# Multiplicity of Positive Solutions for ap-q-Laplacian Type Equation with Critical Nonlinearities

Authors: Tsing-San Hsu; Huei-Li Lin
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We study the effect of the coefficientf
(
x
) of the critical nonlinearity on the number of positive solutions for a p-q-Laplacian equation. Under suitable assumptions for f
(
x
) and g
(
x
), we should prove that for sufficiently small λ
>
0, there exist at least k positive solutions of the following p-q-Laplacian equation, -
Δ
p
u
-
Δ
q
u
=
f
x
u
|
p
*
-
2
u
+
λ
g
x
u
|
r
-
2
u
in
Ω, u
=
0
on
∂
Ω, where Ω
⊂
R
N is a bounded smooth domain, N
>
p, 1
<
q
<
N
(
p
-
1
)
/
(
N
-
1
)
<
p
≤
max
⁡
{
p
,
p
^
*
-
q
/
(
p
-
1
)
}
<
r
<
p
^
*, p
^
*
=
N
p
/
(
N
-
p
) is the critical Sobolev exponent, and Δ
s
u
=
d
i
v
(
|
∇
u
|
s
-
2
∇
u is the s-Laplacian of u.


---

## Body

## 1. Introduction

This paper is concerned with the multiplicity of positive solutions to the followingp-q-Laplacian equation with critical nonlinearities:
(
E
λ
)
-
Δ
p
u
-
Δ
q
u
=
f
(
x
)
|
u
|
p
*
-
2
u
+
λ
g
(
x
)
|
u
|
r
-
2
u
in
Ω
,
u
=
0
on
⁡
∂
Ω
,
where Ω
⊂
R
N is a bounded smooth domain with smooth boundary ∂
Ω, 1
<
q
<
p
<
N, λ
>
0, and Δ
s
u
=
div
⁡
(
|
∇
u
|
s
-
2
∇
u
) is the s-Laplacian of u, and assume that(
H
1
)
1
<
q
<
N
(
p
-
1
)
/
(
N
-
1
)
<
p
≤
max
⁡
{
p
,
p
*
-
q
/
(
p
-
1
)
}
<
r
<
p
*
=
N
p
/
(
N
-
p
);
(
H
2
)
f and g are positive continuous functions in Ω
¯;
(
H
3
)
There existk points a
1
,
a
2
,
…
,
a
k in Ω such that f
(
a
i
) are strict local maxima satisfying
(1)
f
(
a
i
)
=
max
⁡
x
∈
Ω
¯
f
(
x
)
=
1
for
1
≤
i
≤
k
,
and for someβ
>
N
/
(
p
-
1
), f
(
x
)
=
f
(
a
i
)
+
O
(
|
x
-
a
i
|
β
) as x
→
a
i uniformly in i.Problem(
E
λ
) comes, for example, from a general reaction-diffusion system
(2)
u
t
=
div
⁡
[
H
(
u
)
∇
u
]
+
c
(
x
,
u
)
,
where H
(
u
)
=
|
∇
u
|
p
-
2
+
|
∇
u
|
q
-
2. This system has a wide range of applications in physics and related science such as biophysics, plasma physics, and chemical reaction design. In such applications, the function u describes a concentration, the first term on the right-hand side of (2) corresponds to the diffusion with a diffusion coefficient H
(
u
), whereas the second one is the reaction and relates to sources and loss processes. Typically, in chemical and biological applications, the reaction term c
(
x
,
u
) has a polynomial form with respect to the concentration u.The stationary solution of (2) was studied by many authors; that is, many works are considered the solutions of the following problem:
(3)
-
div
⁡
[
H
(
u
)
∇
u
]
=
c
(
x
,
u
)
.
See [1–5] for different c
(
x
,
u
). In the present paper we are concerned with problem (
E
λ
) in a bounded domain with c
(
x
,
u
)
=
f
(
x
)
|
u
|
p
*
-
2
u
+
λ
g
(
x
)
|
u
|
r
-
2
u in (3). Recently, in [6], the authors obtain the existence of cat
Ω
(
Ω
) positive solutions of problem (
E
λ
) for N
>
p
2 and f
(
x
)
≡
g
(
x
)
≡
1 when condition (
H
1
) holds, where cat
Ω
(
Ω
) denotes the Lusternik-Schnirelmann category of Ω in itself.Specially, ifp
=
q, (
E
λ
) can be reduced to the following elliptic problems:
(4)
-
Δ
p
u
=
f
(
x
)
|
u
|
p
*
-
2
u
+
λ
g
(
x
)
|
u
|
r
-
2
u
in
Ω
,
u
=
0
on
⁡
∂
Ω
.After the well-known results of Brézis and Nirenberg [7], who studied (4) in the case of p
=
r
=
2 and f
(
x
)
≡
g
(
x
)
≡
1, a lot of problems involving the critical growth in bounded and unbounded domains have been considered; see, for example, [8–10] and reference therein. In particular, the first multiplicity result for (4) has been achieved by Rey in [11] in the semilinear case. Precisely Rey proved that if N
≥
5, p
=
r
=
2, and f
(
x
)
≡
g
(
x
)
≡
1, for λ small enough, problem (4) has at least cat
Ω
(
Ω
) solutions. Furthermore, Alves and Ding in [12] obtained the existence of cat
Ω
(
Ω
) positive solutions to problem (4) with p
≥
2, r
∈
[
p
,
p
*
), and f
(
x
)
≡
g
(
x
)
≡
1. Finally, we mention that [13] studied (4) when 1
<
r
<
p
<
N and f
,
g are sign-changing and verified the existence of two positive solutions for λ
∈
(
0
,
λ
0
) for some positive constant λ
0.The main purpose of this paper is to analyze the effect of the coefficientf
(
x
) of the critical nonlinearity to prove the multiplicity of positive solutions of problem (
E
λ
) for small λ
>
0. By the similar argument in [14], we can construct the k compact Palais-Smale sequences that are suitably localized in correspondence of k maximum points of f. Under some assumptions (
H
1
)
–
(
H
3
), we could show that there are at least k positive solutions of problem (
E
λ
) for sufficiently small λ
>
0.This paper is organized as follows. First of all, we study the argument of the Nehari manifoldℳ
λ. Next, we prove the existence of a positive solution u
0
∈
ℳ
λ. Finally, we show that the condition (
H
3
) affects the number of positive solution of (
E
λ
); that is, there are at least k critical points u
i
∈
ℳ
λ of J
λ such that J
λ
(
u
i
)
=
α
λ
i ((PS)-value) for 1
≤
i
≤
k.The main results of this paper are given as follows.Theorem 1.
Suppose that (H1)–(H3) hold; then problem(
E
λ
) has a positive solution u
0 in W
0
1
,
p
(
Ω
) for all λ
>
0.Theorem 2.
Suppose that (H1)–(H3) hold; then there exists aλ
0
>
0 such that for any λ
∈
(
0
,
λ
0
), problem (
E
λ
) admits at least k positive solutions in W
0
1
,
p
(
Ω
).


## 2. Preliminaries

In what follows, we denote by∥
·
∥
p
,
|
·
|
p the norms on W
0
1
,
p
(
Ω
) and L
p
(
Ω
), respectively; that is,
(5)
∥
u
∥
p
=
(
∫
Ω
‍
|
∇
u
|
p
d
x
)
1
/
p
,
|
u
|
p
=
(
∫
Ω
‍
|
u
|
p
d
x
)
1
/
p
.
We denote the dual space of W
0
1
,
p
(
Ω
) by W
′
(
Ω
). Set also
(6)
D
1
,
p
(
R
N
)
∶
=
{
u
∈
L
p
*
(
R
N
)
:
∂
u
∂
x
i
∈
L
p
(
R
N
)
for
i
=
1,2
,
…
,
N
}
equipped with the norm
(7)
∥
u
∥
*
=
(
∫
R
N
‍
|
∇
u
|
p
d
x
)
1
/
p
.
We will denote by S the best Sobolev constant as follows:
(8)
S
=
inf
⁡
u
∈
W
0
1
,
p
(
Ω
)
∖
{
0
}
∥
u
∥
p
p
|
u
|
p
*
p
.
It is well known that S is independent of Ω and is never achieved except when Ω
=
R
N (see [15]). Throughout this paper, we denote the Lebesgue measure of Ω by |
Ω
| and denote a ball centered at a
∈
R
N with radius r by B
r
(
a
) and also denote positive constants (possibly different) by C, C
i. O
(
ε
t
) denotes |
O
(
ε
t
)
|
/
ε
t
≤
C, o
(
ε
t
) denotes |
o
(
ε
t
)
|
/
ε
t
→
0 as ε
→
0, and o
n
(
1
) denotes o
n
(
1
)
→
0 as n
→
∞.Associated with(
E
λ
), we consider the energy functional J
λ in W
0
1
,
p
(
Ω
), for each u
∈
W
0
1
,
p
(
Ω
),
(9)
J
λ
(
u
)
=
1
p
∥
u
∥
p
p
+
1
q
∥
u
∥
q
q
-
1
p
*
∫
Ω
‍
f
(
x
)
|
u
|
p
*
d
x
-
1
r
∫
Ω
‍
λ
g
(
x
)
|
u
|
r
d
x
.
It is well known that J
λ is of C
1 in W
0
1
,
p
(
Ω
) and the solutions of (
E
λ
) are the critical points of the energy functional J
λ (see [16]).We define the Nehari manifold(10)
ℳ
λ
∶
=
{
u
∈
W
0
1
,
p
(
Ω
)
∖
{
0
}
:
〈
J
λ
′
(
u
)
,
u
〉
=
0
}
,
where
(11)
〈
J
λ
′
(
u
)
,
u
〉
=
∥
u
∥
p
p
+
∥
u
∥
q
q
-
∫
Ω
‍
f
(
x
)
|
u
|
p
*
d
x
-
∫
Ω
‍
λ
g
(
x
)
|
u
|
r
d
x
=
0
.
The Nehari manifold ℳ
λ contains all nontrivial solutions of (
E
λ
).Note thatJ
λ is not bounded from below in W
0
1
,
p
(
Ω
). From the following lemma, we have that J
λ is bounded from below on the Nehari manifold ℳ
λ.Lemma 3.
Suppose that1
<
q
<
p
<
r
<
p
* and (H2) hold. Then for any λ
>
0, one has that J
λ is bounded from below on ℳ
λ. Moreover, J
λ
(
u
)
>
0 for all u
∈
ℳ
λ.Proof.
Foru
∈
ℳ
λ, (10) leads to
(12)
J
λ
(
u
)
=
(
1
p
-
1
r
)
∥
u
∥
p
p
+
(
1
q
-
1
r
)
∥
u
∥
q
q
+
(
1
r
-
1
p
*
)
∫
Ω
‍
f
(
x
)
|
u
|
p
*
d
x
>
0
.Define(13)
α
λ
∶
=
inf
⁡
u
∈
ℳ
λ
J
λ
(
u
)
.Now we show thatJ
λ possesses the mountain-pass (MP, in short) geometry.Lemma 4.
Suppose1
<
q
<
p
<
r
<
p
* and (H2) holds. Then for any λ
>
0, one has that(i)
there exist positive numbersR and d
0 such that J
λ
(
u
)
≥
d
0 for ∥
u
∥
p
=
R;
(ii)
there existsu
¯
∈
W
0
1
,
p
(
Ω
) such that ∥
u
-
∥
p
>
R and J
λ
(
u
¯
)
<
0.Proof.
(
i
) By (8), the Hölder inequality, and the Sobolev embedding theorem, we have that
(14)
J
λ
(
u
)
≥
1
p
∥
u
∥
p
p
-
1
p
*
∫
Ω
‍
f
(
x
)
|
u
|
p
*
d
x
-
1
r
∫
Ω
‍
λ
g
(
x
)
|
u
|
r
d
x
≥
1
p
∥
u
∥
p
p
-
1
p
*
S
-
p
*
/
p
∥
u
∥
p
p
*
-
1
r
λ
|
g
|
∞
|
Ω
|
(
p
*
-
r
)
/
p
*
S
-
r
/
p
∥
u
∥
p
r
.
Hence, there exist positive R and d
0 such that J
λ
(
u
)
≥
d
0 for ∥
u
∥
=
R.
(
ii
) For any u
∈
W
0
1
,
p
(
Ω
)
∖
{
0
}, from
(15)
J
λ
(
t
u
)
=
t
p
p
∥
u
∥
p
p
+
t
q
q
∥
u
∥
q
q
-
t
p
*
p
*
∫
Ω
‍
f
(
x
)
|
u
|
p
*
d
x
-
t
r
r
∫
Ω
‍
λ
g
(
x
)
|
u
|
r
d
x
,
we have lim
⁡
t
→
∞
J
λ
(
t
u
)
=
-
∞. For fixed some u
∈
W
0
1
,
p
(
Ω
)
∖
{
0
}, there exist t
¯
>
0 such that ∥
t
¯
u
∥
p
>
R and J
λ
(
t
-
u
)
<
0. Let u
¯
=
t
¯
u.Define(16)
ϕ
λ
(
u
)
∶
=
〈
J
λ
′
(
u
)
,
u
〉
.
Then for u
∈
ℳ
λ,
(17)
〈
ϕ
λ
′
(
u
)
,
u
〉
=
p
∥
u
∥
p
p
+
q
∥
u
∥
q
q
-
p
*
∫
Ω
‍
f
(
x
)
|
u
|
p
*
d
x
-
r
∫
Ω
‍
λ
g
(
x
)
|
u
|
r
d
x
=
(
p
*
-
r
)
∫
Ω
‍
λ
g
(
x
)
|
u
|
r
d
x
-
(
p
*
-
p
)
∥
u
∥
p
p
-
(
p
*
-
q
)
∥
u
∥
q
q
=
(
p
-
r
)
∥
u
∥
p
p
+
(
q
-
r
)
∥
u
∥
q
q
+
(
r
-
p
*
)
∫
Ω
‍
f
(
x
)
|
u
|
p
*
d
x
<
0
.Lemma 5.
Suppose that1
<
q
<
p
<
r
<
p
* and (H2) holds. If u
0
∈
ℳ
λ satisfies
(18)
J
λ
(
u
0
)
=
min
⁡
u
∈
ℳ
λ
J
λ
(
u
)
=
α
λ
,
then u
0 is a solution of (
E
λ
).Proof.
By (17), 〈
ϕ
λ
′
(
u
)
,
u
〉
<
0 for u
∈
ℳ
λ. Since J
λ
(
u
0
)
=
min
⁡
u
∈
ℳ
λ
J
λ
(
u
), by the Lagrange multiplier theorem, there is τ
∈
R such that J
λ
′
(
u
0
)
=
τ
ϕ
λ
′
(
u
0
) in W
′
(
Ω
). This implies that
(19)
0
=
〈
J
λ
′
(
u
0
)
,
u
0
〉
=
τ
〈
ϕ
λ
′
(
u
0
)
,
u
0
〉
.
It then follows that τ
=
0 and J
λ
′
(
u
0
)
=
0 in W
′
(
Ω
). Thus, u
0 is a nontrivial solution of (
E
λ
) and J
λ
(
u
0
)
=
α
λ.Lemma 6.
Suppose that1
<
q
<
p
<
r
<
p
* and (H2) holds. For each u
∈
W
0
1
,
p
(
Ω
)
∖
{
0
}, there exists a unique positive number t
u such that t
u
u
∈
ℳ
λ and J
λ
(
t
u
u
)
=
sup
t
≥
0
J
λ
(
t
u
) for any λ
>
0.Proof.
For fixedu
∈
W
0
1
,
p
(
Ω
)
∖
{
0
}, consider
(20)
h
(
t
)
=
J
λ
(
t
u
)
=
t
p
p
∥
u
∥
p
p
+
t
q
q
∥
u
∥
q
q
-
t
p
*
p
*
∫
Ω
‍
f
(
x
)
|
u
|
p
*
d
x
-
t
r
r
∫
Ω
‍
λ
g
(
x
)
|
u
|
r
d
x
.
Since h
(
0
)
=
0, lim
⁡
t
→
∞
h
(
t
)
=
-
∞, by Lemma 4
(
i
), then it is easy to see that there exists a unique positive number t
u such that sup
⁡
t
≥
0
h
(
t
) is achieved at t
u. This means that h
′
(
t
u
)
=
0; that is, t
u
u
∈
ℳ
λ.We will denote byα
~
λ the MP level:
(21)
α
~
λ
∶
=
inf
⁡
u
∈
W
0
1
,
p
(
Ω
)
∖
{
0
}
sup
⁡
t
≥
0
J
λ
(
t
u
)
.Then we have the following result.Lemma 7.
Suppose that1
<
q
<
p
<
r
<
p
* and (H2) holds, then α
λ
=
α
~
λ for any λ
>
0.Proof.
By Lemma6, we have
(22)
α
~
λ
=
inf
⁡
u
∈
W
0
1
,
p
(
Ω
)
∖
{
0
}
sup
⁡
t
≥
0
J
λ
(
t
u
)
=
inf
⁡
t
u
u
∈
ℳ
λ
J
λ
(
t
u
u
)
≥
inf
⁡
u
∈
ℳ
λ
J
λ
(
u
)
=
α
λ
.
On the other hand, for u
∈
ℳ
λ, by Lemma 6, we have t
u
=
1 and J
λ
(
u
)
=
sup
⁡
t
≥
0
J
λ
(
t
u
). Hence,
(23)
α
λ
=
inf
⁡
u
∈
ℳ
λ
J
λ
(
u
)
=
inf
⁡
u
∈
ℳ
λ
sup
⁡
t
≥
0
J
λ
(
t
u
)
≥
inf
⁡
u
∈
W
0
1
,
p
(
Ω
)
∖
{
0
}
sup
⁡
t
≥
0
J
λ
(
t
u
)
=
α
~
λ
.
Now the desired result follows from (22) and (23).Remark 8.
By Lemma7 and the definition, it is apparent that α
λ
1
≤
α
λ
2 if λ
1
≥
λ
2; that is, α
λ is nonincreasing in λ. Moreover, by Lemma 4(i), for any λ
0
>
0, there exists a d
=
d
(
λ
0
), related to the MP geometry, such that
(24)
0
<
d
≤
α
λ
≤
α
0
∀
λ
∈
[
0
,
λ
0
]
.
Here α
0 is the MP level associated to the functional
(25)
J
0
(
u
)
=
1
p
∥
u
∥
p
p
+
1
q
∥
u
∥
q
q
-
1
p
*
∫
Ω
‍
f
(
x
)
|
u
|
p
*
d
x
.


## 3.(PS)
c-Condition in W
0
1
,
p
(
Ω
) for J
λ

First, we define the Palais-Smale (denote by (PS)) sequence, (PS)-value, and (PS)-conditions inW
0
1
,
p
(
Ω
) for J
λ.Definition 9.
(i) Forc
∈
R, a sequence {
u
n
} is a (
PS
)
c-sequence in W
0
1
,
p
(
Ω
) for J
λ if J
λ
(
u
n
)
=
c
+
o
n
(
1
) and J
λ
′
(
u
n
)
=
o
n
(
1
) strongly in W
′
(
Ω
) as n
→
∞.
(ii)c
∈
R is a (PS)-value in W
0
1
,
p
(
Ω
) for J
λ if there exists a (
PS
)
c-sequence in W
0
1
,
p
(
Ω
) for  J
λ.
(iii)J
λ satisfies the (
PS
)
c-condition in W
0
1
,
p
(
Ω
) if every (
PS
)
c-sequence {
u
n
} in W
0
1
,
p
(
Ω
) for J
λ contains a convergent subsequence.Applying Ekeland’s variational principle and using the same argument as in Cao and Zhou [17] or Tarantello [18], we have the following lemma.Lemma 10.
Suppose that1
<
q
<
p
<
r
<
p
* and (H2) holds. Then for any λ
>
0, there exists a (
P
S
)
α
λ-sequence {
u
n
} in ℳ
λ for J
λ.To prove the existence of positive solutions, we claim thatJ
λ satisfies the (
PS
)
c-condition in W
0
1
,
p
(
Ω
) for c
∈
(
0
,
(
1
/
N
)
S
N
/
p
).Lemma 11.
Suppose that1
<
q
<
p
<
r
<
p
*, |
f
|
∞
=
1, and (H2) holds. Then for any λ
>
0, J
λ satisfies the (PS)c-condition in W
0
1
,
p
(
Ω
) for all c
∈
(
0
,
(
1
/
N
)
S
N
/
p
).Proof.
Let{
u
n
}
⊂
W
0
1
,
p
(
Ω
) be a (
PS
)
c-sequence for J
λ which satisfies
(26)
J
λ
(
u
n
)
=
c
+
o
n
(
1
)
,
J
λ
′
(
u
n
)
=
o
n
(
1
)
in
W
′
(
Ω
)
.
Then
(27)
c
+
s
n
+
t
n
∥
u
n
∥
p
≥
J
λ
(
u
n
)
-
1
r
〈
J
λ
′
(
u
n
)
,
u
n
〉
=
(
1
p
-
1
r
)
∥
u
n
∥
p
p
+
(
1
q
-
1
r
)
∥
u
n
∥
q
q
+
(
1
r
-
1
p
*
)
∫
Ω
‍
f
(
x
)
|
u
n
|
p
*
d
x
≥
r
-
p
r
p
∥
u
n
∥
p
p
,
where s
n
=
o
n
(
1
), t
n
=
o
n
(
1
), as n
→
∞. It follows that {
u
n
} is bounded in W
0
1
,
p
(
Ω
). Thus, there exist a subsequence still denoted by {
u
n
} and u
∈
W
0
1
,
p
(
Ω
) such that
(28)
u
n
⇀
u
weakly
in
W
0
1
,
p
(
Ω
)
,
u
n
⟶
u
strongly
in
L
s
(
Ω
)
∀
1
≤
s
<
p
*
,
u
n
⟶
u
a
.
e
.
in
Ω
.
Furthermore, we have that J
λ
′
(
u
)
=
0 in W
′
(
Ω
). By g being continuous on Ω
¯, we get
(29)
λ
∫
Ω
‍
g
(
x
)
|
u
n
|
r
d
x
=
λ
∫
Ω
‍
g
(
x
)
|
u
|
r
d
x
+
o
n
(
1
)
.
Let v
n
=
u
n
-
u. Then by f being positive continuous on Ω
¯ and Brézis-Lieb lemma (see [19]), we obtain
(30)
∥
v
n
∥
p
p
=
∥
u
n
∥
p
p
-
∥
u
∥
p
p
+
o
n
(
1
)
,
∥
v
n
∥
q
q
=
∥
u
n
∥
q
q
-
∥
u
∥
q
q
+
o
n
(
1
)
,
∫
Ω
f
(
x
)
|
v
n
|
p
*
d
x
=
∫
Ω
‍
f
(
x
)
|
u
n
|
p
*
d
x
-
∫
Ω
‍
f
(
x
)
|
u
|
p
*
d
x
+
o
n
(
1
)
.
From (26)–(30), we can deduce that
(31)
1
p
∥
v
n
∥
p
p
+
1
q
∥
v
n
∥
q
q
-
1
p
*
∫
Ω
‍
f
(
x
)
|
v
n
|
p
*
d
x
=
c
-
J
λ
(
u
)
+
o
n
(
1
)
,
(32)
∥
v
n
∥
p
p
+
∥
v
n
∥
q
q
=
∫
Ω
‍
f
(
x
)
|
v
n
|
p
*
d
x
+
o
n
(
1
)
.
Without loss of generality, we may assume that
(33)
∥
v
n
∥
p
p
=
a
+
o
n
(
1
)
,
∥
v
n
∥
q
q
=
b
+
o
n
(
1
)
.
So (32) and |
f
|
∞
=
1 imply that
(34)
∫
Ω
‍
|
v
n
|
p
*
d
x
≥
∫
Ω
‍
f
(
x
)
|
v
n
|
p
*
d
x
=
a
+
b
+
o
n
(
1
)
.
By the Sobolev inequality and (33) and (34), we have ∥
v
n
∥
p
p
≥
S
(
∫
Ω
‍
|
v
n
|
p
*
d
x
)
p
/
p
* and
(35)
a
≥
S
(
a
+
b
)
p
/
p
*
≥
S
a
p
/
p
*
.
If a
>
0, then (35) implies that a
≥
S
N
/
p, combined with (31), (33)–(35) and Lemma 3, 1
<
q
<
p
<
p
*, as n
→
∞; we get
(36)
c
=
a
p
+
b
q
-
a
+
b
p
*
+
J
λ
(
u
)
≥
1
N
a
≥
1
N
S
N
/
p
,
which is a contradiction. So, we have a
=
0; J
λ satisfies the (
PS
)
c-condition in W
0
1
,
p
(
Ω
) for all c
∈
(
0
,
(
1
/
N
)
S
N
/
p
).


## 4. Existence ofk Positive Solutions

In this section, we first give some preliminary notations and useful lemmas.Chooser
0
>
0 small enough such that B
r
0
(
a
i
)
¯
⊂
Ω and B
r
0
(
a
i
)
¯
∩
B
r
0
(
a
j
)
¯
=
⌀ for i
≠
j,  i
,
j
=
1,2
,
…
,
k.Define(37)
Q
i
(
u
)
=
∫
Ω
‍
ϕ
i
(
x
)
|
u
|
p
*
d
x
∫
Ω
‍
|
u
|
p
*
d
x
,
ϕ
i
(
x
)
=
min
⁡
{
1
,
|
x
-
a
i
|
}
,
1
≤
i
≤
k
.
Then we have the following separation result.Lemma 12.
IfQ
i
(
u
)
≤
r
0
/
3 and Q
j
(
u
)
≤
r
0
/
3 for u
∈
W
0
1
,
p
(
Ω
)
∖
{
0
}, then i
=
j.Proof.
For anyu
∈
W
0
1
,
p
(
Ω
)
∖
{
0
} satisfying Q
i
(
u
)
≤
r
0
/
3
(
1
≤
i
≤
k
), we get
(38)
r
0
3
∫
Ω
‍
|
u
|
p
*
d
x
≥
∫
Ω
‍
ϕ
i
(
x
)
|
u
|
p
*
d
x
≥
∫
Ω
∖
B
r
0
(
a
i
)
‍
ϕ
i
(
x
)
|
u
|
p
*
d
x
≥
r
0
∫
Ω
∖
B
r
0
(
a
i
)
‍
|
u
|
p
*
d
x
,
which implies that
(39)
∫
Ω
‍
|
u
|
p
*
d
x
≥
3
∫
Ω
∖
B
r
0
(
a
i
)
‍
|
u
|
p
*
d
x
,
1
≤
i
≤
k
.
Hence, from (39), we obtain
(40)
2
∫
Ω
‍
|
u
|
p
*
d
x
≥
3
(
∫
Ω
∖
B
r
0
(
a
i
)
‍
|
u
|
p
*
d
x
+
∫
Ω
∖
B
r
0
(
a
j
)
‍
|
u
|
p
*
d
x
)
≥
3
∫
Ω
‍
|
u
|
p
*
d
x
if
i
≠
j
,
which is a contradiction.Fori
=
1,2
,
…
,
k, we set
(41)
𝒩
λ
i
≔
{
u
∈
ℳ
λ
:
Q
i
(
u
)
<
r
0
3
}
∂
𝒩
λ
i
∶
=
{
u
∈
ℳ
λ
:
Q
i
(
u
)
=
r
0
3
}
,
and define
(42)
α
λ
i
∶
=
inf
⁡
𝒩
λ
i
J
λ
(
u
)
,
α
~
λ
i
∶
=
inf
⁡
∂
𝒩
λ
i
J
λ
(
u
)
.
Now let us assume that (
H
1
)
–
(
H
3
) hold. From conditions (
H
2
) and (
H
3
), we can choose a ρ
∈
(
0
,
r
0
/
2
) small enough and there exist some positive constants γ
1
,
γ
2 such that for 1
≤
i
≤
k, we have
(43)
⋃
1
≤
i
≤
k
B
2
ρ
(
a
i
)
-
⊂
Ω
,
|
f
(
x
)
-
f
(
a
i
)
|
≤
γ
1
|
x
-
a
i
|
β
∀
x
∈
B
2
ρ
(
a
i
)
¯
,
g
(
x
)
≥
γ
2
∀
x
∈
⋃
1
≤
i
≤
k
B
2
ρ
(
a
i
)
-
,
for some β
>
N
/
(
p
-
1
). For i
∈
{
1,2
,
…
,
k
} and ε
>
0, we define
(44)
u
ε
a
i
(
x
)
=
η
i
(
x
)
(
ε
+
|
x
-
a
i
|
p
/
(
p
-
1
)
)
(
N
-
p
)
/
p
,
v
ε
a
i
(
x
)
=
ε
(
N
-
p
)
/
p
2
u
ε
a
i
(
x
)
,
where η
i
∈
C
0
∞
(
B
2
ρ
(
a
i
)
) is a function such that 0
≤
η
i
(
x
)
≤
1 and η
i
(
x
)
≡
1 on B
ρ
(
a
i
). Then we obtain the following estimates (see [20]):
(45)
∫
Ω
‍
|
u
ε
a
i
|
t
d
x
=
{
K
1
ε
(
N
(
p
-
1
)
-
t
(
N
-
p
)
)
/
p
+
O
(
1
)
,
t
>
N
(
p
-
1
)
p
,
K
1
|
ln
⁡
ε
|
+
O
(
1
)
,
t
=
N
(
p
-
1
)
p
,
O
(
1
)
,
t
<
N
(
p
-
1
)
p
,
(46)
∫
Ω
‍
|
∇
u
ε
a
i
|
t
d
x
=
{
K
2
ε
(
t
+
N
(
p
-
1
)
-
t
N
)
/
p
+
O
(
1
)
,
t
>
N
(
p
-
1
)
p
,
K
2
|
ln
⁡
ε
|
+
O
(
1
)
,
t
=
N
(
p
-
1
)
p
,
O
(
1
)
,
t
<
N
(
p
-
1
)
p
.
From (43)–(46) [13, Lemma 4.2] and conditions (
H
2
)-(
H
3
), we can deduce the following estimates:
(47)
∫
Ω
‍
|
∇
v
ε
a
i
|
p
d
x
=
K
2
+
O
(
ε
(
N
-
p
)
/
p
)
,
∫
Ω
‍
|
∇
v
ε
a
i
|
q
d
x
=
K
2
+
O
(
ε
q
(
N
-
p
)
/
p
2
)
,
(48)
∫
Ω
‍
f
(
x
)
|
v
ε
a
i
|
p
*
d
x
=
K
3
p
*
/
p
+
O
(
ε
N
/
p
)
,
(49)
∫
Ω
‍
|
v
ε
a
i
|
r
d
x
=
K
1
ε
(
(
p
-
1
)
/
p
)
(
N
-
r
(
(
N
-
p
)
/
p
)
)
+
O
(
ε
r
(
N
-
p
)
/
p
2
)
,
where K
1, K
2, and K
3 are positive constants independent of ε, and S
=
K
2
/
K
3  is the best Sobolev constant given in (8).Next, we will investigate the effect of the coefficientf
(
x
) to find some Palais-Smale sequences which are used to prove Theorem 2.Lemma 13.
If (H1)–(H3) hold, then for anyi
∈
{
1,2
,
…
,
k
} and any λ
>
0, there exists a ε
0
>
0 such that for ε
∈
(
0
,
ε
0
) one has
(50)
sup
⁡
t
≥
0
J
λ
(
t
v
ε
a
i
)
<
1
N
S
N
/
p
u
n
i
f
o
r
m
l
y
i
n
i
.
In particular, 0
<
α
λ
≤
α
λ
i
<
(
1
/
N
)
S
N
/
p for all λ
>
0.Proof.
By Lemma6, there exists a t
ε
i
>
0 such that t
ε
i
v
ε
a
i
∈
ℳ
λ. Furthermore,
(51)
Q
i
(
t
ε
i
v
ε
a
i
)
=
∫
Ω
‍
ϕ
i
(
x
)
|
v
ε
a
i
|
p
*
d
x
∫
Ω
‍
|
v
ε
a
i
|
p
*
d
x
=
∫
Ω
ε
‍
ϕ
i
(
a
i
+
ε
y
)
|
η
i
(
a
i
+
ε
y
)
V
(
y
)
|
p
*
d
y
∫
Ω
ε
‍
|
η
i
(
a
i
+
ε
y
)
V
(
y
)
|
p
*
d
y
⟶
ϕ
i
(
a
i
)
=
0
as
ε
⟶
0
,
where Ω
ε
=
{
x
:
ε
x
+
a
i
∈
Ω
} and V
(
y
)
=
1
/
(
1
+
|
y
|
p
/
(
p
-
1
)
)
(
N
-
p
)
/
p. Hence, there exists an ε
1
>
0 small enough such that for any ε
∈
(
0
,
ε
1
), we have
(52)
Q
i
(
t
ε
i
v
ε
a
i
)
<
r
0
3
,
which implies t
ε
i
v
ε
a
i
∈
𝒩
λ
i for any ε
∈
(
0
,
ε
1
), and then
(53)
0
<
α
λ
≤
α
λ
i
≤
J
λ
(
t
ε
i
v
ε
a
i
)
≤
sup
⁡
t
≥
0
J
λ
(
t
t
ε
i
v
ε
a
i
)
=
sup
⁡
t
≥
0
J
λ
(
t
v
ε
a
i
)
.
Set
(54)
h
(
t
)
=
J
λ
(
t
v
ε
a
i
)
=
t
p
p
∥
v
ε
a
i
∥
p
p
d
x
+
t
q
q
∥
v
ε
a
i
∥
q
q
-
t
p
*
p
*
∫
Ω
‍
f
(
x
)
|
v
ε
a
i
|
p
*
d
x
-
t
r
r
∫
Ω
‍
λ
g
(
x
)
|
v
ε
a
i
|
r
d
x
.
Since h
(
0
)
=
0, lim
⁡
t
→
+
∞
h
(
t
)
=
-
∞, then there exists a t
ε such that sup
⁡
t
≥
0
J
λ
(
t
v
ε
a
i
)
=
J
λ
(
t
ε
v
ε
a
i
) hold, and then t
ε satisfies
(55)
0
=
h
′
(
t
ε
)
=
t
ε
p
-
1
∥
v
ε
a
i
∥
p
p
+
t
ε
q
-
1
∥
v
ε
a
i
∥
q
q
-
t
ε
p
*
-
1
∫
Ω
‍
f
(
x
)
|
v
ε
a
i
|
p
*
d
x
-
t
ε
r
-
1
∫
Ω
‍
λ
g
(
x
)
|
v
ε
a
i
|
r
d
x
;
then we have
(56)
∥
v
ε
a
i
∥
p
p
+
t
ε
q
-
p
∥
v
ε
a
i
∥
q
q
>
t
ε
p
*
-
p
∫
Ω
‍
f
(
x
)
|
v
ε
a
i
|
p
*
d
x
.
From (47) and (48), fixing any ε
2
>
0 small enough, there exists T
1
>
0 such that
(57)
t
ε
≤
T
1
for
any
ε
∈
(
0
,
ε
2
)
.
Also, from (55), we obtain
(58)
∥
v
ε
a
i
∥
p
p
<
t
ε
p
*
-
p
∫
Ω
‍
f
(
x
)
|
v
ε
a
i
|
p
*
d
x
+
t
ε
r
-
p
|
g
|
∞
∫
Ω
‍
λ
|
v
ε
a
i
|
r
d
x
.
From (47)–(49) and (58), there exist ε
3
>
0 and T
2
>
0 such that
(59)
t
ε
≥
T
2
for
any
ε
∈
(
0
,
ε
3
)
.
Let ε
4
=
min
⁡
{
ε
1
,
ε
2
,
ε
3
}
>
0; then
(60)
0
<
T
2
≤
t
ε
≤
T
1
∀
ε
∈
(
0
,
ε
4
)
,
where T
1 and T
2 are independent of ε. From [13, Lemma 4.2] and conditions (
H
2
)
-
(
H
3
), we also have
(61)
sup
⁡
t
≥
0
(
t
p
p
∥
v
ε
a
i
∥
p
p
-
t
p
*
p
*
∫
Ω
‍
f
(
x
)
|
v
ε
a
i
|
p
*
d
x
)
=
1
N
S
N
/
p
+
O
(
ε
(
N
-
p
)
/
p
)
.
By (43), (46)–(49), (60), and (61), for ε
∈
(
0
,
ε
4
), we obtain
(62)
h
(
t
ε
)
≤
sup
⁡
t
≥
0
(
t
p
p
∥
v
ε
a
i
∥
p
p
-
t
p
*
p
*
∫
Ω
‍
f
(
x
)
|
v
ε
a
i
|
p
*
d
x
)
+
t
ε
q
q
∥
v
ε
a
i
∥
q
q
-
t
r
r
∫
Ω
‍
λ
g
(
x
)
|
v
ε
a
i
|
r
d
x
≤
1
N
S
N
/
p
+
O
(
ε
(
N
-
p
)
/
p
)
+
T
1
q
q
∥
v
ε
a
i
∥
q
q
-
T
2
r
r
λ
γ
2
∫
Ω
‍
|
v
ε
a
i
|
r
d
x
≤
1
N
S
N
/
p
+
C
1
ε
(
N
-
p
)
/
p
+
C
2
ε
q
(
N
-
p
)
/
p
2
-
C
3
ε
(
p
/
(
p
-
1
)
)
(
N
-
r
(
(
N
-
p
)
/
p
)
)
,
where C
1
,
C
2
,
C
3 are positive constants independent of ε. Since 1
<
q
<
N
(
p
-
1
)
/
(
N
-
1
)
<
p
≤
max
⁡
{
p
,
p
*
-
q
/
(
p
-
1
)
}
<
r
<
p
*, we obtain that
(63)
N
-
p
p
>
q
(
N
-
p
)
p
2
>
p
p
-
1
(
N
-
r
N
-
p
p
)
;
then there exists an ε
0
∈
(
0
,
ε
3
) such that h
(
t
ε
)
=
sup
⁡
t
≥
0
J
λ
(
t
v
ε
a
i
)
<
(
1
/
N
)
S
N
/
p uniformly in i for all ε
∈
(
0
,
ε
0
). Moreover, from (53), we have 0
<
α
λ
≤
α
λ
i
<
(
1
/
N
)
S
N
/
p for all 1
≤
i
≤
k and λ
>
0. This completes the proof.Proof of Theorem1.
From Lemmas5, 10, 11, and 13, we get for all λ
>
0 that there exists a u
0 such that J
λ
′
(
u
0
)
=
0 and J
λ
(
u
0
)
=
α
λ. Set u
+
=
max
⁡
{
u
,
0
}. Replace the terms ∫
Ω
‍
f
(
x
)
|
u
|
p
*
d
x and ∫
Ω
‍
g
(
x
)
|
u
|
r
d
x of the functional J
λ by ∫
Ω
‍
f
(
x
)
u
+
p
*
d
x and ∫
Ω
‍
g
(
x
)
u
+
r
d
x, respectively. It then follows that u
0 is a nonnegative solution of (
E
λ
). Applying the maximum principle, (
E
λ
) admits at least one positive solution u
0 in W
0
1
,
p
(
Ω
).By studying the argument as in [21, Theorem III 3.1] and [22], we obtain the following lemma.Lemma 14.
Let{
u
n
}
⊂
W
0
1
,
p
(
Ω
) be a nonnegative function sequence with |
u
n
|
p
*
=
1 and ∥
u
n
∥
p
p
→
S. Then there exists a sequence (
y
n
,
σ
n
)
∈
Ω
×
R
+ such that
(64)
v
n
(
x
)
∶
=
σ
n
(
N
-
p
)
/
p
u
n
(
σ
n
x
+
y
n
)
contains a convergent subsequence denoted again by {
v
n
} such that
(65)
v
n
⟶
v
i
n
D
1
,
p
(
R
N
)
,
where v
(
x
)
>
0 in R
N. Moreover, we have σ
n
→
0, (
1
/
σ
n
)
dist
⁡
(
y
n
,
∂
Ω
)
→
∞, and y
n
→
y
∈
Ω
¯ as n
→
∞.Lemma 15.
Suppose that (H2) and (H3) hold. Then for anyi
∈
{
1,2
,
…
,
k
}, there exists λ
~
i
>
0 such that
(66)
α
~
λ
i
>
1
N
S
N
/
p
∀
λ
∈
(
0
,
λ
~
i
)
.Proof.
Fixi
∈
{
1,2
,
…
,
k
}. Assume the contrary; that is, there then exists a sequence {
λ
n
} with λ
n
→
0
+ as n
→
∞ such that α
~
λ
n
i
→
c
≤
(
1
/
N
)
S
N
/
p. Consequently, there exists a sequence {
u
n
}
⊂
∂
𝒩
λ
n
i such that, as n
→
∞,
(67)
∥
u
n
∥
p
p
+
∥
u
n
∥
q
q
=
∫
Ω
‍
f
(
x
)
|
u
n
|
p
*
d
x
+
λ
n
∫
Ω
‍
g
(
x
)
|
u
n
|
r
d
x
+
o
n
(
1
)
,
and by Remark 8, we have that there exists a d
>
0 such that
(68)
0
<
d
≤
lim
⁡
⁡
inf
⁡
n
→
∞
α
λ
n
≤
lim
⁡
n
→
∞
J
λ
n
(
u
n
)
=
c
≤
1
N
S
N
/
p
,
where d is independent of λ
n for all n. It then follows easily that {
u
n
} is uniformly bounded in W
0
1
,
p
(
Ω
), and since g
(
x
) is continuous on Ω
¯, we obtain
(69)
λ
n
∫
Ω
‍
g
(
x
)
|
u
n
|
r
d
x
=
o
n
(
1
)
as
n
⟶
∞
.
From (67)–(69), we may assume that there exist a
≥
0 and b
≥
0 such that
(70)
∥
u
n
∥
p
p
=
a
+
o
n
(
1
)
,
∥
u
n
∥
q
q
=
b
+
o
n
(
1
)
.
So (70) and |
f
|
∞
=
1 imply that
(71)
|
u
n
|
p
*
p
*
≥
∫
Ω
‍
f
(
x
)
|
u
n
|
p
*
d
x
=
a
+
b
+
o
n
(
1
)
.
By (70), (71), and the Sobolev inequality, we have
(72)
a
=
lim
⁡
n
→
∞
∥
u
n
∥
p
p
≥
S
lim
⁡
n
→
∞
|
u
n
|
p
*
p
≥
S
lim
⁡
n
→
∞
(
∫
Ω
‍
f
(
x
)
|
u
n
|
p
*
d
x
)
p
/
p
*
≥
S
(
a
+
b
)
p
/
p
*
≥
S
a
p
/
p
*
,
which implies a
=
0 or a
≥
S
N
/
p. If a
=
0, then by (72) we have that b
=
0. From a
=
b
=
0, we can deduce that c
=
0 which is a contradiction. Hence,
(73)
a
≥
S
N
/
p
.
On the other hand, by J
λ
n
(
u
n
)
=
c
+
o
n
(
1
), c
≤
(
1
/
N
)
S
N
/
p, and (69)–(71), we get
(74)
a
p
+
b
q
-
a
+
b
p
*
=
lim
⁡
n
→
∞
J
λ
n
(
u
n
)
=
c
≤
1
N
S
N
/
p
.
This implies that
(75)
a
N
≤
(
a
p
-
a
p
*
)
+
(
b
q
-
b
p
*
)
=
c
≤
1
N
S
N
/
p
.
Hence, together with (73), we get a
=
S
N
/
p and b
=
0, and then, from (71) and (72), we also have
(76)
lim
⁡
n
→
∞
|
u
n
|
p
*
p
*
=
lim
⁡
n
→
∞
∫
Ω
‍
f
(
x
)
|
u
n
|
p
*
d
x
=
a
=
S
N
/
p
.
Set w
n
=
u
n
/
|
u
n
|
p
*; then we have
(77)
|
w
n
|
p
*
=
1
,
lim
⁡
n
→
∞
∥
w
n
∥
p
p
=
lim
⁡
n
→
∞
∥
u
n
∥
p
p
|
u
n
|
p
*
p
=
S
.
Using Lemma 14, there exists a sequence (
y
n
,
σ
n
)
∈
Ω
×
R
+ such that the sequence
(78)
v
n
(
x
)
∶
=
σ
n
(
N
-
p
)
/
p
w
n
(
σ
n
x
+
y
n
)
converges strongly to v
∈
D
1
,
p
(
R
N
), σ
n
→
0, y
n
→
y
∈
Ω
¯, and (
1
/
σ
n
)
dist
⁡
(
y
n
,
∂
Ω
)
→
∞ as n
→
∞.
LetΩ
n
=
{
x
:
σ
n
x
+
y
n
∈
Ω
}. Since σ
n
→
0, y
n
→
y
∈
Ω
¯, and (
1
/
σ
n
)
dist
⁡
(
y
n
,
∂
Ω
)
→
∞ as n
→
∞, then Ω
n
→
R
N as n
→
∞. Observe that Q
i
(
w
n
)
=
Q
i
(
u
n
)
=
r
0
/
3. By the Lebesgue dominated convergence theorem, we have
(79)
r
0
3
=
lim
⁡
n
→
∞
Q
i
(
w
n
)
=
lim
⁡
n
→
∞
∫
Ω
‍
ϕ
i
(
x
)
|
w
n
|
p
*
d
x
∫
Ω
‍
|
w
n
|
p
*
d
x
=
lim
⁡
n
→
∞
∫
Ω
‍
ϕ
i
(
x
)
|
v
n
(
(
x
-
y
n
)
/
σ
n
)
|
p
*
d
x
∫
Ω
‍
|
v
n
(
(
x
-
y
n
)
/
σ
n
)
|
p
*
d
x
=
lim
⁡
n
→
∞
∫
Ω
n
‍
ϕ
i
(
σ
n
x
+
y
n
)
|
v
n
(
x
)
|
p
*
d
x
∫
Ω
n
‍
|
v
n
(
x
)
|
p
*
d
x
=
ϕ
i
(
y
)
,
which implies that y
≠
a
i by the definition of ϕ
i
(
x
). On the other hand, by the Lebesgue dominated convergence theorem again and (76), we get
(80)
1
=
lim
⁡
n
→
∞
∫
Ω
‍
f
(
x
)
|
w
n
(
x
)
|
p
*
d
x
=
lim
⁡
n
→
∞
∫
Ω
n
‍
f
(
σ
n
x
+
y
n
)
|
v
n
(
x
)
|
p
*
d
x
=
f
(
y
)
,
which is impossible, because f
(
x
) is not a constant function by condition (
H
3
).According to Lemma13, we have
(81)
0
<
α
λ
≤
α
λ
i
<
1
N
S
N
/
p
∀
λ
>
0
.
According to Lemma 15, for each i
∈
{
1,2
,
…
,
k
}, there exists λ
~
i
>
0 such that
(82)
α
~
λ
i
>
1
N
S
N
/
p
∀
λ
∈
(
0
,
λ
~
i
)
.
Let λ
0
=
min
⁡
1
≤
i
≤
k
λ
~
i
>
0. Then for each i
∈
{
1,2
,
…
,
k
}, by (81) and (82), we obtain that
(83)
α
λ
i
<
α
~
λ
i
∀
λ
∈
(
0
,
λ
0
)
.
Hence
(84)
α
λ
i
=
inf
⁡
u
∈
𝒩
λ
i
∪
∂
𝒩
λ
i
J
λ
(
u
)
∀
λ
∈
(
0
,
λ
0
)
.Applying Ekeland’s variational principle and using the standard computation, we have the following lemma.Lemma 16.
Ifλ
∈
(
0
,
λ
0
), then for each i
∈
{
1,2
,
…
,
k
}, there exists a (
P
S
)
α
λ
i-sequence {
u
n
i
}
⊂
𝒩
λ
i in W
0
1
,
p
(
Ω
) for J
λ.Proof.
See Cao and Zhou [17] or Tarantello [18].Proof of Theorem2.
By Lemma16, for all λ
∈
(
0
,
λ
0
), there exists a (
PS
)
α
λ
i-sequence {
u
n
i
}
⊂
𝒩
λ
i in W
0
1
,
p
(
Ω
) for J
λ where 1
≤
i
≤
k. From (81), we have
(85)
α
λ
i
∈
(
0
,
1
N
S
N
/
p
)
.
Note that J
λ satisfies the (
PS
)
c-condition for c
∈
(
0
,
(
1
/
N
)
S
N
/
p
). Hence, we obtain that J
λ at least k critical points in ℳ
λ for all λ
∈
(
0
,
λ
0
). Set u
+
=
max
⁡
{
u
,
0
}. Replace the terms ∫
Ω
‍
f
(
x
)
|
u
|
p
*
d
x and ∫
Ω
‍
g
(
x
)
|
u
|
r
d
x of the functional J
λ by ∫
Ω
‍
f
(
x
)
u
+
p
*
d
x and ∫
Ω
‍
g
(
x
)
u
+
r
d
x, respectively. It then follows that (
E
λ
) has k nonnegative solutions. Applying the maximum principle, (
E
λ
) admits at least k positive solutions.



---

*Source: 829069-2014-04-03.xml*