Báo cáo hóa học: " Multiple positive solutions for a class of quasilinear elliptic equations involving concave-convex nonlinearities and Hardy terms" ppt

RESEARC H Open Access
Multiple positive solutions for a class of quasi-
linear elliptic equations involving concave-convex
nonlinearities and Hardy terms
Tsing-San Hsu
Correspondence:
edu.tw
Center for General Education,
Chang Gung University, Kwei-San,
Tao-Yuan 333, Taiwan ROC
Abstract
In this paper, we are concerned with the following quasilinear elliptic equation
−
p
u − μ
|u|
p−2
u
|
x
|
p
= λf (x)|u|
q−2
u + g(x)|u|
p∗−2
u in , u =0on∂
,
where Ω ⊂ ℝ
N
is a smooth domain with smooth boundary ∂Ω such that 0 Î Ω, Δ

p
u
= div(|∇u|
p-2
∇u), 1 <p<N,
μ< ¯μ =(
N−p
p
)
p
, l >0, 1 <q<p, sign-changing weight
functions f and g are continuous functions on
¯

,
¯μ =(
N−p
p
)
p
is the best Hardy
constant and
p
∗
=
Np
N−
p
is the critical Sobolev exponent. By extracting the Palais-Smale
sequence in the Neha ri manifold, the multiplicity of positive solutions to this

equation is verified.
Keywords: Multiple positive solutions, critical Sobolev exponent, concave-convex,
Hardy terms, sign-changing weights
1 Introduction and main results
Let Ω be a smooth domain (not necessarily bounded) in ℝ
N
(N ≥ 3) with smooth
boundary ∂Ω such that 0 Î Ω. We will study the multiplicity of positive solutions for
the following quasilinear elliptic equation
⎧
⎨
⎩
−
p
u − μ
|u|
p−
2
u
|x|
p
= λf (x)|u|
q−2
u + g(x)|u|
p∗−2
u,
u =0,
in 
,
on ∂,

(1:1)
where Δ
p
u = div(|∇u|
p-2
∇u), 1 <p <N,
μ< ¯μ =(
N−p
p
)
p
,
¯
μ
is the best Hardy consta nt,
l > 0, 1 <q<p,
p
∗
=
Np
N−
p
is the critical Sobolev exponent and the weight functions
f
,
g
:
¯
 →
R

are continuous, which change sign on Ω.
Let
D
1,p
0
(
)
be the completion of
C
∞
0
(
)
with respect to the norm
(


|∇u|
p
dx)
1/
p
.
The energy functional of (1.1) is defined on
D
1,p
0
(
)
by

J
λ
(u)=
1
p



|∇u|
p
− μ
|u|
p
|x|
p

dx −
λ
q


f |u|
q
dx −
1
p
∗


g|u|

p
∗
dx
.
Then
J
λ
∈ C
1
(D
1,p
0
(), R
)
.
u
∈ D
1,p
0
()\{0
}
is said to be a solution of (1.1) if
J

λ
(u), v =
0
for all
v ∈ D
1,p

0
(
)
and a solution of (1.1) is a critical point of J
l
.
Hsu Boundary Value Problems 2011, 2011:37
/>© 2011 Hsu; licensee Springer. This is an Op en Access a rticle di stributed und er the terms of t he Creative C ommons Attribution License
( which pe rmits unrestricted use, distribution, and reproduction in any medium, provided
the original work is properly cited.
Problem (1.1) is related to the well-known Hardy inequality [1,2]:


|u|
p
|x|
p
dx ≤
1
¯
μ


|∇u|
p
dx, ∀u ∈ C
∞
0
()
.

By the Hardy inequality,
D
1,p
0
(
)
has the equivalent norm ||u||
μ
, where
||u||
p
μ
=



|∇u|
p
− μ
|u|
p
|x|
p

dx, μ ∈ (−∞, ¯μ)
.
Therefore, for 1 <p<N, and
μ
< ¯
μ

, we can define the best Sobolev constant:
S
μ
()= inf
u∈D
1,p
0
()\{0}



|∇u|
p
− μ
|u|
p
|x|
p

dx
(


|u|
p∗
dx)
p
p∗
.
(1:2)

It is well known that S
μ
(Ω)=S
μ
(ℝ
N
)=S
μ
. Note that S
μ
= S
0
when μ ≤ 0 [3].
Such kind of problem with critical expone nts and nonnegative weight functions has
been extensively studied by many au thors. We refer, e.g., in bounded domains and for
p =2to[4-6]andforp>1 to [7-11], while in ℝ
N
and for p = 2 to [12,13], and for p
>1 to [3,14-17], and the references therein.
In the present paper, our research is mainly related to (1.1) with 1 <q<p<N,the
critical exponent and weight functions f, g that change sign on Ω. When p =2,1<q
<2,
μ ∈ [0, ¯μ
)
, f, g are sign changing and Ω is bounded, [18] studied (1.1) and obtained
that there exists Λ > 0 such that (1.1) has at least two positive solutions for all l Î (0,
Λ). For the case p ≠ 2, [19] studied (1.1) and obtained the multiplicity of positive solu-
tions when 1 <q<p<N, μ =0,f, g are sign changing and Ω is bounded. However,
little has been done for this type of problem (1.1). Re cently, Wang et al . [11] have stu-
died (1.1) in a bounded domain Ω under the assumptions 1 <q<p<N, N>p

2
,
−∞ <
μ
< ¯
μ
and f, g are nonnegative. They also prov ed that there existence of Λ
0
>0
such that for l Î (0, Λ
0
), (1.1) po ssesses at least two positive solutions. In this paper,
we study (1.1) and exten d the results of [11,18,19] to the more general case 1 <q<p
<N,
−∞ <
μ
< ¯
μ
, f, g are sign changing and Ω is a smooth domain (not necessarily
bounded) in ℝ
N
(N ≥ 3). By extracting the P alais-Smale sequence in the Nehari mani-
fold, the existence of at least two positive solutions of (1.1) is verified.
The following assumptions are used in this paper:
(
H
)
μ
< ¯
μ

, l >0, 1 <q<p<N, N ≥ 3.
(f
1
)
f ∈ C(
¯
) ∩ L
q∗
()(q
∗
=
p
∗
p
∗
−
q
)
f
+
= max{f,0}≢ 0inΩ.
(f
2
) There exist b
0
and r
0
>0 such that B(x
0
;2r

0
) ⊂ Ω and f (x) ≥ b
0
for all x Î B(x
0
;
2r
0
)
(g
1
)
g
∈ C
(
¯

)
∩ L
∞
(

)
and g
+
= max{g,0}≢ 0inΩ.
(g
2
) There exist x
0

Î Ω and b >0 such that
|
g|
∞
= g(x
0
)=max
x∈
¯

g(x), g(x) > 0, ∀x ∈ ,
g
(
x
)
= g
(
x
0
)
+ o
(
|x − x
0
|
β
)
as x →
0
where | · |

∞
denotes the L
∞
(Ω) norm.
Hsu Boundary Value Problems 2011, 2011:37
/>Page 2 of 15
Set

1
= 
1
(μ)=

p − q
(p ∗−q)|g
+
|
∞

p−q
p∗−p

p ∗−p
(p ∗−q)|f
+
|
q
∗

S

N
p
2
(p−q)+
q
p
μ
.
(1:3)
The main results of this paper are concluded in the following theorems. When Ω is
an unbounded domain, the conclusions are new to the best of our knowledge.
Theorem 1.1 Suppose
(
H
)
,(f
1
) and (g
1
) hold. Th en,(1.1)has at least one positive
solution for all l Î (0, Λ
1
).
Theorem 1.2 Suppose
(
H
)
,(f
1
)-(g

2
) hold, and g is the constant defined as in Lemma
2.2. If
0 ≤
μ
< ¯
μ
, x
0
=0and b ≥ pg, then (1.1) has at least two positive solutions for
all
λ ∈ (0,
q
p

1
)
.
Theorem 1.3 Suppose
(
H
)
,(f
1
)-(g
2
) hold. If μ <0, x
0
≠ 0,
β ≥

N−p
p
−1
and N ≤ p
2
, then
(1.1) has at least two positive solutions for all
λ ∈ (0,
q
p

1
(0))
.
Remark 1.4 As Ω is a bounded smooth domain and p =2,the results of Theorems
1.1, 1.2 are improvements of the main results of [18].
Remark 1.5 As Ω is a bounded smooth domain and p ≠ 2, μ =0,then the results of
Theorems 1.1, 1.2 in this case are the same as the known results in [19].
Remark 1.6 In this remark, we consider that Ω is a bounded domain. In [11], Wang
et al. considered (1.1) with
μ
< ¯
μ
, l >0 and 1 <q<p<p
2
<N. As
0 ≤
μ
< ¯
μ

and 1
w<q <p <N, the results of Theorems 1.1, 1.2 are improvements of the main results of
[11]. As μ <0and 1<q <p <N ≤ p
2
, Theorem 1.3 is the complement to the results in
[[11], Theorem 1.3].
This paper is organized as follows. Some preliminaries and properties of the Nehari
manifold are established in Sections 2 and 3, and Theorems 1.1-1.3 are proved in Sec-
tions 4-6, respectively. Before ending this section, we explain some notations employed
in this paper . In the following argument, we always employ C and C
i
to denote various
positive constants and omit dx in integr al for convenience. B(x
0
; R) is the ball centered
at x
0
Î ℝ
N
with the radius R>0,
(D
1,p
0
())
−
1
denotes the dual spa ce of
D
1,p
0

(
)
,the
norm in L
p
(Ω) is denoted by |·|
p
, the quantity O(ε
t
) denotes |O(ε
t
)/ε
t
| ≤ C, o(ε
t
) means
|o(ε
t
)/ε
t
| ® 0asε ® 0 and o(1) is a generic infinitesimal value. In particular, the quan-
tity O
1
(ε
t
) means that there exist C
1
, C
2
>0 such that C

1
ε
t
≤ O
1
(ε
t
) ≤ C
2
ε
t
as ε is small
enough.
2 Preliminaries
Throughout this paper, (f
1
) and (g
1
) will be assumed. In this section, we will establish
several preliminary lemmas. To this end, we fi rst recall a result on the extremal func-
tions of S
μ,s
.
Lemma 2.1 [16]Assume that 1 < p < N and
0 ≤
μ
< ¯
μ
. Then, the limiting problem
⎧

⎨
⎩
−
p
u − μ
u
p−
1
|x|
p
= u
p∗−1
, in R
N
\{0},
u ∈
D
1,p
(R
N
), u > 0, in R
N
\{0},
(2:1)
Hsu Boundary Value Problems 2011, 2011:37
/>Page 3 of 15
has positive radial ground states
V
p,μ,ε
(x)=ε

−
N−p
p
U
p,μ

x
ε

= ε
−
N−p
p
U
p,μ

|x|
ε

,forallε>0
,
that satisfy

R
N

|∇V
p,μ,ε
(x)|
p

− μ
|V
p,μ,ε
(x)|
p
|x|
p

=

R
N
|V
p,μ,ε
(x)|
p∗
= S
N
p
μ
.
Furthermore, U
p,μ
(|x|) = U
p,μ
(r) is decreasing and has the following properties:
U
p,μ
(1) =


N( ¯μ − μ)
N − p

1
p∗−p
,
lim
r→0
+
r
a(μ)
U
p,μ
(r)=c
1
> 0, lim
r→0
+
r
a(μ)+1
|U

p,μ
(r)| = c
1
a(μ) ≥ 0,
lim
r→+∞
r
b(μ)

U
p,μ
(r)=c
2
> 0, lim
r→+∞
r
b(μ)+1
|U

p,μ
(r)| = c
2
b(μ) > 0
,
c
3
≤ U
p,μ
(r)(r
a(μ)
δ
+ r
b(μ)
δ
)
δ
≤ c
4
, δ :=

N − p
p
,
where c
i
(i =1,2,3,4)are positive constants depending on N, μ and p, and a(μ) and
b(μ) are the zeros of the function h(t)=(p -1)t
p
-(N - p)t
p-1
+ μ, t ≥ 0, satisfying
0 ≤ a(μ) <
N−p
p
< b(μ) ≤
N−p
p
−1
.
Take r >0 small enough such that B(0; r) ⊂ Ω, and define the function
u
ε
(x)=η(x)V
p,μ,ε
(x)=ε
−
N−p
p
η(x ) U
p,μ


|x|
ε

,
(2:2)
where
η ∈ C
∞
0
(B(0; ρ
)
is a cutoff function such that h(x) ≡ 1in
B(0,
ρ
2
)
.
Lemma 2.2 [9,20]Suppose 1 <p<Nand
0 ≤
μ
< ¯
μ
. Then, the following estimates
hold when ε ® 0.
||u
ε
||
p
μ

= S
N
p
μ
+ O(ε
pγ
),


|u
ε
|
p∗
= S
N
p
μ
+ O(ε
p∗γ
),


|u
ε
|
q
=
⎧
⎪
⎨

⎪
⎩
O
1
(ε
θ
),
O
1
(ε
θ
|)lnε|,
O
1
(ε
qγ
),
N
b(μ)
< q < p∗,
q =
N
b(μ)
,
1 ≤ q <
N
b
(
μ
)

,
where
δ =
N−p
p
,
θ = N −
N−p
p
q
and g = b(μ)-δ.
We also recall the following known result by Ben-Naoum, Troestler and Willem,
which will be employed for the energy functional.
Lemma 2.3 [21]Let Ω be an domain, not necessarily bounded, in ℝ
N
,1≤ p <N,
k
(
x
)
∈ L
p∗
p∗−q
(

)
and
k
(
x

)
∈ L
p∗
p∗−q
(

)
Then, the functional
D
1,p
0
() → R : u →

R
N
k(x)|u|
q
d
x
is well-defined and weakly continuous.
Hsu Boundary Value Problems 2011, 2011:37
/>Page 4 of 15
3 Nehari manifold
As J
l
is not bounded below on
D
1
,p
0

(
)
, we need to study J
l
on the Nehari manifold
N
λ
= {u ∈ D
1,p
0
()\{0} : J

λ
(u), u =0}
.
Note that
N
λ
contains all solutions of (1.1) and
u
∈ N
λ
if and only if
||u||
p
μ
− λ


f |u|

q
−


g|u|
p∗
=0
.
(3:1)
Lemma 3.1 J
l
is coercive and bounded below on
N
λ
.
Proof Suppose
u
∈ N
λ
. From (f
1
), (3.1), the Hölder inequality and Sobolev embedding
theorem, we can deduce that
J
λ
(u)=
p ∗−p
pp∗
||u||
p

μ
− λ
p ∗−q
p ∗ q


f |u|
q
≥
1
N
||u||
p
μ
− λ
p ∗−q
p ∗ q
|f
+
|
q∗
|u|
q
p∗
≥
1
N
||u||
p
μ

− λ
p ∗−q
p
∗
q
|f
+
|
q∗
S
−
q
p
μ
||u||
q
μ
.
(3:2)
Thus, J
l
is coercive and bounded below on
N
λ
. □
Define
ψ
λ
(u)=J


λ
(u), u

. Then, for
u
∈ N
λ
,
ψ

λ
(u), u = p||u||
p
μ
− qλ


f |u|
q
− p∗


g|u|
p∗
=(p − q)||u||
p
μ
− (p ∗−q)



g|u|
p∗
= λ(p
∗
− q)


f |u|
q
− (p ∗−p)||u||
p
μ
.
(3:3)
Arguing as in [22], we split
N
λ
into three parts:
N
+
λ
= {u ∈ N
λ
: ψ

λ
(u), u > 0},
N
0
λ

= {u ∈ N
λ
: ψ

λ
(u), u =0},
N
−
λ
= {u ∈ N
λ
: ψ

λ
(u), u < 0}
.
Lemma 3.2 Suppose u
l
is a local minimizer of J
l
on
N
λ
and
u
λ
/∈ N
0
λ
.

Then,
J

λ
(u
λ
)=
0
in
(D
1,p
0
())
−
1
.
Proof The proof is similar to [[23], Theorem 2.3] and is omitted. □
Lemma 3.3
N
0
λ
=
∅
for all l Î (0, Λ
1
).
Proof We argue by contradiction. Suppose that there exists l Î (0, Λ
1
)suchthat
N

0
λ
=
∅
. Then, the fact
u ∈ N
0
λ
and (3.3) imply that
|
|u||
p
μ
=
p ∗−q
p
−
q


g|u|
p∗
,
and
||u||
p
μ
= λ
p
∗

− q
p
∗
−
p


f |u|
q
.
Hsu Boundary Value Problems 2011, 2011:37
/>Page 5 of 15
By (f
1
), (g
1
), the Hölder inequality and Sobolev embedding theorem, we have that
||u||
μ
≥

p − q
(
p
∗
− q
)
|g
+
|

∞

1
p
∗
−p
S
N
p
2
μ
,
and
|
|u||
μ
≤

λ
p
∗
− q
p
∗
− p
|f
+
|
q
∗

S
μ
−
q
p

1
p−q
.
Consequently,
λ ≥

p − q
(p
∗
− q)|g
+
|
∞

p−q
p
∗
−p

p
∗
− p
(p
∗

− q)|f
+
|
q
∗

S
N
p
2
(p−q)+
q
p
μ
= 
1
,
which is a contradiction. □
For each
u ∈ D
1
,p
0
(
)
with


g|u|
p

∗
>
0
, we set
t
max
=

(p − q)||u||
p
μ
(p
∗
− q)


g|u|
p
∗

1
p
∗
−p
> 0
.
Lemma 3.4 Suppose that l Î (0, Λ
1
) and
u

∈ D
1,p
0
(
)
is a function satisfying with


g|u|
p
∗
>
0
.
(i) If


f |u|
q
≤
0
, then there exists a unique t
-
>t
max
such that
t
−
u ∈ N
−

λ
and
J
λ
(t
−
u)=sup
t
≥
0
J
λ
(tu)
.
(ii) If


f |u|
q
≤
0
, then there exists a unique t
±
such that 0<t
+
<t
max
<t
-
,

t
−
u ∈ N
−
λ
and
t
−
u ∈ N
−
λ
. Moreover,
J
λ
(t
+
u)= inf
0≤t≤t
max
J
λ
(tu), J
λ
(t
−
u)=sup
t
≥
t
+

J
λ
(tu)
.
Proof See Brown-Wu [[24], Lemma 2.6]. □
We remark that it follows Lemma 3.3,
N
λ
=
N
+
λ
∪
N
−
λ
for all l Î (0, Λ
1
). Further-
more, by Lemma 3.4, it follows that
N
+
λ
and
N
−
λ
are nonempty, and by Lemma 3.1, we
may define
α

λ
=inf
u∈N
λ
J
λ
(u), α
+
λ
=inf
u∈N
+
λ
J
λ
(u), α
−
λ
=inf
u∈N
−
λ
J
λ
(u)
.
Lemma 3.5 (i) If l Î (0, Λ
1
), then we have
α

λ
≤ α
+
λ
<
0
.
(ii) If
λ ∈ (0,
q
p

1
)
, then
α
−
λ
> d
0
for some positive constant d
0
.
In particular, for each
λ ∈ (0,
q
p

1
)

, we have
α
λ
= α
+
λ
< 0 <α
−
λ
.
Proof (i) Suppose that
u
∈ N
+
λ
. From (3.3), it follows that
p − q
p
∗
−
q
||u||
p
μ
>


g|u|
p
∗

.
(3:4)
Hsu Boundary Value Problems 2011, 2011:37
/>Page 6 of 15
According to (3.1) and (3.4), we have
J
λ
(u)=

1
p
−
1
q

||u||
p
μ
+

1
q
−
1
p
∗



g|u|

p
∗
<

1
p
−
1
q

+

1
q
−
1
p
∗

p − q
p
∗
− q

||u||
p
μ
= −
p − q
q

N
||u||
p
μ
< 0.
By the definitions of a
l
and
α
+
λ
, we get that
α
λ
≤ α
+
λ
<
0
.
(ii) Suppose
λ ∈ (0,
q
p

1
)
and
u
∈ N

−
λ
. Then, (3.3) implies that
p − q
p
∗
−
q
||u||
p
μ
<


|u|
p
∗
.
(3:5)
Moreover, by (g
1
) and the Sobolev embedding theorem, we have


g|u|
p
∗
≤|g
+
|

∞
S
−
p
∗
p
μ
||u||
p
∗
μ
.
(3:6)
From (3.5) and (3.6), it follows that
|
|u||
μ
>

p − q
(
p
∗
− q
)
|g
+
|
∞


1
p
∗
−p
S
N
p
2
μ
for all u ∈ N
−
λ
.
(3:7)
By (3.2) and (3.7), we get
J
λ
(u) ≥||u||
q
μ

1
N
||u||
p−q
μ
− λ
p
∗
− q

p
∗
q
|f
+
|
q
∗
S
−
q
p
μ

>

p − q
(p
∗
− q)|g
+
|
∞

q
p
∗
−p
S
qN

p
2
μ
⎡
⎣
1
N

p − q
(p
∗
− q)|g
+
|
∞

p−q
p
∗
−p
S
N(p−q)
p
2
μ
−λ
p
∗
− q
p

∗
q
|f
+
|
q
∗
S
−
q
p
μ

which implies that
J
λ
(u) > d
0
for all u ∈ N
−
λ
,
for some positive constant d
0
. □
Remark 3.6 If
λ ∈ (0,
q
p


0
)
, then by Lemmas 3.4 and 3.5, for each
u
∈ D
1,p
0
(
)
with


g|u|
p
∗
>
0
, we can easily deduce that
t
−
u ∈ N
−
λ
and J
λ
(t
−
u)=sup
t
≥

0
J
λ
(tu) ≥ α
−
λ
> 0
.
4 Proof of Theorem 1.1
First, we define the Palais-Sma le (simply by (PS)) sequences, (PS)-values and (PS)-con-
ditions in
D
1,p
0
(
)
for J
l
as follows:
Definition 4.1 (i) For c Î ℝ, a sequence {u
n
} is a (PS)
c
-sequence in
D
1
,p
0
(
)

for J
l
if J
l
(u
n
)=c + o(1) and (J
l
)’(u
n
)=o(1) strongly in
(D
1,p
0
())
−
1
as n ® ∞.
Hsu Boundary Value Problems 2011, 2011:37
/>Page 7 of 15
(ii) c Î ℝ is a (PS)-value in
D
1,p
0
(
)
for J
l
if there exists a (PS)
c

-sequence in
D
1,p
0
(
)
for
J
l
.
(iii) J
l
satisfies the (PS)
c
-condition in
D
1,p
0
(
)
if any (PS)
c
-sequence {u
n
} in
D
1,p
0
(
)

for
J
l
contains a convergent subsequence.
Lemma 4.2 (i) If l Î (0, Λ
1
), then J
l
has a
(PS)
α
λ
-sequence
{
u
n
}
⊂ N
λ
.
(ii) If
λ ∈ (0,
q
p

1
)
, then J
l
has a

(PS)
α
λ
-sequence
{u
n
}⊂N
−
λ
.
Proof The proof is similar to [19,25] and the details are omitted. □
Now, we establish the existence of a local minimum for J
l
on
N
λ
.
Theorem 4 .3 Suppose that N ≥ 3,
μ
< ¯
μ
,1<q <p <N and the conditions (f
1
), (g
1
)
hold. If l Î (0, Λ
1
), then there exists
u

λ
∈ N
+
λ
such that
(i)
J
λ
(u
λ
)=α
λ
= α
+
λ
,
(ii) u
l
is a positive solution of (1.1),
(iii) ||u
l
||
μ
® 0 as l ® 0
+
.
Proof By Lemma 4.2 (i), there exists a minimizing sequence
{
u
n

}
⊂ N
λ
such that
J
λ
(u
n
)=α
λ
+ o(1) and J

λ
(u
n
)=o(1) in (D
1
,p
0
())
−1
.
(4:1)
Since J
l
is coercive on
N
λ
(see Lemma 2.1), we get that (u
n

)isboundedin
D
1,p
0
(
)
.
Passing to a subsequence, there exists
u
λ
∈ D
1,p
0
(
)
such that as n ® ∞
⎧
⎪
⎪
⎨
⎪
⎪
⎩
u
n
 u
λ
weakly in D
1
,p

0
(),
u
n
 u
λ
weakly in L
p
∗
(),
u
n
→ u
λ
strongly in L
r
loc
()forall1≤ r < p
∗
,
u
n
→ u
λ
a.e. in .
(4:2)
By (f
1
) and Lemma 2.3, we obtain
λ



f |u
n
|
q
= λ


f |u
λ
|
q
+ o(1) as n →∞
.
(3)
From (4.1)-(4.3), a standard argument shows that u
l
is a critical point of J
l
. Further-
more, the fact
{
u
n
}
⊂ N
λ
implies that
λ



f |u
n
|
q
=
q(p
∗
− p)
p
(
p
∗
− q
)
||u
n
||
p
μ
−
p
∗
q
p
∗
− q
J
λ

(u
n
)
.
(4:4)
Taking n ® ∞ in (4.4), by (4.1), (4.3) and the fact a
l
< 0, we get
λ


f |u
λ
|
q
≥−
p
∗
q
p
∗
−
q
α
λ
> 0
.
(4:5)
Thus,
u

λ
∈ N
λ
is a nontrivial solution of (1.1).
Next, we prove that u
n
® u
l
strongly in
D
1
,p
0
(
)
and J
l
(u
l
)=a
l
. From (4.3), the fact
u
n
, u
λ
∈ N
λ
and the Fatou’s lemma it follows that
α

λ
≤ J
λ
(u
λ
)=
1
N
||u
λ
||
p
μ
− λ
p
∗
− q
p
∗
q


f |u
λ
|
q
≤ lim inf
n→∞

1

N
||u
n
||
p
μ
− λ
p
∗
− q
p
∗
q


f |u
n
|
q

= lim inf
n
→
∞
J
λ
(u
n
)=α
λ

,
Hsu Boundary Value Problems 2011, 2011:37
/>Page 8 of 15
which implies that J
l
(u
l
)=a
l
and
lim
n→∞
||u
n
||
p
μ
= ||u
λ
||
p
μ
. Standard argument shows
that u
n
® u
l
strongly in
D
1,p

0
(
)
.Moreover,
u
λ
∈ N
+
λ
.Otherwise,if
u
λ
∈ N
−
λ
,by
Lemma 3.4, there exist unique
t
+
λ
and
t
−
λ
such that
t
+
λ
u
λ

∈ N
+
λ
,
t
−
λ
u
λ
∈ N
−
λ
and
t
+
λ
< t
−
λ
=
1
. Since
d
dt
J
λ
(t
+
λ
u

λ
)=0 and
d
2
dt
2
J
λ
(t
+
λ
u
λ
) > 0
,
there exists
¯
t ∈ (t
+
λ
, t
−
λ
)
such that
J
λ
(t
+
λ

u
λ
) < J
λ
(
¯
tu
λ
)
. By Lemma 3.4, we get that
J
λ
(t
+
λ
u
λ
) < J
λ
(
¯
tu
λ
) ≤ J
λ
(t
−
λ
u
λ

)=J
λ
(u
λ
)
,
which is a contradiction. If
u ∈ N
+
λ
,then
|u|∈N
+
λ
,andbyJ
l
(u
l
)=J
l
(|u
l
|) = a
l
,we
get
|
u
λ
|∈N

+
λ
is a local minimum of J
l
on
N
λ
. Then, by Lemma 3.2, we may assume
that u
l
is a nontrivial nonnegative solution of (1.1). By Harnack inequality due to Tru-
dinger [26], we obtain that u
l
>0inΩ. Finally, b y (3.3), the Hölder inequality and
Sobolev embedding theorem, we obtain
||u
λ
||
p−q
μ
<λ
p
∗
− q
p
∗
−
p
|f
+

|
q
∗
S
−
q
p
μ
.
which implies that ||u
l
||
μ
® 0asl ® 0
+
. □
Proof of Theorem 1.1 From Theorem 4.3, it follows that the problem (1.1) has a posi-
tive solution
u
λ
∈ N
+
λ
for all l Î (0, Λ
0
). □
5 Proof of Theorem 1.2
For 1 <p <N and
μ
< ¯

μ
, let
c
∗
=
1
N
|g
+
|
−
N−p
p
∞
S
N
p
μ
.
Lemma 5.1 Suppose {u
n
} is a bounded sequence in
D
1,p
0
(
)
. If { u
n
} is a (PS)

c
-sequence
for J
l
with c Î (0, c
*
), then there exists a subsequence of {u
n
} converging weakly to a
nonzero solution of (1.1).
Proof Let
{u
n
}⊂D
1,p
0
()
bea(PS)
c
-sequence for J
l
with c Î (0, c
*
). Since {u
n
}is
bounded in
D
1,p
0

(
)
, passing to a subsequence if necessary, we may assume that as n
® ∞
⎧
⎪
⎪
⎨
⎪
⎪
⎩
u
n
 u
0
weakly in D
1,p
0
(),
u
n
 u
0
weakly in L
p
∗
(),
u
n
→ u

0
strongly in L
r
loc
()for1≤ r < p
∗
,
u
n
→ u
0
a.e. in .
(5:1)
By (f
1
), (g
1
), (5.1) and Lemma 2.3, we have that
J

λ
(u
0
)=
0
and
λ


f |u

n
|
q
= λ


f |u
0
|
q
+ o(1) as n →∞
.
(5:2)
Next,weverifythatu
0
≢ 0. Arguing by contradiction, we assume u
0
≡ 0. Since
J

λ
(u
n
)=o(1)
as n ® ∞ and {u
n
} is bounded in
D
1,p
0

(
)
,thenby(5.2),wecandeduce
that
Hsu Boundary Value Problems 2011, 2011:37
/>Page 9 of 15
0= lim
n→∞
J

λ
(u
n
), u
n
 = lim
n→∞

||u
n
||
p
μ
−


g|u
n
|
p

∗

.
Then, we can set
lim
n→∞
||u
n
||
p
μ
= lim
n→∞


g|u
n
|
p
∗
= l
.
(5:3)
If l =0,thenwegetc =lim
n®∞
J
l
(u
n
) = 0, which is a con tradiction. Thus , we con-

clude that l > 0. Furthermore, the Sobolev embedding theorem implies that
|
|u
n
||
p
μ
≥ S
μ



g|u
n
|
p
∗

p
p
∗
≥ S
μ



g
|g
+
|

∞
|u
n
|
p
∗

p
p
∗
= S
μ
|g
+
|
−
N−p
N
∞



g|u
n
|
p
∗

p
p

∗
.
Then, as n ® ∞ we have
l = lim
n→∞
||u
n
||
p
μ
≥ S
μ
|g
+
|
−
N−p
N
∞
l
p
p
∗
, which implies that
l ≥|g
+
|
−
N−p
p

∞
S
N
p
μ
.
(5:4)
Hence, from (5.2)-(5.4), we get
c = lim
n→∞
J
λ
(u
n
)
=
1
p
lim
n→∞
||u
n
||
p
μ
−
λ
q
lim
n→∞



f |u
n
|
q
−
1
p
∗
lim
n→∞


g|u
n
|
p
∗
=

1
p
−
1
p
∗

l
≥

1
N
|g
+
|
−
N−p
p
∞
S
N
p
μ
.
This is contrary to c <c
*
. Therefore, u
0
is a nontrivial solution of (1.1). □
Lemma 5.2 Suppose
(
H
)
and (f
1
)-(g
2
) hold. If
0 <
μ

< ¯
μ
, x
0
=0and b ≥ pg, then for
any l >0,there exists
v
λ
∈ D
1,p
0
(
)
such that
sup
t
≥
0
J
λ
(tv
λ
) < c
∗
.
(5:5)
In particular,
α
−
λ

< c
∗
for all l Î (0, Λ
1
).
Proof From [[11], Lemma 5.3], we get that i f ε is small enough, there exist t
ε
> 0 and
the positive constants C
i
(i = 1, 2) independent of ε, such that
sup
t
≥
0
J
λ
(
tu
ε
)
= J
λ
(
t
ε
u
ε
)
an

d
0 < C
1
≤ t
ε
≤ C
2
< ∞
.
(5:6)
Hsu Boundary Value Problems 2011, 2011:37
/>Page 10 of 15
By (g
2
), we conclude that







g(x)|u
ε
|
p
∗
−



g(0)|u
ε
|
p
∗





≤


|g(x) − g(0)||u
ε
|
p
∗
= O


B(0;ρ)
|x|
β
|u
ε
|
p
∗


= O
(
ε
β
)
,
which together with Lemma 2.2 implies that


g(x)|u
ε
|
p
∗
= g(0)S
N
p
μ
+ O(ε
p
∗
γ
)+O(ε
β
)
.
(5:7)
From the fact l >0,1<q <p, b ≥ pg and
max
t≥0


t
p
p
B
1
−
t
p
∗
p
∗
B
2

=
1
N
B
N
p
1
B
−
N−p
p
2
, B
1
> 0, B

2
> 0
,
and by Lemma 2.2, (5.7) and (f
2
), we get
J
λ
(t
ε
u
ε
)=
t
p
ε
p
||u
ε
||
p
μ
−
t
p
∗
ε
p
∗



g|u
ε
|
p
∗
− λ
t
q
ε
q


f |u
ε
|
q
≤
1
N
||u
ε
||
N
p
μ



g|u

ε
|
p
∗

−
N−p
p
− λ
C
q
1
q
β
0


|u
ε
|
q
=
1
N

S
N
p
μ
+ O(ε

pγ
)

N
p

g(0)S
N
p
μ
+ O(ε
p
∗
γ
)+O(ε
β
)

−
N−p
p
− λ
C
q
1
q
β
0



|u
ε
|
q
=
1
N
g(0)
−
N−p
p
S
N
p
μ
+ O(ε
pγ
)+O(ε
β
) − λ
C
q
1
q
β
0


|u
ε

|
q
.
(5:8)
By (5.6) and (5.8), we have that
sup
t
≥
0
J
λ
(tu
ε
) ≤ c
∗
+ O(ε
pγ
)+O(ε
β
) − λ
C
q
1
q
β
0


|u
ε

|
q
.
(5:9)
(i) If
1 < q <
N
b
(
μ
)
, then by Lemma 2.2 and
γ = b(μ) − δ = b(μ) −
N−p
p
> 0
we have
that


|u
ε
|
q
= O
1
(ε
qγ
)
.

Combining this with (5.9), for any l > 0, we can choose ε
l
small enough such that
sup
t
≥
0
J
λ
(tu
ε
λ
) < c
∗
.
(ii) If
N
b
(
μ
)
≤ q <
p
, then by Lemma 2.2 and g > 0 we have that


|u
ε
|
q

=

O
1
(ε
θ
), q >
N
b(μ)
,
O
1
(ε
θ
|lnε|), q =
N
b(μ)
,
Hsu Boundary Value Problems 2011, 2011:37
/>Page 11 of 15
and
pγ = b(μ)p + p − N > N +(1−
N
p
)q = θ
.
Combining this with (5.9), for any l > 0, we can choose ε
l
small enough such that
sup

t
≥
0
J
λ
(tu
ε
λ
) < c
∗
.
From (i) and (ii), (5.5) holds by taking
v
λ
= u
ε
λ
.
In fact, by (f
2
), (g
2
) and the definition of
u
ε
λ
, we have that


f |u

ε
λ
|
q
> 0and


g|u
ε
λ
|
p
∗
> 0
.
From Lemma 3.4, the defi nition of
α
−
λ
and (5.5), for any l Î (0, Λ
0
), there exists
t
ε
λ
>
0
such that
t
ε

λ
u
ε
λ
∈ N
−
λ
and
α
−
λ
≤ J
λ
(t
ε
λ
u
ε
λ
) ≤ sup
t
≥
0
J
λ
(tt
ε
λ
u
ε

λ
) < c
∗
.
The proof is thus complete. □
Now, we establish the existence of a local minimum of J
l
on
N
−
λ
.
Theorem 5.3 Suppose
(
H
)
and (f
1
)-(g
2
) hold. If
0 <
μ
< ¯
μ
, x
0
=0,b ≥ pg and
λ ∈ (0,
q

p

1
)
, then there exists
U
λ
∈ N
−
λ
such that
(i)
J
λ
(U
λ
)=α
−
λ
,
(ii) U
l
is a positive solution of (1.1).
Proof If
λ ∈ (0,
q
p

1
)

,thenbyLemmas3.5(ii), 4.2 (ii) and 5.2, there exists a
{u
n
}⊂N
−
λ
-sequence
{u
n
}⊂N
−
λ
in
D
1,p
0
(
)
for J
l
with
α
−
λ
∈ (0, c
∗
)
.SinceJ
l
is coercive

on
N
−
λ
(see Lemma 3.1), we get that {u
n
} is bounded in
D
1
,p
0
(
)
. From Lemma 5.1,
there exists a subsequence still denoted by {u
n
}andanontrivialsolution
U
λ
∈ D
1
,p
0
(
)
of (1.1) such that u
n
⇀ U
l
weakly in

D
1,p
0
(
)
.
First, we prove that
U
λ
∈ N
−
λ
.Onthecontrary,if
U
λ
∈ N
+
λ
,thenby
N
−
λ
∪{0
}
is
closed in
D
1,p
0
(

)
,wehave||U
l
||
μ
< lim inf
n®∞
||u
n
||
μ
.From(g
2
)andU
l
≢ 0inΩ,
we have


g|U
λ
|
p
∗
>
0
. Thus, by Lemma 3.4, there exists a unique t
l
such that
t

λ
U
λ
∈ N
−
λ
.If
u ∈ N
λ
, then it is easy to see that
J
λ
(u)=
1
N
||u||
p
μ
− λ

p
∗
− q
p
∗
q



f |u|

q
.
(5:10)
From Remark 3.6,
u
n
∈ N
−
λ
and (5.10), we can deduce that
α
−
λ
≤ J
λ
(t
λ
U
λ
) < lim
n
→∞
J
λ
(t
λ
u
n
) ≤ lim
n

→∞
J
λ
(u
n
)=α
−
λ
.
This is a contradiction. Thus,
U
λ
∈ N
−
λ
.
Next, by the same argument as that in Theorem 4.3, we get that u
n
® U
l
strongly in
D
1,p
0
(
)
and
J
λ
(U

λ
)=α
−
λ
>
0
for all
λ ∈ (0,
q
p

1
)
.SinceJ
l
(U
l
)=J
l
(|U
l
|) and
|U
λ
|∈N
−
λ
, by Lemma 3.2, we may assume that U
l
is a nontrivial nonnegative solution

Hsu Boundary Value Problems 2011, 2011:37
/>Page 12 of 15
of (1.1). Finally, by Harnack inequality due to Trudinger [26], we obtain that U
l
is a
positive solution of (1.1). □
Proof o f T heorem 1.2 From Theorem 4.3, we get the first positive solution
u
λ
∈ N
+
λ
for all l Î (0, Λ
0
). From Theorem 5.3, we get the second positive solution
U
λ
∈ N
+
λ
for all
λ ∈ (0,
q
p

0
)
. Since
N
−

λ
∩ N
−
λ
=
∅
, this implies that u
l
and U
l
are distinct. □
6 Proof of Theorem 1.3
In this section, we consider the case μ ≤ 0. In this case, it is well-known S
μ
= S
0
where
S
μ
is defined as in (1.2). Thus, we have
c
∗
=
1
N
|g
+
|
−
N−p

p
∞
S
N
p
0
when μ ≤ 0.
Lemma 6.1 Suppose
(
H
)
and (f
1
)-(g
2
) hold. If N ≤ p
2
, μ <0,x
0
≠ 0 and
β
p
≥˜γ :=
N−p
p
(
p−1
)
, then for any l >0and μ <0,there exists
v

λ,μ
∈ D
1,p
0
(
)
such that
sup
t
≥
0
J
λ
(tv
λ,μ
) < c
∗
.
(6:1)
In particular,
α
−
λ
< c
∗
for all l Î (0, Λ
1
).
Proof Note that S
0

has the following explicit extremals [27]:
V
ε
(x)=
¯
Cε
−
N−p
p
⎛
⎝
1+

|x − x
0
|
ε

p
p−1
⎞
⎠
−
N−p
p
, ∀ε>0, x
0
∈ R
N
,

where
¯
C
>
0
is a particular constant. Take r >0smallenoughsuchthatB(x
0
; r) ⊂
Ω\{0} and set
˜
u
ε
(
x
)
= ϕ
(
x
)
V
ε
(
x
)
,where
ϕ(x) ∈ C
∞
0
(B(x
0

; ρ
)
is a cutoff function such
that (x) ≡ 1inB(x
0
; r/2). Arguing as in Lemma 2.2, we have


|∇
˜
u
ε
|
p
= S
N
p
0
+ O(ε
p ˜γ
)
,
(6:2)


|
˜
u
ε
|

p
∗
= S
N
P
0
++O(ε
p
∗
˜γ
)
,
(6:3)


|
˜
u
ε
|
q
=
⎧
⎪
⎨
⎪
⎩
O
1
(ε

θ
),
O
1
(ε
θ
|lnε|),
O
1
(ε
q ˜γ
),
N(p−1)
N−p
< q < p
∗
,
q =
N(p−1)
N−p
,
1 ≤ q <
N(p−1)
N−
p
,
(6:4)
where
θ = N −
N−p

p
q
.Notethat
β ≥
p
˜
γ
,
p
∗
˜γ>
p
˜
γ
.ArguingasinLemma5.2,we
deduce that there exists
˜
t
ε
satisfying
0 < C
1
≤
˜
t
ε
≤ C
2
, such that
J

λ
(t
˜
u
ε
) ≤ sup
t≥0
J
λ
(t
˜
u
ε
)=J
λ
(
˜
t
ε
˜
u
ε
)
=
˜
t
p
ε
p



|∇
˜
u
ε
|
p
−
˜
t
p
∗
ε
p
∗


g|
˜
u
ε
|
p
∗
− λ
˜
t
q
ε
q



f |
˜
u
ε
|
q
− μ
˜
t
p
ε
p


|
˜
u
ε
|
p
|x|
p
≤
1
N
g(x
0
)

−
N−p
p
S
N
p
μ
+ O(ε
p ˜γ
) − λ
C
q
1
q
β
0


|
˜
u
ε
|
q
− μ||x
0
|−ρ|
−p
C
p

2
p


|
˜
u
ε
|
p
.
(6:5)
Hsu Boundary Value Problems 2011, 2011:37
/>Page 13 of 15
From
(
H
)
, N ≤ p
2
and (6.4), we can deduce that
1 < q ˜γ<p ˜γ =
N − p
p
− 1
≤ p ≤
N( p − 1)
N −
p
and



|
˜
u
ε
|
q
= O
1
(ε
q ˜γ
)and


|
˜
u
ε
|
p
=

O
1
(ε
p
|lnε|),
O
1

(ε
p ˜γ
),
p =
N
(
p−1
)
N−p
,
1 < p <
N(p−1)
N−
p
.
Combining this with (6.5), for any l > 0 and μ < 0, we can choose ε
l,μ
small enough
such that
sup
t
≥
0
J
λ
(t
˜
u
ε
λ,μ

) <
1
N
g(x
0
)
−
N−p
p
S
N
p
0
= c
∗
.
Therefore, (6.1) holds by taking
v
λ,μ
=
˜
u
ε
λ,
μ
.
In fact, by (f
2
), (g
2

) and the definition of
˜
u
ε
λ,
μ
, we have that


f |
˜
u
ε
λ,μ
|
q
> 0and


g|
˜
u
ε
λ,μ
|
p
∗
> 0
.
From Lemma 3.4, the definition of

α
−
λ
and (6.1), for any l Î (0, Λ
0
) and μ <0,there
exists
t
ε
λ,
μ
>
0
such that
t
ε
λ,
μ
˜
u
ε
λ,
μ
∈ N
−
λ
and
α
−
λ

≤ J
λ
(t
ε
λ,μ
˜
u
ε
λ,μ
) ≤ sup
t
≥
0
J
λ
(tt
ε
λ,μ
˜
u
ε
λ,μ
) < c
∗
.
The proof is thus complete. □
Proof of Theorem 1.3 Let Λ
1
(0) be defined as in (1.3). ArguingasinTheorems4.3
and5.3,wecangetthefirstpositivesolution

˜
u
λ
∈ N
+
λ
for all l Î (0, Λ
1
(0)) and the
second positive solution
˜
U
λ
∈ N
−
λ
for all
λ ∈ (0,
q
p

1
(0)
)
.Since
N
+
λ
∩ N
−

λ
=
∅
,this
implies that
˜
u
λ
and
˜
U
λ
are distinct. □
Acknowledgements
The author is grateful for the referee’s valuable suggestions.
Received: 13 April 2011 Accepted: 19 October 2011 Published: 19 October 2011
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Cite this article as: Hsu: Multiple positive solutions for a class of quasi-linear elliptic equations involving
concave-convex nonlinearities and Hardy terms. Boundary Value Problems 2011 2011:37.
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