# Multiplicity of Solutions for an Elliptic Problem with Critical Sobolev-Hardy Exponents and Concave-Convex Nonlinearities

Authors: Juan Li; Yuxia Tong
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

We study the existence of multiple solutions for the following elliptic problem:-
Δ
p
u
-
μ
|
u
|
p
-
2
u
/
|
x
|
p
=
|
u
|
p
*
(
t
)
-
2
/
|
x
|
t
u
+
λ
|
u
|
q
-
2
/
|
x
|
s
u
,
u
∈
W
0
1
,
p
(
Ω
)
. We prove that if 1
≤
q
<
p
<
N, then there is a μ
0, such that for any μ
∈
0
,
μ
0, the above mentioned problem possesses infinitely many weak solutions. Our result generalizes a similar result (Azorero and Alonso, 1991).


---

## Body

## 1. Introduction and Main Results

In this paper, we study the existence of multiple solutions to the following elliptic problem:(1)
-
Δ
p
u
-
μ
|
u
|
p
-
2
u
|
x
|
p
=
|
u
|
p
*
(
t
)
-
2
|
x
|
t
u
+
λ
|
u
|
q
-
2
|
x
|
s
u
,
x
∈
Ω
,
u
(
x
)
=
0
,
x
∈
∂
Ω
,
where Ω
⊂
ℜ
N
(
N
≥
3
) is a smooth bounded domain containing the origin 0, Δ
p
u
=
div
⁡
(
|
∇
u
|
p
-
2
∇
u
) is the p-Laplacian of u, 1
<
p
<
N, 0
≤
μ
<
μ
-
≡
(
(
N
-
p
)
/
p
)
p, 0
≤
t, s
<
p, 1
≤
q
<
p, and p
*
(
t
)
=
p
(
N
-
t
)
/
(
N
-
p
) is the critical Sobolev-Hardy exponent; note that p
*
(
0
)
=
p
*
≡
p
N
/
(
N
-
p
) is the critical Sobolev exponent.Problem (1) is related to the well-known Sobolev-Hardy inequalities [1]:
(2)
(
∫
ℜ
N
‍
|
u
|
p
*
(
t
)
|
x
|
t
d
x
)
p
/
p
*
(
t
)
≤
C
∫
ℜ
N
‍
|
∇
u
|
p
d
x
,
hhhhhhhhhhhhhhhhhhhh
∀
u
∈
W
0
1
,
p
(
Ω
)
.
As t
=
p, p
*
(
t
)
=
p, then the well-known Hardy inequality holds [1, 2]:
(3)
∫
ℜ
N
‍
|
u
|
p
|
x
|
p
d
x
≤
1
μ
-
∫
ℜ
N
‍
|
∇
u
|
p
d
x
,
∀
u
∈
W
0
1
,
p
(
Ω
)
.In this paper, we useL
q
(
Ω
,
|
x
|
s
) to denote the usual weighted L
q
(
Ω
) space with the weight |
x
|
s. In W
0
1
,
p
(
Ω
), for μ
∈
[
0
,
μ
-
), we use the norm
(4)
∥
u
∥
=
∥
u
∥
W
0
1
,
p
(
Ω
)
=
(
∫
Ω
‍
(
|
∇
u
|
p
-
μ
|
u
|
p
|
x
|
p
)
d
x
)
1
/
p
.
By (3), this norm is equivalent to the usual norm (
∫
Ω
‍
|
∇
u
|
p
d
x
)
1
/
p. By the Hardy inequality and the Sobolev-Hardy inequality, for 0
≤
μ
<
μ
-, 0
≤
t
<
p, we can define the following best constants:
(5)
A
μ
,
t
(
Ω
)
=
inf
⁡
u
∈
W
0
1
,
p
(
Ω
)
∖
{
0
}
∥
u
∥
p
(
∫
Ω
‍
(
|
u
|
p
*
(
t
)
/
|
x
|
t
)
d
x
)
p
/
p
*
(
t
)
.
Note that A
μ
,
0 is the best constant in the Sobolev inequality, that is,
(6)
A
μ
,
0
(
Ω
)
=
inf
⁡
u
∈
W
0
1
,
p
(
Ω
)
∖
{
0
}
∫
Ω
‍
(
|
∇
u
|
p
-
μ
(
|
u
|
p
/
|
x
|
p
)
)
d
x
(
∫
Ω
‍
|
u
|
p
*
d
x
)
p
/
p
*
.
The energy functional of (1) is defined as follows:
(7)
I
(
u
)
=
1
p
∫
Ω
‍
(
|
∇
u
|
p
-
μ
|
u
|
p
|
x
|
p
)
d
x
-
1
p
*
(
t
)
∫
Ω
‍
|
u
|
p
*
(
t
)
|
x
|
t
d
x
-
λ
q
∫
Ω
‍
|
u
|
q
|
x
|
s
d
x
.
Then, I
(
u
) is well defined on W
0
1
,
p
(
Ω
) and belongs to C
1
(
W
0
1
,
p
(
Ω
)
,
ℜ
). The solutions of problem (1) are then the critical points of the functional I.In recent years, the quasilinear problems related to Hardy inequality and Sobolev-Hardy inequality have been studied by some authors [3–7]. Ghoussoub and Yuan [5] studied problem (1) with μ
=
0, s
=
0, and p
≤
q
≤
p
* and proved the existence results of positive solutions and sign-changing solutions. Kang in [3, 4] studied (1,1) when p
≤
q
≤
p
*
(
s
) and verified the existence of positive solutions of (1) when the parameters p, q, s, λ, μ satisfy suitable conditions. To the best of our knowledge, there are few results of problem (1) involving the p-sublinear of 1
≤
q
<
p
<
N. We are only aware of the works [6–9] which studied the existence and multiplicity of solution of problem (1) involving weight functions. Azorero and Alonso [9] studied problem (1) with s
=
t
=
μ
=
0, 1
<
q
<
p and proved that there exists λ
1, such that (1) has infinitely many solutions for λ
∈
(
0
,
λ
1
). Hsu [7] studied problem (1) and proved that there exists Λ
0
>
0 such that (1) has at least two positive solutions for λ
∈
(
0
,
Λ
0
). In this paper, we study (1) and extend the results of [7, 9].Throughout this paper, letR
0 be the positive constant such that Ω
⊂
B
(
0
,
R
0
), where B
(
0
,
R
0
)
=
{
x
∈
ℜ
N
:
|
x
|
<
R
0
}. By Holder inequalities, for all u
∈
W
0
1
,
p
(
Ω
), we obtain
(8)
∫
Ω
‍
|
u
|
q
|
x
|
s
d
x
≤
(
∫
B
(
0
,
R
0
)
‍
|
x
|
(
t
q
-
s
p
*
(
t
)
)
/
(
p
*
(
t
)
-
q
)
d
x
)
(
p
*
(
t
)
-
q
)
/
p
*
(
t
)
×
(
∫
Ω
‍
|
u
|
p
*
(
t
)
|
x
|
t
d
x
)
q
/
p
*
(
t
)
≤
(
N
ω
N
∫
0
R
0
‍
r
(
(
t
q
-
s
p
*
(
t
)
)
/
(
p
*
(
t
)
-
q
)
)
+
N
-
1
d
r
)
(
p
*
(
t
)
-
q
)
/
p
*
(
t
)
×
(
∫
Ω
‍
|
u
|
p
*
(
t
)
|
x
|
t
d
x
)
q
/
p
*
(
t
)
=
(
N
ω
N
R
0
(
(
t
q
-
s
p
*
(
t
)
)
/
(
p
*
(
t
)
-
q
)
)
+
N
(
p
*
(
t
)
-
q
)
q
(
t
-
N
)
+
p
*
(
t
)
(
N
-
s
)
)
(
p
*
(
t
)
-
q
)
/
p
*
(
t
)
×
(
∫
Ω
‍
|
u
|
p
*
(
t
)
|
x
|
t
d
x
)
q
/
p
*
(
t
)
=
c
0
(
∫
Ω
‍
|
u
|
p
*
(
t
)
|
x
|
t
d
x
)
q
/
p
*
(
t
)
,
where ω
N
=
2
π
N
/
2
/
N
Γ
(
N
/
2
) is the volume of the unit ball in ℜ
N. The following inequality comes from the paper [7]:
(9)
∫
Ω
‍
|
u
|
q
|
x
|
s
d
x
≤
(
N
ω
N
R
0
N
-
s
N
-
s
)
(
p
*
(
s
)
-
q
)
/
p
*
(
s
)
A
μ
,
s
-
q
/
p
∥
u
∥
q
.Now we are ready to state our main results.Theorem 1.
IfΩ
⊂
ℜ
N is a bounded domain in ℜ
N, and 1
≤
q
<
p
<
N, then there is a μ
0
>
0 such that problem (1) possesses infinitely many weak solutions in W
0
1
,
p
(
Ω
) for any μ
∈
(
0
,
μ
0
).


## 2.  The Palais-Smale Condition

LetX be a Banach space and X
-
1 be the dual space of X. The functional I
∈
C
1
(
X
,
ℜ
) is said to satisfy the Palais-Smale condition at level c ((
P
S
)
c), if any sequence (
u
n
)
⊂
X satisfying
(10)
I
(
u
n
)
⟶
c
,
I
′
(
u
n
)
⟶
0
strongly
in
X
-
1
as
n
⟶
∞
contains a subsequence converging in X to a critical point of the functional I. In this paper, we will take X
=
W
0
1
,
p
(
Ω
).Lemma 2.
Let{
u
n
}
⊂
W
0
1
,
p
(
Ω
) be a Palais-Smale sequence for I defined by (7), that is,
(11)
I
(
u
n
)
⟶
c
,
(12)
I
′
(
u
n
)
⟶
0
i
n
W
-
1
,
p
′
(
Ω
)
,
1
p
+
1
p
′
=
1
.
If 1
≤
q
<
p and c
<
(
p
-
t
)
/
p
(
N
-
t
)
(
A
μ
,
t
)
(
N
-
t
)
/
(
p
-
t
)
-
K
λ
p
*
(
t
)
/
(
p
*
(
t
)
-
q
), and K depends on p, q, N, then, there exists a subsequence {
u
n
k
}
⊂
{
u
n
}, strongly convergent in W
0
1
,
p
(
Ω
).Proof.
By (11) and (12), it is easy to prove that the sequence u
n is bounded in W
0
1
,
p
(
Ω
). Passing to a subsequence if necessary, we may assume that, as n
→
∞,
(13)
u
n
⇀
u
weakly
in
W
0
1
,
p
(
Ω
)
,
u
n
⇀
u
weakly
in
L
p
*
(
t
)
(
Ω
,
|
x
|
-
t
)
,
u
n
⇀
u
weakly
in
L
p
(
Ω
,
|
x
|
-
p
)
,
u
n
⟶
u
strongly
in
L
q
(
Ω
,
|
x
|
-
s
)
,
u
n
⟶
u
strongly
in
L
p
(
Ω
)
,
u
n
⟶
u
a
.
e
.
in
Ω
.
Then, u
∈
W
0
1
,
p
(
Ω
) is a solution of problem (1). By the concentration compactness principle (see [10, 11]), there exists a subsequence, still denoted by u
n, at the most countable set ȷ, a set of different points {
x
j
}
j
∈
ȷ
⊂
Ω
∖
{
0
}, sets of nonnegative real numbers {
μ
~
j
}
j
∈
ȷ
∪
{
0
}, {
υ
~
j
}
j
∈
ȷ
∪
{
0
}, and nonnegative real numbers τ
~
0 and γ
~
0, such that
(14)
|
∇
u
n
|
p
⇀
d
μ
~
≥
|
∇
u
|
p
+
∑
‍
j
∈
ȷ
μ
~
j
δ
x
j
+
μ
~
0
δ
0
,
|
u
n
|
p
*
⇀
d
ν
~
=
|
u
|
p
*
+
∑
‍
j
∈
ȷ
ν
~
j
δ
x
j
+
ν
~
0
δ
0
,
|
u
n
|
p
*
(
t
)
|
x
|
t
⇀
d
τ
~
=
|
u
|
p
*
(
t
)
|
x
|
t
+
τ
~
0
δ
0
,
|
u
n
|
p
|
x
|
p
⇀
d
γ
~
=
|
u
|
p
|
x
|
p
+
γ
~
0
δ
0
,
where δ
x is the Dirac mass at x.
Case 1 (t
=
s
=
0  and p
*
(
t
)
=
p
*). We claim that ȷ is finite, and, for any j
∈
ȷ, either
(15)
ν
~
j
=
0
or
ν
~
j
≥
(
A
μ
,
0
)
N
/
p
.
In fact, letε
>
0 be small enough such that 0
∈
¯
B
(
x
j
,
ε
) and B
(
x
i
,
ε
)
⋂
‍
B
(
x
j
,
ε
)
=
∅ for i
≠
j, i
,
j
∈
ȷ. We consider φ
j
∈
C
0
∞
(
ℜ
N
), such that
(16)
φ
j
≡
1
on
⁡
B
(
x
j
,
ε
2
)
,
φ
j
≡
0
on
⁡
B
(
x
j
,
ε
)
c
,
h
h
h
|
∇
φ
j
|
≤
4
ε
,
h
h
h
0
≤
φ
j
≤
1
.
It is clear that the sequence {
φ
j
u
n
} is bounded in W
0
1
,
p
(
Ω
). Note that
(17)
〈
I
′
(
u
n
)
,
u
n
φ
j
〉
=
∫
Ω
‍
|
∇
u
n
|
p
φ
j
d
x
+
∫
Ω
‍
u
n
|
∇
u
n
|
p
-
2
∇
u
n
∇
φ
j
d
x
-
μ
∫
Ω
‍
|
u
n
|
p
|
x
|
p
φ
j
d
x
-
∫
Ω
‍
|
u
n
|
p
*
φ
j
d
x
-
λ
∫
Ω
‍
|
u
n
|
q
φ
j
d
x
.
By (13), (16), and the Holder inequality, we obtain
(18)
0
≤
lim
⁡
ε
→
0
lim
⁡
n
→
∞
|
∫
Ω
‍
u
n
|
∇
u
n
|
p
-
2
∇
u
n
∇
φ
j
d
x
|
≤
C
lim
⁡
ε
→
0
(
∫
B
(
x
j
,
ε
)
‍
|
u
|
p
*
d
x
)
1
/
p
*
=
0
,
lim
⁡
ε
→
0
lim
⁡
n
→
∞
|
∫
Ω
‍
|
u
n
|
p
|
x
|
p
φ
j
d
x
|
≤
lim
⁡
ε
→
0
lim
⁡
n
→
∞
|
∫
B
ε
(
x
j
)
|
u
n
|
p
(
|
x
j
|
-
ε
)
p
φ
j
d
x
|
=
0
.
From (12)~(18), we get that
(19)
0
=
lim
⁡
ε
→
0
lim
⁡
n
→
∞
〈
I
′
(
u
n
)
,
u
n
φ
j
〉
≥
μ
~
j
-
ν
~
j
.
By the Sobolev inequality, A
0,0
ν
j
p
/
p
*
≤
μ
~
j, hence, we deduce that
(20)
ν
~
j
=
0
or
ν
~
j
≥
(
A
0,0
)
N
/
p
,
which implies that ȷ is finite.
Now we consider the possibility of concentration at the origin. Letε
>
0 be small enough such that x
j
∈
¯
B
(
0
,
ε
), ∀
j
∈
ȷ. Take φ
0
∈
C
0
∞
(
ℜ
N
) such that
(21)
φ
0
≡
1
on
⁡
B
(
0
,
ε
2
)
,
φ
0
≡
0
on
⁡
B
(
0
,
ε
)
c
,
|
∇
φ
0
|
≤
4
ε
.
By (13) and (14), we also get that
(22)
0
=
lim
⁡
ε
→
0
lim
⁡
n
→
∞
〈
I
′
(
u
n
)
,
u
n
φ
0
〉
=
lim
⁡
ε
→
0
(
∫
Ω
‍
φ
0
d
μ
~
-
∫
Ω
‍
φ
0
d
γ
~
-
∫
Ω
‍
φ
0
d
ν
~
-
λ
∫
Ω
‍
φ
0
|
u
|
q
|
x
|
s
d
x
)
≥
μ
~
0
-
μ
γ
~
0
-
ν
~
0
.
By the definition of A
μ
,
0, we deduce that
(23)
A
μ
,
0
ν
~
0
p
/
p
*
≤
μ
~
0
-
μ
γ
~
0
.
From (22) we have
(24)
A
μ
,
0
ν
~
0
p
/
p
*
≤
μ
~
0
-
μ
γ
~
0
≤
ν
~
0
,
which implies that ν
~
0
=
0 or
(25)
ν
~
0
≥
(
A
μ
,
0
)
N
/
p
.
We will prove that (25) and ν
~
j
≥
(
A
0,0
)
N
/
p are not possible. By (13) and (14),
(26)
c
=
lim
⁡
n
→
∞
I
(
u
n
)
=
lim
⁡
n
→
∞
{
I
(
u
n
)
-
1
p
〈
I
′
(
u
n
)
,
u
n
〉
}
=
lim
⁡
n
→
∞
{
1
N
∫
Ω
‍
|
u
n
|
p
*
d
x
+
λ
(
1
p
-
1
q
)
∫
Ω
‍
|
u
n
|
q
d
x
}
≥
1
N
(
∫
Ω
‍
|
u
|
p
*
d
x
+
∑
‍
j
∈
ȷ
ν
~
j
+
ν
~
0
)
+
λ
(
1
p
-
1
q
)
∫
Ω
‍
|
u
|
q
d
x
≥
1
N
∫
Ω
‍
|
u
|
p
*
d
x
+
1
N
min
⁡
{
(
A
μ
,
0
)
N
/
p
,
(
A
0,0
)
N
/
p
}
+
λ
(
1
p
-
1
q
)
∫
Ω
‍
|
u
|
q
d
x
≥
1
N
∫
Ω
‍
|
u
|
p
*
d
x
+
1
N
(
A
μ
,
0
)
N
/
p
+
λ
(
1
p
-
1
q
)
∫
Ω
‍
|
u
|
q
d
x
.
By applying the Holder inequality at (26), we have
(27)
c
≥
1
N
(
A
μ
,
0
)
N
/
p
+
1
N
∫
Ω
‍
|
u
|
p
*
d
x
-
λ
(
1
q
-
1
p
)
|
Ω
|
(
p
*
-
q
)
/
p
*
(
∫
Ω
‍
|
u
|
p
*
d
x
)
q
/
p
*
.
Let f
1
(
x
)
=
c
1
x
p
*
-
λ
c
2
x
q, c
1
=
1
/
N, c
2
=
(
1
/
q
)
-
(
1
/
p
). This function obtains its absolute minimum (for x
>
0) at point x
0
=
(
λ
c
2
q
/
p
*
c
1
)
1
/
(
p
*
-
q
). That is,
(28)
f
1
(
x
)
≥
f
1
(
x
0
)
=
-
K
1
λ
p
*
/
(
p
*
-
q
)
,
where
(29)
K
1
=
c
2
p
/
(
p
*
-
q
)
c
1
-
q
/
(
p
*
-
q
)
×
(
(
q
p
*
)
q
/
(
p
*
-
q
)
-
(
q
p
*
)
p
*
/
(
p
*
-
q
)
)
>
0
,
because of 1
<
q
<
p
<
N
p
/
(
N
-
p
). But this result contradicts the hypothesis. Then, ν
~
j
=
0
∀
j
∈
ȷ
∪
{
0
} and we conclude.
Case 2 (0
<
t
<
p,   then p
<
p
*
(
t
)
<
p
*). We only need to consider the possibility of concentration at the origin. Let ε
>
0 be small enough such that B
(
0
,
ε
)
⊂
Ω. Take φ
0 a smooth cut-off function centered at the origin such that 0
≤
φ
0
≤
1, φ
0
=
1 for |
x
|
≤
ε
/
2, φ
0
=
0 for |
x
|
≥
ε, and |
∇
φ
0
|
≤
4
/
ε. By (13) and (14), we get that
(30)
0
=
lim
⁡
ε
→
0
lim
⁡
n
→
∞
〈
I
′
(
u
n
)
,
u
n
φ
0
〉
=
lim
⁡
ε
→
0
(
∫
Ω
‍
φ
0
d
μ
~
-
μ
∫
Ω
‍
φ
0
d
γ
~
-
∫
Ω
‍
φ
0
d
τ
~
-
λ
∫
Ω
‍
φ
0
|
u
|
q
|
x
|
s
d
x
)
≥
μ
~
0
-
μ
γ
~
0
-
τ
~
0
.
By the definition of A
μ
,
t, we deduce that
(31)
A
μ
,
t
τ
~
0
p
/
p
*
(
t
)
≤
μ
~
0
-
μ
γ
~
0
From (30), we have
(32)
A
μ
,
t
τ
~
0
p
/
p
*
(
t
)
≤
τ
~
0
,
which implies that τ
~
0
=
0 or
(33)
τ
~
0
≥
(
A
μ
,
t
)
(
N
-
t
)
/
(
p
-
t
)
.
We will prove (33) is not possible. From the above arguments and (8), we conclude that
(34)
c
=
lim
⁡
n
→
∞
J
(
u
n
)
=
lim
⁡
n
→
∞
{
J
(
u
n
)
-
1
p
〈
J
(
u
n
)
,
u
n
〉
}
≥
p
-
t
p
(
N
-
t
)
∫
Ω
‍
|
u
|
p
*
(
t
)
|
x
|
t
d
x
+
p
-
t
p
(
N
-
t
)
(
A
μ
,
t
)
(
N
-
t
)
/
(
p
-
t
)
+
λ
c
0
(
1
q
-
1
p
)
(
∫
Ω
‍
|
u
|
p
*
(
t
)
|
x
|
t
d
x
)
q
/
p
*
(
t
)
.
Let f
2
(
x
)
=
c
3
x
p
*
(
t
)
-
λ
c
4
x
q, c
3
=
(
p
-
t
)
/
p
(
N
-
t
), c
4
=
c
0
(
(
1
/
q
)
-
(
1
/
p
)
). This function obtains its absolute minimum at point x
0
=
(
λ
c
4
q
/
p
*
(
t
)
c
3
)
1
/
(
p
*
-
q
). That is,
(35)
f
2
(
x
)
≥
f
2
(
x
0
)
=
-
K
2
λ
p
*
(
t
)
/
(
p
*
(
t
)
-
q
)
.
But this result contradicts the hypothesis. Hence, up to a subsequence, we obtain that u
n
→
u strongly in W
0
1
,
p
(
Ω
).
Thus, the proof of the Lemma is completed.


## 3. Existence of Infinitely Many Solutions

In this section, we will prove our main result of Theorem1. We first recall some concepts and results in minimax theory.LetX be a Banach space, and Σ denote all closed subsets of X
-
{
0
} which are symmetric with respect to the origin. For A
∈
Σ, we define the genus γ
(
A
) by
(36)
γ
(
A
)
=
min
⁡
{
k
∈
N
:
∃
ϕ
∈
C
(
A
;
R
k
∖
{
0
}
)
,
ϕ
(
x
)
=
-
ϕ
(
-
x
)
}
,
if the minimum exists, and if such a minimum does not exist, then we define γ
(
A
)
=
∞. The main properties of the genus are contained in the following lemma (see [12] for the details).Lemma 3.
LetA
,
B
∈
Σ. Then one has the following.(1)
If there existsf
∈
C
(
A
,
B
), odd, then γ
(
A
)
≤
(
B
).
(2)
IfA
⊂
B, then γ
(
A
)
≤
γ
(
B
).
(3)
If there exists an odd homeomorphism betweenA and B, then γ
(
A
)
=
γ
(
B
).
(4)
IfS
N
-
1 is the sphere in ℜ
N, then γ
(
S
N
-
1
)
=
N.
(5)
Considerγ
(
A
∪
B
)
≤
γ
(
A
)
+
γ
(
B
).
(6)
Ifγ
(
B
)
<
+
∞, then γ
(
A
-
B
¯
)
≥
γ
(
A
)
-
γ
(
B
).
(7)
IfA is compact, then γ
(
A
)
<
+
∞, and there exists δ
>
0 such that γ
(
N
δ
(
A
)
)
=
γ
(
A
), where N
δ
(
A
)
=
{
x
∈
X
:
d
(
x
,
A
)
≤
δ
}.
(8)
IfX
0 is a subspace of X with codimension k, and γ
(
A
)
<
k, then A
∩
X
0
≠
∅.LetX be a Banach space and E be a C
1 functional on X. Denote E
c
=
{
u
∈
X
∣
E
(
u
)
≤
c
}, Σ
k
=
{
C
⊂
W
0
1
,
p
(
Ω
)
-
{
0
}
,
C
i
s
c
l
o
s
e
d
,
C
=
-
C
,
γ
(
C
)
≥
k
}.Given the functionalI, under the hypothesis 1
<
q
<
p
<
n, using Sobolev’s equality and (9), we obtain
(37)
I
(
u
)
≥
1
p
∥
u
∥
p
-
1
p
*
(
t
)
A
μ
,
t
p
*
(
t
)
/
p
∥
u
∥
p
*
(
t
)
-
λ
c
0
q
∥
u
∥
q
.
If we define for x
≥
0
(38)
h
(
x
)
=
1
p
x
p
-
1
p
*
(
t
)
A
μ
,
t
p
*
(
t
)
/
p
x
p
*
(
t
)
-
λ
c
0
q
x
q
,
then
(39)
I
(
u
)
≥
h
(
∥
u
∥
)
.
Because p
*
(
t
)
>
p and h
(
x
)
→
-
∞,  as x
→
+
∞, it is easy to see that there exists 0
<
μ
0
≤
1 such that, if 0
<
μ
≤
μ
0, h attains its positive maximum.From the structure ofh
(
x
), we see that there are constants 0
<
R
0
<
R
1, such that h
(
R
0
)
=
h
(
R
1
)
=
0, h
(
R
)
≤
0 if R
<
R
0, h
(
R
)
>
0 if R
0
<
R
<
R
1, and h
(
R
)
<
0 if R
>
R
1. Following [9], let τ
:
R
+
→
[
0,1
]
∈
C
∞ be nonincreasing, such that
(40)
τ
(
x
)
=
{
1
,
if
0
≤
x
≤
R
0
,
0
,
if
x
≥
R
1
,
and let φ
(
u
)
=
τ
(
∥
u
∥
); we consider the truncated functional
(41)
J
(
u
)
=
1
p
∫
Ω
‍
(
|
∇
u
|
p
-
μ
u
p
|
x
|
p
)
d
x
-
1
p
*
(
t
)
∫
Ω
‍
|
u
|
p
*
(
t
)
φ
(
u
)
|
x
|
t
d
x
-
λ
q
∫
Ω
‍
|
u
|
q
|
x
|
s
d
x
.
Similar to (39), we have J
(
u
)
≥
h
-
(
∥
u
∥
), where
(42)
h
-
(
x
)
=
1
p
x
p
-
τ
(
x
)
p
*
(
t
)
A
μ
,
t
p
*
(
t
)
/
p
x
p
*
(
t
)
-
λ
c
0
q
x
q
.
Clearly, h
-
(
x
)
≥
h
(
x
) for x
≥
0 and h
-
(
x
)
=
h
(
x
) if 0
≤
x
≤
R
0, h
-
(
x
)
≥
h
(
x
)
≥
0, if R
0
<
R
<
R
1, and if x
>
R
1, h
-
(
x
)
=
(
1
/
p
)
x
p
-
(
λ
c
0
/
q
)
x
q is strictly increasing, and so h
-
(
x
)
>
0, if x
>
R
1. Consequently, h
-
(
x
)
≥
0 for x
≥
R
0.Lemma 4.
(
1
)  Consider J
∈
C
1
(
W
0
1
,
p
(
Ω
)
,
R
).
(
2
) If J
(
u
)
≤
0, then ∥
u
∥
≤
R
0 and I
(
v
)
=
J
(
v
) for all v in a small enough neighborhood of u.
(
3
) There exists λ
0
>
0, such that if 0
<
λ
<
λ
0, then J verifies a local Palais-Smale condition for c
≤
0.Proof.
(1) and (2) are immediate. To prove (3), observe that all Palais-Smale sequences forJ with c
≤
0 must be bounded; then, by Lemma 2, if λ verifies (
(
p
-
t
)
/
p
(
N
-
t
)
)
(
A
μ
,
t
)
(
N
-
t
)
/
(
p
-
t
)
-
K
λ
p
*
(
t
)
/
(
p
*
(
t
)
-
q
)
≥
0, there exists a convergent subsequence.Now, we use the idea in [9] to construct negative critical values of J via genus.Lemma 5.
Givenn
∈
N, there is an ε
(
n
)
>
0, such that
(43)
γ
(
{
u
∈
W
0
1
,
p
(
Ω
)
:
J
(
u
)
≤
-
ε
(
n
)
}
)
≥
n
.Proof.
Fixn; let E
n be an n-dimensional subspace of W
0
1
,
p
(
Ω
); we take u
n
∈
E
n with norm ∥
u
n
∥
=
1 for 0
<
ρ
<
R
0; we have
(44)
J
(
ρ
u
n
)
=
I
(
ρ
u
n
)
=
1
p
ρ
p
-
1
p
*
(
t
)
ρ
p
*
(
t
)
×
∫
Ω
‍
|
u
n
|
p
*
(
t
)
|
x
|
t
d
x
-
λ
q
ρ
q
∫
Ω
‍
|
u
n
|
q
|
x
|
s
d
x
.
Since E
n is a space of finite dimension, all the norms in E
n are equivalent. If we define
(45)
α
n
=
inf
⁡
{
∫
Ω
‍
|
u
|
p
*
(
t
)
|
x
|
t
d
x
:
u
∈
E
n
,
∥
u
∥
=
1
}
>
0
,
β
n
=
inf
⁡
{
∫
Ω
‍
|
u
|
q
|
x
|
s
d
x
:
u
∈
E
n
,
∥
u
∥
=
1
}
>
0
,
we have
(46)
J
(
ρ
u
n
)
≤
1
p
ρ
p
-
α
n
p
*
(
t
)
ρ
p
*
(
t
)
-
λ
β
n
q
ρ
q
,
and we can choose ε (which depends on n), and η
<
R
0, such that J
(
η
u
)
≤
-
ε if u
∈
E
n and ∥
u
∥
=
1.
LetS
η
=
{
u
∈
W
0
1
,
p
(
Ω
)
:
∥
u
∥
=
η
}. Consider S
η
∩
E
n
⊂
{
u
∈
W
0
1
,
p
(
Ω
)
:
J
(
u
)
≤
-
ε
}; therefore, by Lemma 3, we see that
(47)
γ
(
{
u
∈
W
0
1
,
p
(
Ω
)
:
J
(
u
)
≤
-
ε
(
n
)
}
)
≥
γ
(
S
η
∩
E
n
)
=
n
.
We are now in a position to prove our main result.Proof of Theorem1.
LetΣ
k
=
{
C
⊂
W
0
1
,
p
(
Ω
)
-
{
0
}
,
C
i
s
c
l
o
s
e
d
,
C
=
-
C
,
γ
(
C
)
≥
k
}, c
k
=
inf
⁡
C
∈
Σ
k
sup
⁡
u
∈
C
J
(
u
), K
c
=
{
u
∈
W
0
1
,
p
(
Ω
)
,
J
′
(
u
)
=
0
,
J
(
u
)
=
c
}, and suppose that 0
<
λ
<
λ
0 where λ
0 is the constant given by Lemma 4. We claim that if k
,
r
∈
N are such that c
=
c
k
=
c
k
+
1
=
·
·
·
=
c
k
+
r, then γ
(
K
c
)
≥
r
+
1.
In fact, denoteJ
-
ε
=
{
u
∈
W
0
1
,
p
(
Ω
)
:
J
(
u
)
≤
-
ε
}; by Lemma 5, we see that for any k
∈
N, there is a ε
(
k
)
>
0, such that γ
(
J
-
ε
(
k
)
)
≥
k. Since J is continuous and even, J
-
ε
(
k
)
∈
Σ
n and c
k
≤
-
ε
(
n
)
<
0. As J is bounded from below, we see that c
k
>
-
∞ for all k
∈
N.
Suppose thatc
=
c
k
=
c
k
+
1
=
·
·
·
=
c
k
+
r
<
0; then J satisfies (
P
S
)
c condition by Lemma 2, and it is easy to see that K
c is a compact set.
Ifγ
(
K
c
)
≤
r, then there is a closed and symmetric set U with K
c
⊂
U and γ
(
U
)
≤
r by Lemma 3. Since c
<
0, we can also assume that the closed set U
⊂
J
0. Since J satisfies (
P
S
)
c condition for c
<
0, by the Deformation Lemma, there is an odd homeomorphism,
(48)
η
:
W
0
1
,
p
(
Ω
)
⟶
W
0
1
,
p
(
Ω
)
such that η
(
J
c
+
δ
-
U
)
⊂
J
c
-
δ for some δ with 0
<
δ
<
-
c.
Sincec
=
c
k
+
r
=
inf
⁡
C
∈
Σ
k
sup
⁡
u
∈
C
J
(
u
), there exists an A
∈
Σ
k
+
r, such that
(49)
sup
⁡
u
∈
C
J
(
u
)
<
c
+
δ
,
i
.
e
.
,
A
⊂
J
c
+
δ
,
(50)
η
(
A
-
U
)
⊂
η
(
J
c
+
δ
-
U
)
⊂
J
c
-
δ
.
But by Lemma3 and γ
(
U
)
≤
r, we have
(51)
γ
(
A
-
U
¯
)
≥
γ
(
A
)
-
γ
(
U
)
≥
n
,
γ
(
η
A
-
U
¯
)
γ
(
A
-
U
¯
)
≥
n
.
Hence, η
A
-
U
¯
∈
Σ
k and sup
⁡
u
∈
η
A
-
U
¯
≥
c
k
=
c, which contradicts to (50). So we have proved that γ
(
K
c
)
≥
r
+
1.
Now if for allk
∈
N, we have Σ
k
+
1
⊂
Σ
k, c
k
≤
c
k
+
1
<
0. If all c
k are distinct, then γ
(
K
c
k
)
≥
1, and we see that {
c
k
} is a sequence of distinct critical values of J; if for some k
0, there is a r
≥
1 such that
(52)
c
=
c
k
0
=
c
k
0
+
1
=
·
·
·
=
c
k
0
+
r
,
then
(53)
γ
(
K
c
k
0
)
≥
r
+
1
,
which shows that K
c
k
0 contains infinitely many distinct elements.
SinceJ
(
u
)
=
I
(
u
) if J
(
u
)
<
0, we see that there are infinitely many critical points of I
(
u
). The theorem is proved.



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*Source: 180105-2014-03-06.xml*