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Hindawi Publishing Corporation
Boundary Value Problems
Volume 2010, Article ID 856932, 18 pages
doi:10.1155/2010/856932
Research Article
Multiple Positive Solutions for a Class of
Concave-Convex Semilinear Elliptic Equations in
Unbounded Domains with Sign-Changing Weights
Tsing-San Hsu
Center for General Education, Chang Gung University, Kwei-Shan, Tao-Yuan 333, Taiwan
Correspondence should be addressed to Tsing-San Hsu,
Received 8 September 2010; Accepted 18 October 2010
Academic Editor: Julio Rossi
Copyright q 2010 Tsing-San Hsu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We study the existence and multiplicity of positive solutions for the following Dirichlet equations:
−Δu u λax|u|
q−2
u bx|u|
p−2
u in Ω, u 0on∂Ω,whereλ>0, 1 <q<2 <p<2
∗
2
∗
2N/N − 2 if N ≥ 3; 2
∗
∞ if N 1, 2, Ω is a smooth unbounded domain in
N
, ax, bx
satisfy suitable conditions, and ax maybe change sign in Ω.
1. Introduction and Main Results
In this paper, we deal with the existence and multiplicity of positive solutions for the
following semilinear elliptic equation:
−Δu u λa
x
|
u
|
q−2
u b
x
|
u
|
p−2
u in Ω,
u>0inΩ,
u 0on∂Ω,
E
λa,b
where λ>0, 1 <q<2 <p<2
∗
2
∗
2N/N −2 if N ≥ 3, 2
∗
∞ if N 1, 2, Ω ⊂
N
is an
unbounded domain, and a, b are measurable functions and satisfy the following conditions:
A1 a ∈ CΩ ∩ L
q
∗
Ω q
∗
p/p −q with a
max{a, 0}
/
≡0inΩ.
B1 b ∈ CΩ ∩ L
∞
Ω and b
max{b, 0}
/
≡0inΩ.
2 Boundary Value Problems
Semilinear elliptic equations with concave-convex nonlinearities in bounded domains
are widely studied. For example, Ambrosetti et al. 1 considered the following equation:
−Δu λu
q−1
u
p−1
in Ω,
u>0inΩ,
u 0on∂Ω,
E
λ
where λ>0, 1 <q<2 <p<2
∗
. They proved that there exists λ
0
> 0suchthatE
λ
admits at
least two positive solutions for all λ ∈ 0,λ
0
and has one positive solution for λ λ
0
and no
positive solution for λ>λ
0
. Actually, Adimurthi et al. 2, Damascelli et al. 3,Ouyangand
Shi 4, and Tang 5 proved tha t there exists λ
0
> 0suchthatE
λ
in the unit ball B
N
0; 1
has exactly two positive solutions for λ ∈ 0,λ
0
and has exactly one positive solution for
λ λ
0
and no positive solution exists for λ>λ
0
. For more general results of E
λ
involving
sign-changing weights in bounded domains, see Ambrosetti et al. 6, Garcia Azorero et al.
7,BrownandWu8, Brown and Zhang 9, Cao and Zhong 10, de Figueiredo et al. 11,
and their references. However, little has been done for this type of problem in unbounded
domains. For Ω
N
, we are only aware of the works 12–15 which studied the existence
of solutions for some related concave-convex elliptic problems not involving sign-changing
weights.
Wu in 16 has studied the multiplicity of positive solutions for the following equation
involving sign-changing weights:
−Δu u f
λ
x
u
q−1
g
μ
x
u
p−1
in
N
,
u>0in
N
,
u ∈ H
1
N
,
E
f
λ
,g
μ
where 1 <q<2 <p<2
∗
, the parameters λ, μ ≥ 0. He also assumed that f
λ
xλf
xf
−
x
is sign-changing and g
μ
xaxμbx,wherea and b satisfy suitable conditions, and
proved E
f
λ
,g
μ
has at least four positive solutions.
When ΩΩ
× Ω
⊂
N−1
, N ≥ 2 is an infinite strip domain, Wu in 17 considered
E
λa,b
not involving sign-changing weights assuming that 0
/
≤ a ∈ L
2/2−q
Ω,0 ≤ b ∈
CΩ satisfies lim
|x
N
|→∞
bx
,x
N
1inΩ and there exist δ>0and0<C
0
< 1suchthat
bx
,x
N
≥ 1 − C
0
e
−2
√
1θ
1
δ|x
N
|
for all x x
,x
N
∈ Ω,whereθ
1
is the first eigenvalue of the
Dirichlet problem −Δ in Ω
. The author proved that there exists a positive constant Λ
0
such
that for λ ∈ 0, Λ
0
, E
λa,b
possesses at least two positive solutions.
Miotto and Miyagaki in 18 have studied E
λa,b
in ΩΩ
× , under the assumption
that a ∈ L
γ/γ−q
Ω q<γ≤ 2
∗
with a
/
≡0anda
−
is bounded and has a compact support
in Ω and 0 ≤ b ∈ L
∞
Ω satisfies lim
|x
N
|→∞
bx
,x
N
1andthereexistsC
0
> 0suchthat
bx
,x
N
≥ 1 − C
0
e
−2
√
1θ
1
|x
N
|
for all x x
,x
N
∈ Ω,whereθ
1
is the first eigenvalue of the
Dirichlet problem −Δ in Ω
. It was obtained there existence of Λ
0
> 0suchthatforλ ∈ 0, Λ
0
,
E
λa,b
possesses at least two positive solutions.
In a recent work 19, Hsu and Lin have studied E
λa,b
in
N
under the assumptions
A1-A2, B1,andΩ
b
. They proved that there exists a constant Λ
0
> 0suchthatfor
Boundary Value Problems 3
λ ∈ 0, q/2Λ
0
, E
λa,b
possesses at least two positive solutions. The main aim of this paper
is to study E
λa,b
on the general unbounded domains see the condition Ω
b
and extend
the results of 19 to more general unbounded domains. We will apply arguments similar
to those used in 20 and prove the existence and multiplicity of positive solutions by using
Ekeland’s variational principle 21.
Set
Λ
0
2 −q
p − q
b
L
∞
2−q/p−2
p − 2
p − q
a
L
q
∗
S
p
Ω
p2−q/2p−2q/2
> 0,
1.1
where b
L
∞
sup
x∈Ω
b
x, a
L
q
∗
Ω
|a
|
q
∗
dx
1/q
∗
,andS
p
Ω is the best Sobolev
constant for the imbedding of H
1
0
Ω into L
p
Ω. Now, we state the first main result about
the existence of positive solution of E
λa,b
.
Theorem 1.1. Assume that (A1) and (B1) hold. If λ ∈ 0, Λ
0
,thenE
λa,b
admits at least one
positive solution.
Associated with E
λa,b
, we consider the energy functional J
λa,b
in H
1
0
Ω:
J
λa,b
u
1
2
u
2
H
1
−
λ
q
Ω
a
x
|
u
|
q
dx −
1
p
Ω
b
x
|
u
|
p
dx,
1.2
where u
H
1
Ω
∇u|
2
u
2
dx
1/2
. By Rabinowitz 22,PropositionB.10, J
λa,b
∈
C
1
H
1
0
Ω, . It is well known that the solutions of E
λa,b
are the critical points of the energy
functional J
λa,b
in H
1
0
Ω.
Under the assumptions A1, B1,andλ>0, E
λa,b
can be regarded as a perturbation
problem of the following semilinear elliptic equation:
−Δu u b
x
u
p−1
in Ω,
u>0inΩ,
u 0on∂Ω,
E
b
where bx ∈ CΩ ∩L
∞
Ω and bx > 0forallx ∈ Ω.WedenotebyS
b
p
Ω the best constant
which is given by
S
b
p
Ω
inf
u∈H
1
0
Ω\
{
0
}
u
2
H
1
Ω
b
x
|
u
|
p
dx
2/p
. 1.3
4 Boundary Value Problems
A typical approach for solving problem of this kind is to use the following Minimax method:
α
b
Γ
Ω
inf
γ∈ΓΩ
max
t∈0,1
J
b
0
γ
t
,
1.4
where
Γ
Ω
γ ∈ C
0, 1
,H
1
0
Ω
: γ
0
0,γ
1
e
, 1.5
J
b
0
e0ande
/
0. By the Mountain Pass Lemma due to Ambrosetti and Rabinowitz 23,
we called the nonzero critical point u ∈ H
1
0
Ω of J
b
0
a ground state solution of E
b
in Ω if
J
b
0
uα
b
Γ
Ω. We remark that the ground state solutions of E
b
in Ω can also be obtained
by the Nehari minimization problem
α
b
0
Ω
inf
v∈M
b
0
Ω
J
b
0
v
,
1.6
where M
b
0
Ω {u ∈ H
1
0
Ω \{0} : u
2
H
1
Ω
bx|u|
p
dx}.NotethatM
b
0
Ω contains every
nonzero solution of E
b
in Ω,
α
b
Γ
Ω
α
b
0
Ω
p − 2
2p
S
b
p
Ω
p/p−2
> 0
1.7
see Willem 24,andifbx ≡ b
∞
> 0 is a constant, then J
b
0
and α
b
0
Ω replace J
0
and α
∞
0
Ω,
respectively.
The existence of ground state solutions of E
b
is affected by the shape of the domain
Ω and bx that satisfies some suitable conditions and has been the focus of a great deal of
research in recent years. By the Rellich compactness theorem and the Minimax method, it is
easy to obtain a ground state solution for E
b
in bounded domains. When Ω is an unbounded
domain and bx ≡ b
∞
, the existence of ground state solutions has been established by several
authors under various conditions. We mention, in particular, results by Berestycki and Lions
25, Lien et al. 26, Chen and Wang 27, and Del Pino and Felmer 28, 29.In25, Ω
N
.
Actually, Kwong 30 proved that the positive solution of E
b
in
N
is unique. In 26,forΩ
is a periodic domain. In 26, 27,thedomainΩ is required to satisfy
Ω
1
ΩΩ
1
∪ Ω
2
,whereΩ
1
, Ω
2
are domains in
N
and Ω
1
∩ Ω
2
is bounded;
Ω
2
α
∞
0
Ω < min{α
∞
0
Ω
1
,α
∞
0
Ω
2
}.
In 28, 29 for 1 ≤ l ≤ N − 1,
N
l
×
N−l
. For a point x ∈
N
,wehavex y, z,
where y ∈
l
and z ∈
N−l
.Lety ∈
l
,wedenotebyΩ
y
⊂
N−l
the projection of Ω onto
N−l
,
that is,
Ω
y
z ∈
N−l
:
y, z
∈ Ω
. 1.8
Boundary Value Problems 5
The domain Ω satisfies the following conditions:
Ω
3
Ωis a smooth subset of
N
and the projections Ω
y
are bounded uniformly in y ∈
l
;
Ω
4
there exists a nonempty closed set D ⊂
N−l
such that D ⊂ Ω
y
for all y ∈
l
;
Ω
5
for each δ>0, there exists R
0
> 0suchthat
Ω
y
⊂
z ∈
N−l
:dist
z, D
<δ
1.9
for all |y|≥R
0
.
When bx
/
≡b
∞
and bx ∈ CΩ ∩ L
∞
Ω, the existence of ground state solutions
of E
b
has been established by the condition bx ≥ b
∞
and the existence of ground state
solutions of limit equation
−Δu u b
∞
u
p−1
in Ω,
u>0inΩ,
u 0on∂Ω.
E
b
∞
In order to get the second positive solution of E
λa,b
, we need some additional
assumptions for ax, bx,andΩ. We assume the following conditions on ax, bx,and
Ω:
Ω
b
bx > 0forallx ∈ Ω and E
b
in Ω has a ground state solution w
0
such that
J
b
0
w
0
α
b
0
Ω.
A2
Ω
ax|w
0
|
q
dx > 0wherew
0
is a positive ground state solution of E
b
in Ω.
Theorem 1.2. Assume that (A1)-(A2), (B1), and (Ω
b
)hold.Ifλ ∈ 0, q/2Λ
0
, E
λa,b
admits at
least two positive solutions.
Throughout this paper, A1 and B1 will be assumed. H
1
0
Ω denotes the standard
Sobolev space, whose norm ·
H
1
is induced by the standard inner product. The dual space of
H
1
0
Ω will be denoted by H
−1
Ω. ·, ·denote the usual scalar product in H
1
0
Ω.Wedenote
the norm in L
s
Ω by ·
L
s
for 1 ≤ s ≤∞. o
n
1 denotes o
n
1 → 0asn →∞. C, C
i
will
denote various positive constants, the exact values of which are not important. This paper is
organized as follows. In Section 2, we give some properties of Nehari manifold. In Sections 3
and 4, we complete proofs of Theorems 1.1 and 1.2.
2. Nehari Manifold
In this section, we will give some properties of Nehari manifold. As the energy functional
J
λa,b
is not bounded below on H
1
0
Ω, it is useful to consider the functional on the Nehari
manifold
M
λa,b
Ω
u ∈ H
1
0
Ω
\
{
0
}
:
J
λa,b
u
,u
0
. 2.1
6 Boundary Value Problems
Thus, u ∈M
λa,b
Ω if and only if
J
λa,b
u
,u
u
2
H
1
− λ
Ω
a
x
|
u
|
q
dx −
Ω
b
x
|
u
|
p
dx 0.
2.2
Note that M
λa,b
Ω contains every nonzero solution of E
λa,b
. Moreover, we have the
following results.
Lemma 2.1. The energy functional J
λa,b
is coercive and bounded below on M
λa,b
Ω.
Proof. If u ∈M
λa,b
Ω,thenbyA1, 2.2,H
¨
older and Sobolev inequalities
J
λa,b
u
p − 2
2p
u
2
H
1
− λ
p − q
pq
Ω
a
x
|
u
|
q
dx 2.3
≥
p − 2
2p
u
2
H
1
− λ
p − q
pq
S
p
Ω
−q/2
a
L
q
∗
u
q
H
1
. 2.4
Thus, J
λa,b
is coercive and bounded below on M
λa,b
Ω.
Define
ψ
λa,b
u
J
λa,b
u
,u
. 2.5
Then for u ∈M
λa,b
Ω,
ψ
λa,b
u
,u
2
u
2
H
1
− λq
Ω
a
x
|
u
|
q
dx −p
Ω
b
x
|
u
|
p
dx
2 −q
u
2
H
1
−
p − q
Ω
b
x
|
u
|
p
dx
2.6
λ
p − q
Ω
a
x
|
u
|
q
dx −
p − 2
u
2
H
1
. 2.7
Similar to the method used in Tarantello 20, we split M
λa,b
Ω into three parts:
M
λa,b
Ω
u ∈M
λa,b
Ω
:
ψ
λa,b
u
,u
> 0
,
M
0
λa,b
Ω
u ∈M
λa,b
Ω
:
ψ
λa,b
u
,u
0
,
M
−
λa,b
Ω
u ∈M
λa,b
Ω
:
ψ
λa,b
u
,u
< 0
.
2.8
Then, we have the following results.
Boundary Value Problems 7
Lemma 2.2. Assume that u
λ
is a local minimizer for J
λa,b
on M
λa,b
Ω and u
λ
/
∈M
0
λa,b
Ω.Then
J
λa,b
u
λ
0 in H
−1
Ω.
Proof. Our proof is almost the same as that in Brown and Zhang 9,Theorem2.3or see
Binding et al. 31.
Lemma 2.3. We have the following.
i If u ∈M
λa,b
Ω ∪M
0
λa,b
Ω,then
Ω
ax|u|
q
dx > 0;
ii If u ∈M
−
λa,b
Ω,then
Ω
bx|u|
p
dx > 0.
Proof. The proof is immediate from 2.6 and 2.7.
Moreover, we have the following result.
Lemma 2.4. If λ ∈ 0, Λ
0
,thenM
0
λa,b
Ω ∅ where Λ
0
isthesameasin1.1.
Proof. Suppose the contrary. Then there exists λ ∈ 0, Λ
0
such that M
0
λa,b
Ω
/
∅.Thenfor
u ∈M
0
λa,b
Ω by 2.6 and Sobolev inequality, we have
2 − q
p − q
u
2
H
1
Ω
b
x
|
u
|
p
dx ≤
b
L
∞
S
p
Ω
−p/2
u
p
H
1
2.9
and so
u
H
1
≥
2 −q
p − q
b
L
∞
1/p−2
S
p
Ω
p/2p−2
.
2.10
Similarly, using 2.7 and H
¨
older and Sobolev inequalities, we have
u
2
H
1
λ
p − q
p − 2
Ω
a
x
|
u
|
q
dx ≤ λ
p − q
p −2
a
L
q
∗
S
p
Ω
−q/2
u
q
H
1
,
2.11
which implies
u
H
1
≤
λ
p − q
p − 2
a
L
q
∗
1/2−q
S
p
Ω
−q/22−q
.
2.12
Hence, we must have
λ ≥
2 −q
p − q
b
L
∞
2−q/p−2
p − 2
p − q
a
L
q
∗
S
p
Ω
p2−q/2p−2q/2
Λ
0
,
2.13
which is a contradiction. This completes the proof.
8 Boundary Value Problems
For each u ∈ H
1
0
Ω with
Ω
bx|u|
p
dx > 0, we write
t
max
u
2 −q
u
2
H
1
p − q
Ω
b
x
|
u
|
p
dx
1/p−2
> 0.
2.14
Then the following lemma holds.
Lemma 2.5. Let λ ∈ 0, Λ
0
.Foreachu ∈ H
1
0
Ω with
Ω
bx|u|
p
dx > 0, we have the following.
i If
Ω
ax|u|
q
dx ≤ 0, then there is a unique t
−
t
−
u >t
max
u such that t
−
u ∈M
−
λa,b
Ω
and
J
λa,b
t
−
u
sup
t≥0
J
λa,b
tu
.
2.15
ii If
Ω
ax|u|
q
dx > 0, then there are unique
0 <t
t
u
<t
max
u
<t
−
t
−
u
2.16
such that t
u ∈M
λa,b
Ω, t
−
u ∈M
−
λa,b
Ω,and
J
λa,b
t
u
inf
0≤t≤t
max
u
J
λa,b
tu
,J
λa,b
t
−
u
sup
t≥0
J
λa,b
tu
.
2.17
Proof. TheproofisalmostthesameasthatinWu32, Lemma 5 andisomittedhere.
3. Proof of Theorem 1.1
First, we remark that it follows Lemma 2.4 that
M
λa,b
Ω
M
λa,b
Ω
∪M
−
λa,b
Ω
3.1
for all λ ∈ 0, Λ
0
. Furthermore, by Lemma 2.5 it follows that M
λa,b
Ω and M
−
λa,b
Ω are
nonempty, and by Lemma 2.1 we may define
α
λa,b
inf
u∈M
λa,b
Ω
J
λa,b
u
; α
λa,b
inf
u∈M
λa,b
Ω
J
λa,b
u
; α
−
λa,b
inf
u∈M
−
λa,b
Ω
J
λa,b
u
.
3.2
Then we get the following result.
Theorem 3.1. We have the following.
i If λ ∈ 0, Λ
0
,thenwehaveα
λa,b
< 0.
ii If λ ∈ 0, q/2Λ
0
,thenα
−
λa,b
>d
0
for some d
0
> 0.
In particular, for each λ ∈ 0, q/2Λ
0
,wehaveα
λa,b
α
λa,b
.
Boundary Value Problems 9
Proof. i Let u ∈M
λa,b
Ω.By2.6,
2 −q
p − q
u
2
H
1
>
Ω
b
x
|
u
|
p
dx
3.3
and so
J
λ
u
1
2
−
1
q
u
2
H
1
1
q
−
1
p
Ω
b
x
|
u
|
p
dx <
1
2
−
1
q
1
q
−
1
p
2 −q
p − q
u
2
H
1
−
p − 2
2 −q
2pq
u
2
H
1
< 0.
3.4
Therefore, α
λa,b
< 0.
ii Let u ∈M
−
λa,b
Ω.By2.6,
2 −q
p − q
u
2
H
1
<
Ω
b
x
|
u
|
p
dx.
3.5
Moreover, by B1 and Sobolev inequality theorem,
Ω
b
x
|
u
|
p
dx ≤
b
L
∞
S
p
Ω
−p/2
u
p
H
1
.
3.6
This implies
u
H
1
>
2 −q
p − q
b
L
∞
1/p−2
S
p
Ω
p/2p−2
∀ u ∈M
−
λa,b
Ω
.
3.7
By 2.4 and 3.7,wehave
J
λa,b
u
≥
u
q
H
1
p − 2
2p
u
2−q
H
1
− λ
p − q
pq
S
p
Ω
−q/2
a
L
q
∗
>
2 −q
p − q
b
L
∞
q/p−2
S
p
Ω
pq/2p−2
×
⎡
⎣
p − 2
2p
S
p
Ω
p2−q/2p−2
2 −q
p − q
b
L
∞
2−q/p−2
− λ
p − q
pq
S
p
Ω
−q/2
a
L
q
∗
⎤
⎦
.
3.8
10 Boundary Value Problems
Thus, if λ ∈ 0, q/2Λ
0
,then
J
λa,b
u
>d
0
∀ u ∈M
−
λa,b
Ω
,
3.9
for some positive constant d
0
. This completes the proof.
We define the Palais-Smale simply by PS sequences, PS-values, and PS-
conditions in H
1
0
Ω for J
λa,b
as follows.
Definition 3.2. (i) For c ∈
, a sequence {u
n
} is a PS
c
-sequence in H
1
0
Ω for J
λa,b
if J
λa,b
u
n
c o
n
1 and J
λa,b
u
n
o
n
1 strongly in H
−1
Ω as n →∞.
(ii) c ∈
is a PS-value in H
1
0
Ω for J
λa,b
if there exists a PS
c
-sequence in H
1
0
Ω for
J
λa,b
.
(iii) J
λa,b
satisfies the PS
c
-condition in H
1
0
Ω if any PS
c
-sequence {u
n
} in H
1
0
Ω for
J
λa,b
contains a convergent subsequence.
Now, we use the Ekeland variational principle 21 to get the following results.
Proposition 3.3. (i) If λ ∈ 0, Λ
0
, then there exists a PS
α
λa,b
-sequence {u
n
}⊂M
λa,b
Ω in H
1
0
Ω
for J
λa,b
.
(ii) If λ ∈ 0, q/2Λ
0
, then there exists a PS
α
−
λa,b
-sequence {u
n
}⊂M
−
λa,b
Ω in H
1
0
Ω for
J
λa,b
.
Proof. TheproofisalmostthesameasthatinWu32,Proposition9.
Now, we establish the existence of a local minimum for J
λa,b
on M
λa,b
Ω.
Theorem 3.4. Assume (A1) and (B1) hold. If λ ∈ 0, Λ
0
,thenJ
λa,b
has a minimizer u
λ
in M
λa,b
Ω
and it satisfies the following.
i J
λa,b
u
λ
α
λa,b
α
λa,b
.
ii u
λ
is a positive solution of E
λa,b
in Ω.
iii u
λ
H
1
→ 0 as λ → 0
.
Proof. By Proposition 3.3i, there is a minimizing sequence {u
n
} for J
λa,b
on M
λa,b
Ω such
that
J
λa,b
u
n
α
λa,b
o
n
1
,
J
λa,b
u
n
o
n
1
in H
−1
Ω
.
3.10
Since J
λ
is coercive on M
λa,b
Ω see Lemma 2.1,wegetthat{u
n
} is bounded in H
1
0
Ω.
Going if necessary to a subsequence, we can assume that there exists u
λ
∈ H
1
0
Ω such that
u
n
u
λ
weakly in H
1
0
Ω
,
u
n
−→ u
λ
almost every where in Ω,
u
n
−→ u
λ
strongly in L
s
loc
Ω
∀ 1 ≤ s<2
∗
.
3.11
By A1,Egorovtheorem,andH
¨
older inequality, we have
λ
Ω
a
x
|
u
n
|
q
dx λ
Ω
a
x
|
u
λ
|
q
dx o
n
1
as n −→ ∞.
3.12
Boundary Va lue Problems 11
First, we claim that u
λ
is a nonzero solution of E
λa,b
.By3.10 and 3.11,itiseasytosee
that u
λ
is a solution of E
λa,b
.Fromu
n
∈M
λa,b
Ω and 2.3,wededucethat
λ
Ω
a
x
|
u
n
|
q
dx
q
p − 2
2
p − q
u
n
2
H
1
−
pq
p − q
J
λa,b
u
n
. 3.13
Let n →∞in 3.13;by3.10, 3.12,andα
λa,b
< 0, we get
λ
Ω
a
x
|
u
λ
|
q
dx ≥−
pq
p − q
α
λa,b
> 0.
3.14
Thus, u
λ
∈M
λa,b
Ω is a nonzero solution of E
λa,b
. Now we prove that u
n
→ u
λ
strongly in
H
1
0
Ω and J
λa,b
u
λ
α
λa,b
.By3.13,ifu ∈M
λa,b
Ω,then
J
λa,b
u
p − 2
2p
u
2
H
1
−
p − q
pq
λ
Ω
a
x
|
u
|
q
dx.
3.15
In order to prove that J
λa,b
u
λ
α
λa,b
,itsuffices to recall that u
n
,u
λ
∈M
λa,b
Ω,by3.15
and by applying Fatou’s lemma to get
α
λa,b
≤ J
λa,b
u
λ
p − 2
2p
u
λ
2
H
1
−
p − q
pq
λ
Ω
a
x
|
u
λ
|
q
dx
≤ lim inf
n →∞
p − 2
2p
u
n
2
H
1
−
p − q
pq
λ
Ω
a
x
|
u
n
|
q
dx
≤ lim inf
n →∞
J
λa,b
u
n
α
λa,b
.
3.16
This implies that J
λa,b
u
λ
α
λa,b
and lim
n →∞
u
n
2
H
1
u
λ
2
H
1
.Letv
n
u
n
− u
λ
;thenby
Br
´
ezis and Lieb, lemma 33 implies that
u
n
2
H
1
u
n
2
H
1
−
u
λ
2
H
1
o
n
1
.
3.17
Therefore, u
n
→ u
λ
strongly in H
1
0
Ω. Moreover, we have u
λ
∈M
λa,b
Ω. On the contrary, if
u
λ
∈M
−
λa,b
Ω, then by Lemma 2.5, there are unique t
0
and t
−
0
such that t
0
u
λ
∈M
λa,b
Ω and
t
−
0
u
λ
∈M
−
λa,b
Ω.Inparticular,wehavet
0
<t
−
0
1. Since
d
dt
J
λa,b
t
0
u
λ
0,
d
2
dt
2
J
λa,b
t
0
u
λ
> 0,
3.18
12 Boundary Value Problems
there exists t
0
< t ≤ t
−
0
such that J
λa,b
t
0
u
λ
<J
λa,b
tu
λ
. By Lemma 2.5,
J
λa,b
t
0
u
λ
<J
λa,b
tu
λ
≤ J
λa,b
t
−
0
u
λ
J
λa,b
u
λ
, 3.19
which is a contradiction. Since J
λa,b
u
λ
J
λa,b
|u
λ
| and |u
λ
|∈M
λa,b
Ω, by Lemma 2.2 we
may assume that u
λ
is a nonzero nonnegative solution of E
λa,b
. By Harnack inequality 34,
we deduce that u
λ
> 0inΩ. Finally, by 2.3 and H
¨
older and Sobolev inequalities,
u
λ
2−q
H
1
<λ
p − q
p − 2
a
L
q
∗
S
p
Ω
−q/2
3.20
and so u
λ
H
1
→ 0asλ → 0
.
Now, we begin the proof of Theorem 1.1.ByTheorem3.4,weobtainthatE
λa,b
has a
positive solution u
λ
in H
1
0
Ω.
4. Proof of Theorem 1.2
In this section, we will establish the existence of the second positive solution of E
λa,b
by
proving that J
λa,b
satisfies the PS
α
−
λa,b
-condition.
Lemma 4.1. Assume that (A1) and (B1) hold. If {u
n
}⊂H
1
0
Ω is a PS
c
-sequence for J
λa,b
,then
{u
n
} is bounded in H
1
0
Ω.
Proof. We argue by contradiction. Assume that u
n
H
1
→∞.Letu
n
u
n
/u
n
H
1
.Wemay
assume that u
n
u weakly in H
1
0
Ω.Thisimpliesthatu
n
→ u strongly in L
s
loc
Ω for all
1 ≤ s<2
∗
.ByA1,Egorovtheorem,andH
¨
older inequality, we have
λ
q
Ω
a
x
|
u
n
|
q
dx
λ
q
Ω
a
x
|
u
|
q
dx o
n
1
.
4.1
Since {u
n
} is a PS
c
-sequence for J
λa,b
and u
n
H
1
→∞, there hold
1
2
u
n
2
H
1
−
λ
u
n
q−2
H
1
q
Ω
a
x
|
u
n
|
q
dx −
u
n
p−2
H
1
p
Ω
b
x
|
u
n
|
p
dx o
n
1
,
4.2
u
n
2
H
1
− λ
u
n
q−2
H
1
Ω
a
x
|
u
n
|
q
dx −
u
n
p−2
H
1
Ω
b
x
|
u
n
|
p
dx o
n
1
.
4.3
From 4.1–4.3, we can deduce that
u
n
2
H
1
2
p − q
q
p − 2
u
n
q−2
λ
Ω
a
x
|
u
|
q
dx o
n
1
. 4.4
Boundary Va lue Problems 13
Since 1 <q<2andu
n
H
1
→∞, 4.4 implies
u
n
2
H
1
−→ 0asn −→ ∞,
4.5
which contradicts with the fact that u
n
H
1
1foralln.
We assume the condition Ω
b
holds and recall
S
b
p
Ω
inf
u∈H
1
0
Ω\
{
0
}
u
2
H
1
Ω
b
x
|
u
|
p
dx
2/p
.
4.6
Lemma 4.2. Assume that (A1), (B1), and (Ω
b
)hold.If{u
n
}⊂H
1
0
Ω is a PS
c
-sequence for J
λa,b
with c ∈ 0,α
b
0
Ω, then there exists a subsequence of {u
n
} converging weakly to a nonzero solution
of E
λa,b
.
Proof. Let {u
n
}⊂H
1
0
Ω be a PS
c
-sequence for J
λa,b
with c ∈ 0,α
b
0
Ω.Weknowfrom
Lemma 4.1 that {u
n
} is bounded in H
1
0
Ω, and then there exist a subsequence of {u
n
} still
denoted by {u
n
} and u
0
∈ H
1
0
Ω such that
u
n
u
0
weakly in H
1
0
Ω
,
u
n
−→ u
0
almost every where in Ω,
u
n
−→ u
0
strongly in L
s
loc
Ω
∀ 1 ≤ s<2
∗
.
4.7
It is easy to see that J
λa,b
u
0
0, and by A1, Egorov theorem, and H
¨
older inequality, we
have
λ
Ω
a
x
|
u
n
|
q
dx λ
Ω
a
x
|
u
0
|
q
dx o
n
1
.
4.8
Next we verify that u
0
/
≡0. Arguing by contradiction, we assume u
0
≡ 0. Setting
l lim
n →∞
Ω
b
x
|
u
n
|
p
dx.
4.9
Since J
λa,b
u
n
o
n
1 and {u
n
} is bounded, then by 4.8, we can deduce that
0 lim
n →∞
J
λa,b
u
n
,u
n
lim
n →∞
u
n
2
H
1
−
Ω
b
x
|
u
n
|
p
dx
lim
n →∞
u
n
2
H
1
− l,
4.10
14 Boundary Value Problems
that is,
lim
n →∞
u
n
2
H
1
l.
4.11
If l 0, then we get c lim
n →∞
J
λa,b
u
n
0, which contradicts with c>0. Thus we
conclude that l>0. Furthermore, by the definition of S
b
p
Ω we obtain
u
n
2
H
1
≥ S
b
p
Ω
Ω
b
x
|
u
n
|
p
dx
2/p
.
4.12
Then as n →∞we have
l lim
n →∞
u
n
2
H
1
≥ S
b
p
Ω
l
2/p
,
4.13
which implies that
l ≥ S
b
p
Ω
p/p−2
.
4.14
Hence, from 1.7 and 4.8–4.14 we get,
c lim
n →∞
J
λa,b
u
n
1
2
lim
n →∞
u
n
2
H
1
−
λ
q
lim
n →∞
Ω
a
x
|
u
n
|
q
dx −
1
p
lim
n →∞
Ω
b
x
|
u
n
|
p
dx
1
2
−
1
p
l ≥
p − 2
2p
S
b
p
Ω
p/p−2
α
b
0
Ω
.
4.15
This is a contradiction to c<α
b
0
Ω. Therefore u
0
is a nonzero solution of E
λa,b
.
Lemma 4.3. Assume that (A1)-(A2), (B1), and (Ω
b
)hold.Letw
0
be a positive ground state solution
of E
b
;then
i sup
t≥0
J
λa,b
tw
0
<α
b
0
Ω for all λ>0;
ii α
−
λa,b
<α
b
0
Ω for all λ ∈ 0, Λ
0
.
Proof. i First, we consider the functional Q : H
1
0
Ω → defined by
Q
u
1
2
u
2
H
1
−
1
p
Ω
b
x
|
u
|
p
dx ∀ u ∈ H
1
0
Ω
.
4.16
Boundary Va lue Problems 15
Then, from 1.3 and 1.7,weconcludethat
sup
t≥0
Q
tw
0
p − 2
2p
⎛
⎝
w
0
2
H
1
Ω
b
x
|
w
0
|
p
dx
2/p
⎞
⎠
p/p−2
p − 2
2p
S
b
p
Ω
p/p−2
α
b
0
Ω
,
4.17
where the following fact has been used:
sup
t≥0
t
2
2
A −
t
p
p
B
p − 2
2p
A
B
2/p
p/p−2
where A, B > 0. 4.18
Using the definitions of J
λa,b
, w
0
and bx > 0forallx ∈ Ω,foranyλ>0, we have
J
λa,b
tw
0
−→ −∞ as t −→ ∞. 4.19
From this we know that there exists t
0
> 0suchthat
sup
t≥0
J
λa,b
tw
0
sup
0≤t≤t
0
J
λa,b
tw
0
.
4.20
By the continuity of J
λa,b
tw
0
as a function of t ≥ 0andJ
λa,b
00, we can find some
t
1
∈ 0,t
0
such that
sup
0≤t≤t
1
J
λa,b
tw
0
<α
b
0
Ω
.
4.21
Thus, we only need to show that
sup
t
1
≤t≤t
0
J
λa,b
tw
0
<α
b
0
Ω
.
4.22
To this end, by A2 and 4.17,wehave
sup
t
1
≤t≤t
0
J
λa,b
tw
0
≤ sup
t≥0
Q
tw
0
−
t
q
1
q
Ω
a
x
|
w
0
|
q
dx < α
b
0
Ω
.
4.23
Hence i holds.
ii By A1, A2, and the definition of w
0
,wehave
Ω
b
x
|
w
0
|
p
dx > 0,
Ω
a
x
|
w
0
|
q
dx > 0.
4.24
16 Boundary Value Problems
Combining this with lemma 2.5ii, from the definition of α
−
λa,b
and part i,forallλ ∈ 0, Λ
0
,
we obtain that there exists t
0
> 0suchthatt
0
w
0
∈M
−
λa,b
Ω and
α
−
λa,b
≤ J
λa,b
t
0
w
0
≤ sup
t≥0
J
λa,b
tw
0
<α
b
0
Ω
.
4.25
Therefore, ii holds.
Now, we establish the existence of a local minimum of J
λ
on M
−
λa,b
Ω.
Theorem 4.4. Assume that (A1)-(A2), (B1), and (Ω
b
)hold.Ifλ ∈ 0, q/2Λ
0
,thenJ
λa,b
has a
minimizer U
λ
in M
−
λa,b
Ω and it satisfies the following.
i J
λa,b
U
λ
α
−
λa,b
.
ii U
λ
is a positive solution of E
λa,b
in Ω.
Proof. If λ ∈ 0, q/2Λ
0
, then by Theorem 3.1ii,Proposition3.3ii, and Lemma 4.3ii,
there exists a PS
α
−
λa,b
-sequence {u
n
}⊂M
−
λa,b
Ω in H
1
0
Ω for J
λa,b
with α
−
λa,b
∈ 0,α
b
0
Ω.
From Lemma 4.2, there exist a subsequence still denoted by {u
n
} and a nonzero solution
U
λ
∈ H
1
0
Ω of E
λa,b
such that u
n
U
λ
weakly in H
1
0
Ω. Now we prove that u
n
→ U
λ
strongly in H
1
0
Ω and J
λa,b
U
λ
α
−
λa,b
.By3.15,ifu ∈M
λa,b
Ω,then
J
λa,b
u
p − 2
2p
u
2
H
1
−
p − q
pq
λ
Ω
a
x
|
u
|
q
dx.
4.26
First, we prove that U
λ
∈M
−
λa,b
Ω. On the contrary, if U
λ
∈M
λa,b
Ω,thenbyM
−
λa,b
Ω being
closed in H
1
0
Ω,wehaveU
λ
2
H
1
< lim inf
n →∞
u
n
2
H
1
. From Lemma 2.3i and bx > 0for
all x ∈ Ω,weget
Ω
a
x
|
U
λ
|
q
dx > 0,
Ω
b
x
|
U
λ
|
p
dx > 0.
4.27
By Lemma 2.5ii, there exists a unique t
−
λ
such that t
−
λ
U
λ
∈M
−
λa,b
Ω.Sinceu
n
∈M
−
λa,b
Ω,
J
λa,b
u
n
≥ J
λa,b
tu
n
for all t ≥ 0andby4.26,wehave
α
−
λa,b
≤ J
λa,b
t
−
λ
U
λ
< lim
n →∞
J
λa,b
t
−
λ
u
n
≤ lim
n →∞
J
λa,b
u
n
α
−
λa,b
4.28
and this is a contradiction. In order to prove that J
λa,b
U
λ
α
−
λa,b
,itsuffices to recall that u
n
,
U
λ
∈M
−
λa,b
for all n,by4.26 and applying Fatou’s lemma to get
α
−
λa,b
≤ J
λa,b
U
λ
p − 2
2p
U
λ
2
H
1
−
p − q
pq
λ
Ω
a
x
|
U
λ
|
q
dx
≤ lim inf
n →∞
p − 2
2p
u
n
2
H
1
−
p − q
pq
λ
Ω
a
x
|
u
n
|
q
dx
≤ lim inf
n →∞
J
λa,b
u
n
α
−
λa,b
.
4.29
Boundary Va lue Problems 17
This implies that J
λa,b
U
λ
α
−
λa,b
and lim
n →∞
u
n
2
H
1
U
λ
2
H
1
.Letv
n
u
n
− U
λ
;thenby
Br
´
ezis and Lieb, lemma 33 implies that
v
n
2
H
1
u
n
2
H
1
−
U
λ
2
H
1
o
n
1
.
4.30
Therefore, u
n
→ U
λ
strongly in H
1
0
Ω.
Since J
λa,b
U
λ
J
λa,b
|U
λ
| and |U
λ
|∈M
−
λa,b
Ω, by Lemma 2.2 we may assume that
U
λ
is a nonzero nonnegative solution of E
λa,b
. Finally, By the Harnack inequality 34 we
deduce that U
λ
> 0inΩ.
Now, we complete the pr oof of Theorem 1.2:byTheorems3.4, 4.4,weobtainthat
E
λa,b
has two positive solutions u
λ
and U
λ
such that u
λ
∈M
λa,b
Ω, U
λ
∈M
−
λa,b
Ω.Since
M
λa,b
Ω ∩M
−
λa,b
Ω ∅, this implies that u
λ
and U
λ
are distinct.
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