σ
,
then,
(30)
∥
u
∥
≥
(
p
-
q
K
(
σ
-
q
)
S
σ
σ
/
p
)
1
/
(
σ
-
p
)
.
On the other hand, from condition (A), (8) and (26) we have(31)
∥
u
∥
p
=
λ
σ
-
q
σ
-
p
∫
Ω
a
(
x
)
|
u
|
q
d
x
≤
|
λ
|
σ
-
q
σ
-
p
S
q
-
q
/
p
∥
u
∥
q
.
So,
(32)
∥
u
∥
≤
(
|
λ
|
σ
-
q
σ
-
p
S
q
-
q
/
p
)
1
/
(
p
-
q
)
.
Combining (30) and (32), we obtain λ
0
≤
|
λ
|, which is a contradiction.
By Lemma8, for 0
<
|
λ
|
<
λ
0, we write N
λ
=
N
λ
+
∪
N
λ
- and define
(33)
θ
λ
=
inf
⁡
u
∈
N
λ
J
λ
(
u
)
,
θ
λ
+
=
inf
⁡
u
∈
N
λ
+
J
λ
(
u
)
,
θ
λ
-
=
inf
⁡
u
∈
N
λ
-
J
λ
(
u
)
.
Then, we have the following.Lemma 9.
If0
<
|
λ
|
<
λ
0, then
(34)
θ
λ
≤
θ
λ
+
<
0
,
θ
λ
-
>
d
0
for some d
0
>
0 depending on p
,
q
,
σ
,
K
,
λ
,
S
q, and S
σ.Proof.
Letu
∈
N
λ
+. Then, from (23) we have
(35)
p
-
q
σ
-
q
∥
u
∥
p
>
∫
Ω
F
(
x
,
u
)
d
x
.
So
(36)
J
λ
(
u
)
=
q
-
p
p
q
∥
u
∥
p
+
σ
-
q
σ
q
∫
Ω
F
(
x
,
u
)
d
x
<
(
q
-
p
p
q
+
σ
-
q
σ
q
p
-
q
σ
-
q
)
∥
u
∥
p
=
-
(
p
-
q
)
(
σ
-
p
)
p
q
σ
∥
u
∥
p
<
0
.
Thus, from the definition of θ
λ and θ
λ
+, we can deduce that θ
λ
≤
θ
λ
+
<
0.
Now, letu
∈
N
λ
-. Then, using (6) and (8) we obtain
(37)
p
-
q
σ
-
q
∥
u
∥
p
<
∫
Ω
F
(
x
,
u
)
d
x
≤
K
S
σ
-
σ
/
p
∥
u
∥
σ
,
this implies that
(38)
∥
u
∥
>
(
p
-
q
σ
-
q
S
σ
σ
/
p
K
)
1
/
(
σ
-
p
)
,
∀
u
∈
N
λ
-
.
In addition, by (18) and (38)
(39)
J
λ
(
u
)
≥
σ
-
p
p
σ
∥
u
∥
p
-
|
λ
|
S
q
-
q
/
p
σ
-
p
σ
q
∥
u
∥
q
≥
∥
u
∥
q
[
σ
-
p
p
σ
∥
u
∥
p
-
q
-
|
λ
|
S
q
-
q
/
p
σ
-
q
σ
q
]
>
(
p

-

q
σ

-

q
S
σ
σ
/
p
K
)
q
/
(
σ
-
p
)
(
σ
-
p
p
σ
(
p

-

q
σ

-

q
S
σ
σ
/
p
K
)
(
p
-
q
)
/
(
σ
-
p
)
-
|
λ
|
S
q
-
q
/
p
σ
-
q
σ
q
σ
-
p
p
σ
(
p
-
q
σ
-
q
S
σ
σ
/
p
K
)
(
p
-
q
)
/
(
σ
-
p
)
)
.
Thus, since 0
<
|
λ
|
<
λ
0, we conclude that J
λ
>
d
0 for some d
0
>
0. This completes the proof.Foru
∈
E with ∫
Ω
‍
F
(
x
,
u
)
d
x
>
0, set
(40)
T
=
(
(
p
-
q
)
∥
u
∥
p
(
σ
-
q
)
∫
Ω
‍
F
(
x
,
u
)
d
x
)
1
/
(
σ
-
p
)
>
0
.
Then, the following lemma holds.Lemma 10.
For eachu
∈
E with ∫
Ω
‍
F
(
x
,
u
)
d
x
>
0, one has the following:(i)
ifλ
∫
Ω
‍
a
(
x
)
|
u
|
q
d
x
≤
0, then there exists unique t
-
>
T such that t
-
u
∈
N
λ
- and
(41)
J
λ
(
t
-
u
)
=
sup
⁡
t
≥
0
J
λ
(
t
u
)
;
(ii)
ifλ
∫
Ω
‍
a
(
x
)
|
u
|
q
d
x
>
0, then there are unique 0
<
t
+
<
T
<
t
- such that (
t
-
u
,
t
+
u
)
∈
N
λ
-
×
N
λ
+ and
(42)
J
λ
(
t
-
u
)
=
sup
⁡
t
≥
0
J
λ
(
t
u
)
,
J
λ
(
t
+
u
)
=
inf
⁡
0
≤
t
≤
T
J
λ
(
t
u
)
.Proof.
We fixu
∈
E with ∫
Ω
‍
F
(
x
,
u
)
d
x
>
0 and we let
(43)
m
(
t
)
=
t
p
-
q
∥
u
∥
p
-
t
σ
-
q
∫
Ω
F
(
x
,
u
)
d
x
for
t
≥
0
.
Then, it is easy to check that m
(
t
) achieves its maximum at T. Moreover,
(44)
m
(
T
)
=
∥
u
∥
q
[
(
p
-
q
σ
-
q
)
(
p
-
q
)
(
σ
-
p
)
-
(
p
-
q
σ
-
q
)
(
σ
-
q
)
/
(
σ
-
p
)
]
×
(
∥
u
∥
σ
∫
Ω
‍
F
(
x
,
u
)
d
x
)
(
p
-
q
)
/
(
σ
-
p
)
≥
∥
u
∥
q
(
σ
-
p
σ
-
q
)
(
(
σ
-
q
)
S
σ
σ
/
p
K
(
p
-
q
)
)
(
p
-
q
)
/
(
σ
-
p
)
.
(i) We suppose thatλ
∫
Ω
‍
a
(
x
)
|
u
|
q
d
x
≤
0. Since m
(
0
)
=
0
,
m
(
t
)
→
-
∞ as t
→
∞, m
′
(
t
)
>
0 for t
<
T and m
′
(
t
)
<
0 for t
>
T. There is a unique t
-
>
T such that m
(
t
-
)
=
λ
∫
Ω
‍
a
(
x
)
|
u
|
q
d
x
≤
0.
Now, it follows from (14) and (27) that
(45)
ϕ
λ
′
(
t
-
u
)
t
-
u
=
(
t
-
)
1
+
q
m
′
(
t
-
)
<
0
,
J
λ
′
(
t
-
u
)
t
-
u
=
(
t
-
)
q
(
m
(
t
-
)
-
λ
∫
Ω
‍
a
(
x
)
|
u
|
q
d
x
)
=
0
.
Hence, t
-
u
∈
N
λ
-. On the other hand, it is easy to see that for all t
>
T
(46)
d
2
d
t
2
J
λ
(
t
u
)
<
0
,
d
d
t
J
λ
(
t
u
)
=
0
for
t
=
t
-
.
Thus, J
λ
(
t
-
u
)
=
sup
⁡
t
≥
0
J
λ
(
t
u
).
(ii) We suppose thatλ
∫
Ω
‍
a
(
x
)
|
u
|
q
d
x
>
0. Then, by (A), (8) and the fact that |
λ
|
<
λ
0 we obtain
(47)
m
(
0
)
=
0
<
λ
∫
Ω
‍
a
(
x
)
|
u
|
q
d
x
≤
|
λ
|
S
q
-
q
/
p
∥
u
∥
q
<
m
(
T
)
.
Then, there are unique t
+ and t
- such that 0
<
t
+
<
T
<
t
-, m
(
t
+
)
=
λ
∫
Ω
‍
a
(
x
)
|
u
|
q
d
x
=
m
(
t
-
), and m
′
(
t
-
)
<
0
<
m
′
(
t
+
). We have (
t
-
u
,
t
+
u
)
∈
N
λ
-
×
N
λ
+, and
(48)
J
λ
(
t
+
u
)
≤
J
λ
(
t
u
)
≤
J
λ
(
t
-
u
)
∀
t
∈
[
t
+
,
t
-
]
,
J
λ
(
t
+
u
)
≤
J
λ
(
t
u
)
∀
0
≤
t
≤
t
+
.
Thus,
(49)
J
λ
(
t
-
u
)
=
sup
⁡
t
≥
0
J
λ
(
t
u
)
,
J
λ
(
t
+
u
)
=
inf
⁡
0
≤
t
≤
T
J
λ
(
t
u
)
.
This completes the proof.For eachu
∈
E with λ
∫
Ω
‍
a
(
x
)
|
u
|
q
d
x
>
0, set
(50)
T
~
=
(
(
σ
-
q
)
λ
∫
Ω
a
(
x
)
|
u
|
q
d
x
(
σ
-
p
)
∥
u
∥
p
)
1
/
(
p
-
q
)
>
0
.
Then we have the following.Lemma 11.
For eachu
∈
E with λ
∫
Ω
‍
a
(
x
)
|
u
|
q
d
x
>
0, one has the following:(i)
if∫
Ω
‍
F
(
x
,
u
)
d
x
≤
0, then there exists a unique 0
<
t
+
<
T
~ such that t
+
u
∈
N
λ
+ and
(51)
J
λ
(
t
+
u
)
=
inf
⁡
t
≥
0
J
λ
(
t
u
)
;
(ii)
if∫
Ω
‍
F
(
x
,
u
)
d
x
>
0, then there are unique 0
<
t
+
<
T
~
<
t
- such that (
t
-
u
,
t
+
u
)
∈
N
λ
-
×
N
λ
+ and
(52)
J
λ
(
t
-
u
)
=
sup
⁡
t
≥
0
J
λ
(
t
u
)
,
J
λ
(
t
+
u
)
=
inf
⁡
0
≤
t
≤
T
~
J
λ
(
t
u
)
.Proof.
Foru
∈
E with λ
∫
Ω
‍
a
(
x
)
|
u
|
q
d
x
>
0, we can take
(53)
m
~
(
t
)
=
t
p
-
σ
∥
u
∥
p
-
λ
t
q
-
σ
∫
Ω
‍
a
(
x
)
|
u
|
q
d
x
for
t
>
0
,
and similar to the argument in Lemma 9, we obtain the results of Lemma 10.Proposition 12.
(i) There exist minimizing sequences{
u
n
+
} in N
λ
+ such that
(54)
J
λ
(
u
n
+
)
=
θ
λ
+
+
∘
(
1
)
,
J
λ
′
(
u
n
+
)
=
∘
(
1
)
in
E
-
1
.
(ii) There exist minimizing sequences{
u
n
-
} in N
λ
- such that
(55)
J
λ
(
u
n
-
)
=
θ
λ
-
+
∘
(
1
)
,
J
λ
′
(
u
n
-
)
=
∘
(
1
)
in
E
-
1
.Proof.
The proof is almost the same as that in Wu [14, Proposition 9] and is omitted here.


## 3. Proof of Our Result

Throughout this section, the normL
s is denoted by ∥
·
∥
s for 1
≤
s
≤
∞ and the parameter λ satisfies 0
<
|
λ
|
<
λ
0.Theorem 13.
If0
<
|
λ
|
<
λ
0, then, problem (1) has a positive solution u
0
+ in N
λ
+ such that
(56)
J
λ
(
u
0
+
)
=
θ
λ
=
θ
λ
+
.Proof.
By Proposition12(i), there exists a minimizing sequence {
u
n
+
} for J
λ on N
λ
+ such that
(57)
J
λ
(
u
n
+
)
=
θ
λ
+
+
o
(
1
)
,
J
λ
′
(
u
n
+
)
=
o
(
1
)
in
E
-
1
.
Then by Lemma 5, there exists a subsequence {
u
n
} and u
0
+ in E such that
(58)
u
n
⇀
u
0
+
weakly
in
E
,
u
n
⟶
u
0
+
strongly
in
L
q
(
Ω
)
and
in
L
σ
(
Ω
)
.
This implies that ∫
Ω
‍
a
(
x
)
|
u
n
|
q
d
x
→
∫
Ω
‍
a
(
x
)
|
u
0
+
|
q
d
x as n
→
∞.
Next, we will show that(59)
∫
Ω
‍
F
(
x
,
u
n
)
d
x
⟶
∫
Ω
F
(
x
,
u
0
+
)
d
x
as
n
⟶
∞
.
By Lemma 3, we have
(60)
∂
F
(
x
,
u
n
)
∂
u

∈

L
γ

(
Ω
)
,
∂
F
(
x
,
u
n
)
∂
u

⟶

∂
F
(
x
,
u
0
+
)
∂
u
in
L
γ

(
Ω
)
,
where γ
=
σ
/
(
σ
-
1
). On the other hand, it follows from the Hölder inequality that
(61)
∫
Ω
‍
|
u
n
∂
F
(
x
,
u
n
)
∂
u
-
u
0
+
∂
F
(
x
,
u
0
+
)
∂
u
|
d
x
≤
∫
Ω
‍
|
(
u
n
-
u
0
+
)
∂
F
(
x
,
u
n
)
∂
u
|
d
x
+
∫
Ω
‍
|
u
0
+
|
|
∂
F
(
x
,
u
n
)
∂
u
-
∂
F
(
x
,
u
0
+
)
∂
u
|
d
x
≤
∥
u
n
-
u
0
+
∥
σ
∥
∂
F
(
x
,
u
n
)
∂
u
∥
γ
+
∥
u
0
+
∥
σ
∥
∂
F
(
x
,
u
n
)
∂
u
-
∂
F
(
x
,
u
0
+
)
∂
u
∥
γ
⟶
0
as
n
⟶
∞
.
Hence, ∫
Ω
‍
F
(
x
,
u
n
)
d
x
→
∫
Ω
‍
F
(
x
,
u
0
+
)
d
x as n
→
∞.
By (57) and (58) it is easy to prove that u
0
+ is a weak solution of (1).
Since(62)
J
λ
(
u
n
)
=
σ
-
p
p
σ
∥
u
n
∥
p
-
λ
σ
-
q
q
σ
∫
Ω
‍
a
(
x
)
|
u
n
|
q
d
x
≥
-
λ
σ
-
q
q
σ
∫
Ω
‍
a
(
x
)
|
u
n
|
q
d
x
then by (57) and Lemma 9, we have J
λ
(
u
n
)
→
θ
λ
<
0 as n
→
∞. Letting n
→
∞, we obtain
(63)
λ
∫
Ω
‍
a
(
x
)
|
u
0
+
|
q
d
x
>
0
.
Now, we aim to prove that u
n
→
u
0
+ strongly in E and J
λ
(
u
0
+
)
=
θ
λ.
Using the fact thatu
0
+
∈
N
λ and by Fatou's lemma, we get
(64)
θ
λ
≤
J
λ
(
u
0
+
)
=
1
p
∥
u
0
+
∥
p
-
1
σ
∫
Ω
‍
F
(
x
,
u
0
+
)
d
x
-
λ
q
∫
Ω
‍
a
(
x
)
|
u
0
+
|
q
d
x
≤
liminf
⁡
n
→
∞
(
1
p
∥
u
n
∥
p
-
1
σ
∫
Ω
‍
F
(
x
,
u
n
)
d
x
-
λ
q
∫
Ω
a
(
x
)
|
u
n
|
q
d
x
)
≤
liminf
⁡
n
→
∞
J
λ
(
u
n
)
=
θ
λ
.
This implies that
(65)
J
λ
(
u
0
+
)
=
θ
λ
,
lim
⁡
n
→
∞
∥
u
n
∥
p
=
∥
u
0
+
∥
p
.
Let u
~
n
=
u
n
-
u
0
+; then by Brézis-Lieb Lemma [3] we obtain
(66)
lim
⁡
n
→
∞
(
∥
u
n
∥
p
-
∥
u
~
n
∥
p
)
=
∥
u
0
+
∥
p
.
Therefore, u
n
→
u
0
+ strongly in E.
Moreover, we haveu
0
+
∈
N
λ
+. In fact, if u
0
+
∈
N
λ
- then, there exist t
0
+
,
t
0
- such that t
0
-
u
0
+
∈
N
λ
- and t
0
+
u
0
+
∈
N
λ
+. In particular we have t
0
+
<
t
0
-
=
1. Since
(67)
d
2
d
t
2
J
λ
(
t
0
+
u
0
+
)
>
0
,
d
d
t
J
λ
(
t
0
+
u
0
+
)
=
0
,
there exists t
0
+
<
t
~
<
t
0
- such that J
λ
(
t
0
+
u
0
+
)
<
J
λ
(
t
~
u
0
+
). By Lemma 10, we have
(68)
J
λ
(
t
0
+
u
0
+
)
<
J
λ
(
t
~
u
0
+
)
≤
J
λ
(
t
0
-
u
0
+
)
=
J
λ
(
u
0
+
)
,
which is a contradiction.
Finally, by (63) we may assume that u
0
+ is a nontrivial nonnegative solution of problem (1).Theorem 14.
If0
<
|
λ
|
<
λ
0, then, problem (1) has a positive solution u
0
- in N
λ
- such that
(69)
J
λ
(
u
0
-
)
=
θ
λ
-
.Proof.
By Proposition12(ii), there exists a minimizing sequence {
u
n
} for J
λ on N
λ
- such that
(70)
J
λ
(
u
n
)
=
θ
λ
-
+
o
(
1
)
,
J
λ
′
(
u
n
)
=
o
(
1
)
in
E
-
1
,
(71)
u
n
⇀
u
0
-
weakly
in
E
,
u
n
⟶
u
0
-
strongly
in
L
q
(
Ω
)
and
in
L
σ
(
Ω
)
.
Moreover, by (23) we obtain
(72)
∫
Ω
‍
F
(
x
,
u
n
)
d
x
>
p
-
q
σ
-
q
∥
u
n
∥
p
.
So, by (38) and (72) there exists a positive constant C
~ such that
(73)
∫
Ω
‍
F
(
x
,
u
n
)
>
C
~
.
This implies that
(74)
∫
Ω
‍
F
(
x
,
u
0
-
)
≥
C
~
.
By (70) and (71), we obtain clearly that u
0
- is a weak solution of (1).
Now, we aim to prove thatu
n
→
u
0
- strongly in E. Supposing otherwise, then
(75)
∥
u
0
-
∥
<
liminf
⁡
n
→
∞
∥
u
n
∥
.
By Lemma 9, there is a unique t
0
- such that t
0
-
u
0
-
∈
N
λ
-. Since u
n
∈
N
λ
-, J
λ
(
u
n
)
≥
J
λ
(
t
u
n
) for all t
≥
0, we have
(76)
J
λ
(
t
0
-
u
0
-
)
<
lim
⁡
n
→
∞
J
λ
(
t
0
-
u
n
)
≤
lim
⁡
n
→
∞
J
λ
(
u
n
)
=
θ
λ
-
,
which is a contradiction. Hence u
n
→
u
0
- strongly in E.
This imply that(77)
J
λ
(
u
n
)
⟶
J
λ
(
u
0
-
)
=
θ
λ
-
as
n
⟶
∞
.
By Lemma 5 and (74) we may assume that u
0
- is a nontrivial solution of problem (1).
Now, we begin to show the proof of Theorem1: by Theorem 13, we obtain that for all 0
<
λ
<
λ
0, problem (1) has a nontrivial solution u
0
+
∈
N
λ
+. On the other hand, from Theorem 14, we get the second solution u
0
-
∈
N
λ
-. Since N
λ
-
∩
N
λ
+
=
∅, then u
0
- and u
0
+ are distinct.



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*Source: 461965-2014-02-12.xml*