Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 579481, 19 pages
doi:10.1155/2012/579481
Research Article
Multiplicity of Positive Solutions for
Weighted Quasilinear Elliptic Equations Involving
Critical Hardy-Sobolev Exponents and
Concave-Convex Nonlinearities
Tsing-San Hsu and Huei-Li Lin
Center for General Education, Chang Gung University, Kwei-Shan, Taiwan
Correspondence should be addressed to Tsing-San Hsu, tshsu@mail.cgu.edu.tw
Received 24 September 2011; Revised 20 December 2011; Accepted 16 January 2012
Academic Editor: Martin D. Schechter
Copyright q 2012 T.-S. Hsu and H.-L. Lin. 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.
By variational methods and some analysis techniques, the multiplicity of positive solutions
is obtained for a class of weighted quasilinear elliptic equations with critical Hardy-Sobolev
exponents and concave-convex nonlinearities.
1. Introduction and Main Results
Let Ω be a smooth bounded domain in RN N ≥ 3 and 0 ∈ Ω. We will study the multiplicity
of positive solutions for the following quasilinear elliptic problem:
− div |x|
−ap
|∇u|
p−2
∇u − μ
|u|p−2 u
∗
|u|p a,b−2 u
bp∗ a,b
|x|pa1
|x|
u 0 on ∂Ω,
λ
|u|q−2 u
|x|dp
∗ a,d
in Ω,
1.1
where λ > 0, 1 < p < N, 0 ≤ μ < μ, μ N − p/p − ap , 0 ≤ a < N − p/p, a ≤ b, d < a 1,
1 ≤ q < p, p∗ a, d Np/N − pa 1 − d is the critical Sobolev-Hardy exponent. Note that
p∗ 0, 0 p∗ Np/N − p is the critical Sobolev exponent.
2
Abstract and Applied Analysis
1,p
In this paper, W Wa Ω denotes the space obtained as the completion of C0∞ Ω
with respect to the norm Ω |x|−ap |∇u|p dx1/p . The energy functional of 1.1 is defined on W
by
1
Jλ u p
−ap
|x|
Ω
|∇u| − μ
|u|p
p
dx −
|x|pa1
1
p∗ a, b
|u|p
Ω
|x|
∗
a,b
dx −
bp∗ a,b
λ
q
|u|q
Ω
∗
|x|dp a,d
dx.
1.2
Then Jλ ∈ C1 W, R. u ∈ W \ {0} is said to be a solution of 1.1 if Jλ u, v
0 for all v ∈ W
and a solution of 1.1 is a critical point of Jλ . By the standard elliptic regularity argument,
we deduce that u ∈ C1 Ω \ {0}.
Problem 1.1 is related to the following Hardy inequality 1:
RN
|u|p
|x|
∗
p/p∗ a,b
a,b
bp∗ a,b
≤C
dx
RN
|x|−ap |∇u|p dx,
∀u ∈ C0∞ RN ,
1.3
which is also called the general or weighted Hardy-Sobolev inequality. For the sharp
constants and extremal functions, see 2, 3. If b a 1, then p∗ a, b p and the following
general or weighted Hardy inequality holds 1, 4:
|u|p
RN
|u|
dx ≤
pa1
1
μ
RN
|x|−ap |∇u|p dx,
∀u ∈ C0∞ RN ,
1.4
where μ N − p/p − ap is the best Hardy constant.
In the space W, we employ the following norm if μ < μ:
u uW Ω
|x|
−ap
|u|p
p
|∇u| − μ
|x|pa1
1/p
dx
.
1.5
By 1.4 it is equivalent to the usual norm Ω |x|−ap |∇u|p dx1/p of the space W. According to
1.4, we can define the following best constant for 0 ≤ a < N − p/p, a ≤ b < a 1 and μ < μ:
Sμ,a,b Ω inf u∈W\{0} p
Ω |u|
up
∗ a,b
/|x|bp
∗ a,b
dx
p/p∗ a,b .
1.6
From Kang 5, Lemma 2.2, Sμ,a,b is independent of Ω ⊂ RN . Thus, we will simply denote
that Sμ,a,b Ω Sμ,a,b RN Sμ,a,b .
Abstract and Applied Analysis
3
When a 0, we set s dp∗ 0, d and t bp∗ 0, b, then 1.1 is equivalent to the
following quasilinear elliptic equations:
∗
|u|p−2 u |u|p t−2 u
|u|q−2 u
λ
− div |∇u|p−2 ∇u − μ
|x|p
|x|s
|x|t
u 0 on ∂Ω,
in Ω,
1.7
where λ > 0, 1 < p < N, 0 ≤ μ < μ N − p/pp , 0 ≤ s, t < p, 1 ≤ q < p and p∗ t pN − t/N − p.
Such kind of problem relative with 1.7 has been extensively studied by many
authors. When p 2, people have paid much attention to the existence of solutions for
singular elliptic problems see 6–16 and their references therein, besides, in the most of
these papers, the operator −Δ − μ/|x|2 with Sobolev-Hardy critical exponents the case that
t 0 has been considered. Some authors also studied the singular problems with SobolevHardy critical exponents the case that t /
0 see 17–22 and their references therein. In
23, 24, the authors deal with doubly-critical exponents.
When p /
2. The quasilinear problems related to Hardy inequality and Sobolev-Hardy
inequality have been studied by some authors 25–32. Here we recall the work in 25, where
the extremal functions for the best Sobolev constant Sμ,0,0 were studied. The results can be
employed to study the problems with critical Sobolev exponents and Hard terms, see 25,
28. In 26 it is investigated in RN a quasilinear elliptic equation involving doubly critical
exponents by the concentration compactness principle 33, 34.
We should note that the nonlinearities of problems studied in 11–14, 26, 28, 31 are
all not sublinear or p-sublinear near the origin. To the best of our knowledge, there are few
results of problem 1.7 with nonlinearities being p-sublinear near the origin for 1 < p < N.
We are only aware of the works 20, 30, 32 which studied the existence and multiplicity of
positive solution of problem 1.7 with 1 ≤ q < p < N. In this paper, we study 1.1 and
extend the results of 20, 30, 32 to the case a /
0 and 1 ≤ q < p < N.
For 0 ≤ a < N − p/p, a ≤ b < a 1, and 0 ≤ μ < μ, consider the following limiting
problem:
− div |x|
−ap
|∇u|
p−2
∇u − μ
|u|p−2 u
|x|
pa1
|u|p
∗
a,b−2
u
bp∗ a,b
|x|
u > 0 in R \ {0},
1,p
u ∈ Wa RN ,
N
in RN \ {0},
1.8
1,p
where Wa RN is the space obtained as the completion of C0∞ RN with respect to the
norm RN |x|−ap |∇u|p dx1/p . From 5, Lemma 2.2, we know 1.8 has a unique ground state
solution Up,μ satisfying
Up,μ 1 1/p∗a,b−p
p∗a, b μ − μ
p
1.9
4
Abstract and Applied Analysis
ε x ε−N−p/p−a Up,μ x/ε for some ε > 0, that
and all ground states must be of the form U
is,
|∇u|p − μ|u|p /|x|pa1 dx
inf
p/p∗ a,b
1,p
∗
p∗ a,b
u∈Wa RN \{0}
/|x|bp a,b dx
Ω |u|
Sμ,a,b
RN
1.10
ε . Moreover, Up,μ is radially symmetric and possesses the following properis achieved by U
ties:
lim r αμ Up,μ r c1 > 0,
r →0
lim r βμ Up,μ r c2 > 0,
r → ∞
lim r αμ1 Up,μ
r c1 α μ ≥ 0,
r →0
lim r βμ1 Up,μ
r c2 β μ > 0,
1.11
r → ∞
where ci i 1, 2 are positive constants and αμ, βμ are the zeros of the function
fτ p − 1 τ p − N − pa 1 τ p−1 μ,
τ ≥ 0, 0 ≤ μ < μ,
1.12
which satisfy 0 ≤ αμ < N − pa 1/p < βμ < N − pa 1/p − 1. Furthermore, there
exist the positive constants c3 c3 μ, p, a, b and c4 c4 μ, p, a, b such that
δ
c3 ≤ Up,μ x |x|αμ/δ |x|βμ/δ ≤ c4 ,
δ
N − pa 1
.
p
1.13
Throughout this paper, let R0 be the positive constant such that Ω ⊂ B0; R0 , where
B0; R0 {x ∈ RN : |x| < R0 }. By Hölder and Sobolev-Hardy inequalities, for all u ∈ W, we
obtain
Ω
|u|q
|x|dp
∗ a,d
≤
B0;R0 ≤
⎛
−dp∗ a,d
p∗ a,d−q/p∗ a,d |x|
Ω
r
0
−dp∗ a,dN−1
dr
∗
|x|dp
p∗ a,d−q/p∗ a,d
R0
NωN
|u|p
a,d
q/p∗ a,d
∗ a,d
Sμ,a,d
−q/p
⎞p∗ a,d−q/p∗ a,d
N−dp∗ a,d
NωN R0
−q/p
⎠
≤⎝
Sμ,a,d
uq .
∗
N − dp a, d
uq
1.14
Abstract and Applied Analysis
5
Set
Λ0 p−q
∗
p a, b − q
p−q/p∗ a,b−p ⎛
⎞−p∗ a,d−q/p∗ a,d
N−dp∗ a,d
p∗ a, b − p ⎝ NωN R0
⎠
p∗ a, b − q
N − dp∗ a, d
q/p p∗ a,bp−q/pp∗ a,b−p
Sμ,a,b
× Sμ,a,d
,
1.15
where ωN 2π N/2 /NΓN/2 is the volume of the unit ball in RN .
Furthermore, from 0 ≤ a < N − p/p and a ≤ d < a 1, we can deduce that
0<p
N−p
− a pd N − pa 1 − d < N,
p
1.16
which implies
pN
> p,
N − pa 1 − d
N−p
Np
N − dp∗ a, d − a > 0.
N − pa 1 − d
p
p∗ a, d 1.17
Combining these with 1 ≤ q < p, we get Λ0 > 0.
We are now ready to state our main results.
Theorem 1.1. Assume that N ≥ 3, 0 ≤ μ < μ, 0 ≤ a < N − p/p, a ≤ b, d < a 1, and
1 ≤ q < p < N. Then one has the following results.
i If λ ∈ 0, Λ0 , then 1.1 has at least one positive solution in W.
ii If λ ∈ 0, q/pΛ0 , then 1.1 has at least two positive solutions in W.
Remark 1.2. In 5, Kang considered 1.1 with p-sublinear perturbation of p ≤ q < p∗ a, d.
Via variational methods, he proved the existence of positive solutions of 1.1 when the
parameters a, b, d, p, q, λ, μ satisfy suitable conditions. But the existence of positive solutions
for 1.1 involving the p-sublinear of 1 ≤ q < p < N is not considered. In this paper, we will
give a complement result.
This paper is organized as follows. In Sections 2 and 3, we give some preliminaries
and some properties of Nehari manifold. In Section 4, we complete proofs of Theorem 1.1. At
the end of this section, we explain some notations employed. In the following discussions, we
will denote various positive constants as C, Ci and omit dx in the integral for convenience. We
denote B0; R as a ball centered at the origin with radius R, and ωN 2π N/2 /NΓN/2 is the
volume of the unit ball B0; 1 in RN . We denote the norm in Lr Ω by | · |r for 1 ≤ r ≤ ∞, and
Lr Ω, |x|−s , 1 ≤ r < ∞ is the closure of C0∞ Ω with the norm |·|Lr Ω,|x|−s Ω |x|−s |·|r 1/r . W −1
denoting the dual space of W. Oεt denotes |Oεt /εt | ≤ C, and oεt means |oεt /εt | → 0
as ε → 0. By o1 we always mean it is a generic infinitesimal value.
6
Abstract and Applied Analysis
2. Nehari Manifold
Since the functional Jλ is not bounded from below on W, we will work on Nehari manifold.
For λ > 0 we define
Nλ u ∈ W \ {0} : Jλ u, u 0 .
2.1
We recall that any nonzero solutions of 1.1 belong to Nλ . Moreover, by definition, we have
that u ∈ Nλ if and only if
u / 0,
p
u −
1
p∗ a, b
|u|p
Ω
∗
a,b
bp∗ a,b
|x|
−
λ
q
|u|q
Ω
∗
|x|dp a,d
0.
2.2
The following result is concerned with the behavior of Jλ on Nλ .
Lemma 2.1. Jλ is coercive and bounded from below on Nλ .
Proof. If u ∈ Nλ , then by 1.14 and 2.2, we get
∗
p a, b − q
p∗ a, b − p
|u|q
p
Jλ u ∗
u − λ
dp∗ a,d
p a, bp
p∗ a, bq
Ω |x|
⎛
⎞p∗ a,d−q/p∗ a,d
N−dp∗ a,d
∗
p a, b − q ⎝ NωN R0
p∗ a, b − p
p
⎠
≥ ∗
u − λ
p a, bp
p∗ a, bq
N − dp∗ a, d
−q/p
× Sμ,a,d
uq .
2.3
2.4
Since 0 ≤ a < N − p/p, a ≤ b, d < a 1 and 1 ≤ q < p < p∗ a, b, it follows that Jλ is coercive
and bounded from below on Nλ .
Define ψλ : W → R, by ψλ u Jλ u, u
, that is,
ψλ u up −
|u|p
Ω
∗
|x|bp
a,b
∗ a,b
−λ
|u|q
Ω
|x|dp
∗ a,d
2.5
.
Note that ψλ is of class C1 with
ψλ u, u pup − p∗ a, b
|u|p
Ω
∗
|x|bp
a,b
∗ a,b
− qλ
|u|q
Ω
|x|dp
∗ a,d
.
2.6
Abstract and Applied Analysis
7
Furthermore, if u ∈ Nλ , then by 2.2, we have that
ψλ u, u p − q up − p∗ a, b − q
|u|p
∗
a,b
∗
|x|bp a,b
|u|q
p
∗
∗
.
p − p a, b u − q − p a, b λ
dp∗ a,d
Ω |x|
Ω
2.7
2.8
Now we split Nλ into three sets:
Nλ u ∈ Nλ
N0λ u ∈ Nλ
N−λ u ∈ Nλ
: ψλ u, u > 0 ,
: ψλ u, u 0 ,
: ψλ u, u < 0 .
2.9
The following result shows that minimizers on Nλ are the “usual” critical points for
Jλ .
Lemma 2.2. Suppose u0 is a local minimizer of Jλ on Nλ and u0 ∈
/ N0λ . Then, Jλ u0 0 in W −1 .
Proof. See 30, Lemma 2.2.
Motivated by the above result, we will get conditions for N0λ ∅.
Lemma 2.3. N0λ ∅ for all λ ∈ 0, Λ0 .
Proof. We argue by contradiction. Suppose that there exists a λ ∈ 0, Λ0 such that N0λ /
∅. Let
u ∈ N0λ be arbitrary, then by 2.2, 2.7, and 2.8, we have that
p∗ a, b − q
0 < u p−q
|u|p
p
Ω
p∗ a, b − q
0 < u λ ∗
p a, b − p
|x|
a,b
bp∗ a,b
,
|u|q
p
Ω
∗
|x|dp
∗ a,d
2.10
.
By 1.14, 2.10, and Sobolev-Hardy inequality, we get
u ≥
p−q
p∗ a, b − q
1/p∗ a,b−p
Sμ,a,b
p∗ a,b/pp∗ a,b−p
,
⎛
⎞p∗ a,d−q/p∗ a,dp−q
N−dp∗ a,d
∗
−q/pp−q
p a, b − q 1/p−q ⎝ NωN R0
⎠
.
Sμ,a,d
u ≤ λ ∗
∗
p a, b − p
N − dp a, d
2.11
8
Abstract and Applied Analysis
Hence we must have
λ≥
p−q
∗
p a, b − q
p−q/p∗ a,b−p ⎛
⎞−p∗ a,d−q/p∗ a,d
N−dp∗ a,d
p∗ a, b − p ⎝ NωN R0
⎠
p∗ a, b − q
N − dp∗ a, d
q/p p∗ a,bp−q/pp∗ a,b−p
× Sμ,a,d
Sμ,a,b
Λ0 ,
2.12
which is a contradiction.
For each u ∈ W \ {0}, let
τmax 1/p∗ a,b−p
p − q up
.
∗
∗
∗
p a, b − q Ω |u|p a,b /|x|bp a,b
2.13
Lemma 2.4. If λ ∈ 0, Λ0 , then for each u ∈ W \ {0}, the set {τu : τ > 0} intersects Nλ exactly
twice. More specifically, there exist a unique τ − τ − u > 0 such that τ − u ∈ N−λ and a unique
τ τ u > 0 such that τ u ∈ Nλ . Moreover, τ < τmax < τ − and
Jλ τ u inf Jλ τu,
0≤τ≤τmax
Jλ τ − u sup Jλ τu.
2.14
τ≥τmax
Proof. The proof is similar to that of 29, Lemma 2.7 and is omitted.
We remark that by Lemma 2.3 we have, Nλ Nλ ∪ N−λ for all λ ∈ 0, Λ0 . Furthermore,
by Lemma 2.4 it follows that Nλ and N−λ are non-empty and by Lemma 2.1 we may define
αλ inf Jλ u,
u∈Nλ
Theorem 2.5.
αλ inf Jλ u,
α−λ inf− Jλ u.
u∈Nλ
u∈Nλ
2.15
i If λ ∈ 0, Λ0 , then one has αλ ≤ αλ < 0.
ii If λ ∈ 0, q/pΛ0 , then α−λ > d0 for some positive constant d0 .
In particular, for each λ ∈ 0, q/pΛ0 , one has αλ αλ < 0 < α−λ .
Proof. i Let u ∈ Nλ . By 2.7,
p−q
up >
∗
p a, b − q
|u|p
Ω
∗
a,b
∗
|x|bp a,b
,
2.16
Abstract and Applied Analysis
9
and so also using 2.2,
∗
1
1
1 1
|u|p a,b
−
− ∗
up bp∗ a,b
p q
q p a, b
Ω |x|
p−q
1
1
1 1
−
− ∗
<
up
p q
q p a, b
p∗ a, b − q
p − q p∗ a, b − p
−
up < 0.
pqp∗ a, b
Jλ u 2.17
Therefore, from the definition of αλ and αλ , we can deduce that αλ ≤ αλ < 0.
ii Let u ∈ N−λ . By 2.7,
p−q
up <
∗
p a, b − q
|u|p
Ω
∗
a,b
∗
|x|bp a,b
2.18
.
Moreover, by Sobolev-Hardy inequality,
|u|p
Ω
|x|
∗
a,b
bp∗ a,b
−p∗ a,b/p
∗
≤ Sμ,a,b
up a,b .
2.19
This implies
u >
p−q
∗
p a, b − q
1/p∗ a,b−p
Sμ,a,b
p∗ a,b/pp∗ a,b−p
∀u ∈ N−λ .
2.20
By 2.4 and 2.20, we have
⎡
∗
∗
p a, b − q
⎢ p a, b − p
Jλ u ≥ uq ⎣ ∗
up−q − λ
p a, bp
p∗ a, bq
⎤
⎞p∗ a,d−q/p∗ a,d
N−dp∗ a,d
NωN R0
−q/p ⎥
⎠
Sμ,a,d
×⎝
⎦
N − dp∗ a, d
⎛
>
q/p∗ a,b−p
qp∗ a,b/pp∗ a,b−p
p−q
Sμ,a,b
∗
p a, b − q
⎡
p−q/p∗ a,b−p
∗
p∗ a,bp−q/pp∗ a,b−p
p−q
⎢ p a, b − p
×⎣ ∗
Sμ,a,b
p a, bp
p∗ a, b − q
⎤
⎞p∗ a,d−q/p∗ a,d
⎛
N−dp∗ a,d
−q/p ⎥
p a, b − q ⎝ NωN R0
⎠
−λ
Sμ,a,d
⎦
p∗ a, bq
N − dp∗ a, d
∗
10
Abstract and Applied Analysis
q/p∗ a,b−p
qp∗ a,b/pp∗ a,b−p p∗ a, b − q
q
p−q
Λ0 − λ
Sμ,a,b
p
p∗ a, b − q
p∗ a, bq
⎞p∗ a,d−q/p∗ a,d
⎛
N−dp∗ a,d
NωN R0
−q/p
⎠
.
×⎝
Sμ,a,d
N − dp∗ a, d
2.21
Thus, if λ ∈ 0, q/pΛ0 , then
Jλ u > d0
∀u ∈ N−λ ,
2.22
for some positive constant d0 .
Remark 2.6. If λ ∈ 0, q/pΛ0 , then by Lemma 2.4 and Theorem 2.5, for each u ∈ W \ {0}, we
can easily deduce that
τ − u ∈ N−λ ,
Jλ τ − u supJλ τu ≥ α−λ > 0.
2.23
τ≥0
3. Proof of the Main Results
First, we define the Palais-Smale simply by PS sequences, PS-values, and PSconditions in W for Jλ as follows.
Definition 3.1. i For c ∈ R, a sequence {un } is a PSc -sequence in W for Jλ if Jλ un c o1
and Jλ un o1 strongly in W −1 as n → ∞.
ii c ∈ R is a PS-value in W for Jλ if there exists a PSc -sequence in W for Jλ .
iii Jλ satisfies the PSc -condition in W if any PSc -sequence {un } in W for Jλ
contains a convergent subsequence.
Now, we use the Ekeland variational principle 35 to get the following results.
Proposition 3.2. i If λ ∈ 0, Λ0 , then Jλ has a PSαλ -sequence {un } ⊂ Nλ .
ii If λ ∈ 0, q/pΛ0 , then Jλ has a PSα− -sequence {un } ⊂ N−λ .
λ
Proof. The proof is similar to 29, Proposition 3.3 and the details are omitted.
Now, we establish the existence of a local minimum for Jλ on Nλ .
Theorem 3.3. Assume that N ≥ 3, 0 ≤ μ < μ, 0 ≤ a < N − p/p, a ≤ b, d < a 1, and
1 ≤ q < p < N. If λ ∈ 0, Λ0 , then there exists uλ ∈ Nλ such that
i Jλ uλ αλ αλ ,
ii uλ is a positive solution of 1.1,
iii uλ → 0 as λ → 0 .
Abstract and Applied Analysis
11
Proof. By Proposition 3.2i, there exists a minimizing sequence {un } ⊂ Nλ such that
Jλ un o1
Jλ un αλ o1,
in W −1 .
3.1
Since Jλ is coercive on Nλ see Lemma 2.1, we get that {un } is bounded in W. From
∗
5, Lemma 2.1, we deduce that the embedding W → Lr Ω, |x|−dp a,d is compact for
∗
1 ≤ r < p a, d. Thus, there exists uλ ∈ W, passing to a subsequence if necessary, using
similar arguments found in 27, 36, then one can get that as n → ∞
un uλ weakly in W,
∗
un −→ uλ strongly in Lq Ω, |x|−dp a,d
for 1 ≤ q < p,
un −→ uλ a.e. in Ω,
∇un −→ ∇uλ a.e. in Ω,
un
uλ
a1 weakly in Lp Ω,
a1
|x|
|x|
∗
p∗ a,b−2
un
|un |
|uλ |p a,b−2 uλ
v −→
v,
∗
∗
|x|bp a,b
|x|bp a,b
Ω
Ω
3.2
∀v ∈ W.
Consequently, passing to the limit in Jλ un , v
, by 3.1 and 3.2, as n → ∞, we have
Ω
|uλ |p−2 uλ v
|∇uλ |p−2 ∇uλ ∇v
−
μ
|x|ap
|x|pa1
−
|uλ |p
Ω
∗
a,b−2
|x|
uλ v
bp∗ a,b
−λ
|uλ |q−2 uλ v
Ω
|x|dp
∗ a,d
0,
3.3
for all v ∈ W. That is, Jλ uλ , v
0. Thus uλ is a weak solution of 1.1. Furthermore, from
un ∈ Nλ and 2.3, we deduce that
λ
|un |q
Ω
|x|dp
∗ a,d
q p∗ a, b − p
p∗ a, bq
Jλ un .
∗
un p − ∗
p a, b − q
p p a, b − q
3.4
Let n → ∞ in 3.4, by 3.1 and 3.2, and since αλ < 0 by i of Theorem 2.5, we get
λ
|uλ |q
Ω
|x|dp
∗ a,d
≥−
p∗ a, bq
αλ > 0.
p∗ a, b − q
Thus uλ /
≡ 0, and since Jλ uλ 0, it follows that uλ ∈ Nλ and, in particular, Jλ uλ ≥ αλ .
3.5
12
Abstract and Applied Analysis
Next, we will show, up to a subsequence, that un → uλ strongly in W and Jλ uλ αλ .
From the fact un , u ∈ Nλ , 2.3 and the Fatou’s lemma, it follows that
∗
p a, b − q
p∗ a, b − p
|uλ |q
p
−
λ
u
λ
dp∗ a,d
pp∗ a, b
p∗ a, bq
Ω |x|
∗
p∗ a, b − p
p a, b − q
|un |q
p
≤ lim inf
un − λ
dp∗ a,d
n→∞
p∗ a, bp
p∗ a, bq
Ω |x|
αλ ≤ Jλ uλ 3.6
lim infJλ un αλ ,
n→∞
which implies that Jλ uλ αλ and limn → ∞ un p uλ p . Standard argument shows that
un → uλ strongly in W. Moreover, uλ ∈ Nλ . Otherwise, if uλ ∈ N−λ , by Lemma 2.4, there exist
unique τλ and τλ− such that τλ uλ ∈ Nλ , τλ− uλ ∈ N−λ and τλ < τλ− 1. Since
d Jλ τλ uλ 0,
dτ
d2 Jλ τλ uλ > 0,
dτ 2
3.7
there exists τ ∈ τλ , τλ− such that Jλ τλ uλ < Jλ τuλ . By Lemma 2.4 we get that
Jλ τλ uλ < Jλ τuλ ≤ Jλ τλ− uλ Jλ uλ ,
3.8
which is a contradiction. Since Jλ uλ Jλ |uλ | and |uλ | ∈ Nλ , by Lemma 2.2, we may assume
that uλ is a nontrivial nonnegative solution of 1.1. By 5, Lemma 2.3, it follows that uλ > 0
in Ω. Finally, by 1.14 and 2.8, we obtain
⎛
⎞p∗ a,d−q/p∗ a,d
N−dp∗ a,d
−q/p
p∗ a, b − q ⎝ NωN R0
⎠
<λ ∗
,
Sμ,a,d
p a, b − p
N − dp∗ a, d
uλ p−q
3.9
which implies that uλ → 0 as λ → 0 .
Next, we will establish the existence of the second positive solution of 1.1 by proving
that Jλ satisfies the PSα− -condition.
λ
Lemma 3.4. Let {un } be a bounded sequence in W. If {un } is a PSc -sequence for Jλ with
c∈
0,
p∗ a,b/p∗ a,b−p
p∗ a, b − p ,
S
μ,a,b
p∗ a, bp
3.10
then there exists a subsequence of {un } converging weakly to a nonzero solution of 1.1.
Proof. Let {un } ⊂ W be a PSc -sequence for Jλ with c ∈ 0, p∗ a, b −
∗
∗
p/p∗ a, bp Sμ,a,b p a,b/p a,b−p . Since {un } is bounded in W and the embedding
Abstract and Applied Analysis
13
∗
W → Lr Ω, |x|−dp a,d is compact for 1 ≤ r < p∗ a, d see 5, Lemma 2.1, thus passing
to a subsequence if necessary, we may assume that as n → ∞
un u0 weakly in W,
∗
Ω, |x|−bp a,b
∗
un −→ u0 strongly in Lq Ω, |x|−dp a,d
for 1 ≤ q < p,
un u0 weakly in Lp
∗
a,b
3.11
un −→ u0 a.e. in Ω.
Using the same argument in Theorem 3.3, we deduce that Jλ u0 0 and
λ
|un |q
Ω
|x|dp
∗ a,d
λ
|u0 |q
Ω
o1.
∗
|x|dp a,d
3.12
Next we verify that u0 /
≡ 0. Arguing by contradiction, we assume u0 ≡ 0. Set
l lim
n→∞
|un |p
∗
a,b
bp∗ a,b
|x|
Ω
3.13
.
Since Jλ u0 0 and {un } is bounded in W, then by 3.12, we can deduce that
0
lim J un , un
n→∞ λ
!
|un |p
p
lim
n→∞
un −
|x|bp
Ω
∗
a,b
lim un p − l,
∗ a,b
n→∞
3.14
that is,
lim un p l.
3.15
n→∞
If l 0, then by 3.12–3.15, we get
c lim Jλ un lim
n→∞
n→∞
1
un p −
p
|un |p
Ω
∗
a,b
∗
|x|bp a,b
−λ
|un |q
Ω
|x|dp
∗ a,d
0,
3.16
which contradicts c > 0. Thus we conclude that l > 0. Furthermore, the Sobolev-Hardy
inequality implies that
|un |p
p
un ≥ Sμ,a,b
Ω
|x|bp
∗
a,b
p/p∗ a,b
∗ a,b
.
3.17
∗
Then as n → ∞, we have l limn → ∞ un p ≥ Sμ,a,b lp/p a,b , which implies that
p∗ a,b/p∗ a,b−p
l ≥ Sμ,a,b
.
3.18
14
Abstract and Applied Analysis
Hence, from 3.12–3.18 we get
c lim Jλ un n→∞
∗
1
1
|un |p a,b λ
|un |q
lim un p − ∗
lim
lim
−
pn→∞
p a, b n → ∞ Ω |x|bp∗ a,b q n → ∞ Ω |x|dp∗ a,d
1
1
−
l
p p∗ a, b
p∗ a,b/p∗ a,b−p
p∗ a, b − p ≥ ∗
.
Sμ,a,b
p a, bp
3.19
This contradicts the definition of c. Therefore, u0 is a nontrivial solution of 1.1.
Lemma 3.5. If N ≥ 3, 0 ≤ μ < μ, 0 ≤ a < N − p/p, a ≤ b, d < a 1, and 1 ≤ q < p < N, then
for any λ > 0, there exists vλ ∈ W such that
supJλ τvλ <
τ≥0
p∗ a,b/p∗ a,b−p
p∗ a, b − p .
Sμ,a,b
p∗ a, bp
In particular, α−λ < p∗ a, b − p/p∗ a, bpSμ,a,b p
∗
a,b/p∗ a,b−p
3.20
for all λ ∈ 0, Λ0 .
Proof. Let Up,μ be a ground state solution of 1.8, ρ > 0 small enough such that B0; ρ ⊂
ε x Ω, η ∈ C0∞ Ω, 0 ≤ ηx ≤ 1, ηx 1 for |x| < ρ/2, ηx 0 for |x| ≥ ρ. Set U
−N−p/p−a
Up,μ x/ε and uε x ηxUε x, ε > 0. Then, following the same lines as in 5,
ε
we get the following estimates as ε → 0:
p∗ a,b/p∗ a,b−p
O εβμppa1−N ,
uε p Sμ,a,b
|uε |p
Ω
|x|
∗
a,b
bp∗ a,b
|uε |q
Ω
|x|dp
∗ a,d
p∗ a,b/p∗ a,b−p
∗
Sμ,a,b
O εβμbp a,b−N ,
⎧
∗
⎪
⎪
CεN−dp a,d−qδ ,
⎪
⎪
⎪
⎪
⎪
⎨
≥ Cεqβμ−δ |ln ε|,
⎪
⎪
⎪
⎪
⎪
⎪
qβμ−δ
⎪
,
⎩Cε
N − dp∗ a, d
< q < p∗ a, d,
β μ
N − dp∗ a, d
,
q
β μ
∗
N − dp a, d
,
1≤q<
β μ
3.21
3.22
3.23
where bμ is given in the introduction satisfying δ N − pa 1/p < bμ < N − pa 1/p − 1.
Abstract and Applied Analysis
15
Now we consider the following functions:
∗
τ p a,b
τp
uε p − ∗
p
p a, b
gτ Jλ τuε |uε |p
∗
a,b
∗
−λ
τq
q
|x|bp a,b
∗
∗
τp
τ p a,b
|uε |p a,b
gτ .
uε p − ∗
p
p a, b Ω |x|bp∗ a,b
Ω
|uε |q
Ω
,
∗
|x|dp a,d
3.24
Using the definitions of g and uε , we get
gτ Jλ τuε ≤
τp
uε p ,
p
∀τ ≥ 0, λ > 0.
3.25
Combining this with 3.21, let ε ∈ 0, 1, then there exists τ0 ∈ 0, 1 independent of ε such
that
sup gτ <
0≤τ≤τ0
p∗ a,b/p∗ a,b−p
p∗ a, b − p ,
Sμ,a,b
∗
p a, bp
∀λ > 0, ∀ε ∈ 0, 1.
3.26
On the other hand, by the fact for B1 , B2 > 0
max
τ≥0
∗
τp
τ p a,b
B1 − ∗
B2
p
p a, b
p∗ a, b − p
∗
∗
∗
B1 p a,b/p a,b−p B2 −p/p a,b−p ,
∗
p a, bp
3.27
and by 3.21 and 3.22, we can get that
−p/p∗ a,b−p
∗
p∗ a, b − p
|uε |p a,b
p∗ a,b/p∗ a,b−p
max gτ ∗
uε bp∗ a,b
τ≥0
p a, bp
Ω |x|
p∗ a,b/p∗ a,b−p
p∗ a,b/p∗ a,b−p
p∗ a, b − p ∗
O εβμppa1−N
Sμ,a,b
p a, bp
−p/p∗ a,b−p
p∗ a,b/p∗ a,b−p
∗
× Sμ,a,b
O εβμbp a,b−N
p∗ a,b/p∗ a,b−p
p∗ a, b − p ∗
O εβμppa1−N .
Sμ,a,b
p a, bp
3.28
Hence as λ > 0, 1 ≤ q < p, by 3.28 we have that
τq
supgτ sup gτ − λ
q
τ≥τ0
τ≥τ0
|uε |q
Ω
|x|dp
∗ a,d
τ
p∗ a,b/p∗ a,b−p
p∗ a, b − p ≤ ∗
O εβμppa1−N − λ 0
Sμ,a,b
p a, bp
q
q
Ω
|uε |q
|x|dp
∗ a,d
.
3.29
16
Abstract and Applied Analysis
i If 1 ≤ q < N − dp∗ a, d/βμ, then by 3.23 we have that
|uε |q
Ω
|x|dp
∗ a,d
≥ Cεqβμ−δ
and since bμ > δ N − pa 1/p, then
β μ p pa 1 − N p β μ − δ > q β μ − δ .
3.30
3.31
Combining this with 3.26 and 3.29, for any λ > 0, we can choose ελ small enough such
that
sup Jλ τuελ <
τ≥0
p∗ a,b/p∗ a,b−p
p∗ a, b − p .
Sμ,a,b
∗
p a, bp
3.32
ii If N − dp∗ a, d/βμ ≤ q < p, then by 3.23 and bμ > δ N − pa 1/p we
have that
⎧
N−dp∗ a,d−qδ
⎪
⎪
,
⎨Cε
N − dp∗ a, d
< q < p∗ a, d,
|uε |
β μ
≥
dp∗ a,d
N − dp∗ a, d
⎪
⎪
Ω |x|
⎩Cεqβμ−δ |ln ε|, q ,
β μ
N − dp∗ a, d − qδ ≤ qβ μ − qδ < p β μ − δ β μ p pa 1 − N.
q
3.33
Combining this with 3.26 and 3.29, for any λ > 0, we can choose ελ small enough such
that
sup Jλ τuελ <
τ≥0
p∗ a,b/p∗ a,b−p
p∗ a, b − p .
Sμ,a,b
∗
p a, bp
3.34
From i and ii, 3.20 holds by taking vλ uελ .
From Lemma 2.4, the definition of α−λ , and 3.20, for any λ ∈ 0, Λ0 , we obtain that
there exists τλ− > 0 such that τλ− vλ ∈ N−λ and
p∗ a,b/p∗ a,b−p
p∗ a, b − p .
α−λ ≤ Jλ τλ− vλ ≤ supJλ τvλ < ∗
Sμ,a,b
p
bp
a,
t≥0
3.35
The proof is thus complete.
Now, we establish the existence of a local minimum of Jλ on N−λ .
Theorem 3.6. Assume that N ≥ 3, 0 ≤ μ < μ, 0 ≤ a < N − p/p, a ≤ b, d < a 1 and
1 ≤ q < p < N. If λ ∈ 0, q/pΛ0 , then there exists Uλ ∈ N−λ such that
i Jλ Uλ α−λ ,
ii Uλ is a positive solution of 1.1.
Abstract and Applied Analysis
17
Proof. If λ ∈ 0, q/pΛ0 , then by Theorem 2.5 ii, Proposition 3.2 ii, and Lemma 3.5,
there exists a PSα− -sequence {un } ⊂ N−λ in W for Jλ with α−λ ∈ 0, p∗ a, b −
∗
λ
∗
p/p∗ a, bpSμ,a,b p a,b/p a,b−p . Since Jλ is coercive on Nλ see Lemma 2.1, we get that
{un } is bounded in W. From Lemma 3.4, there exists a subsequence still denoted by {un } and
a nontrivial solution Uλ ∈ W of 1.1 such that un Uλ weakly in W.
First, we prove that Uλ ∈ N−λ . Arguing by contradiction, we assume Uλ ∈ Nλ . Since
N−λ is closed in W, we have Uλ < lim infn → ∞ un . Thus, by Lemma 2.4, there exists a
unique τλ− such that τλ− Uλ ∈ N−λ . If u ∈ Nλ , then it is easy to see that
∗
p a, b − q
p∗ a, b − p
|u|q
p
Jλ u ∗
.
u − λ
∗
dp∗ a,d
p a, bp
p a, bq
Ω |x|
3.36
From Remark 2.6, un ∈ N−λ , Uλ < lim infn → ∞ un , and 3.36, we can deduce that
α−λ ≤ Jλ τλ− Uλ < lim Jλ τλ− un ≤ lim Jλ un α−λ .
n→∞
n→∞
3.37
This is a contradiction. Thus, Uλ ∈ N−λ .
Next, by the same argument as that in Theorem 3.3, we get that un → Uλ strongly in
W and Jλ Uλ α−λ > 0 for all λ ∈ 0, q/pΛ0 . Since Jλ Uλ Jλ |Uλ | and |Uλ | ∈ N−λ , by
Lemma 2.2, we may assume that Uλ is a nontrivial nonnegative solution of 1.1. Finally, by
5, Lemma 2.3, we obtain that Uλ is a positive solution of 1.1.
Now, we complete the proof of Theorem 1.1. The part i of Theorem 1.1 immediately
follows from Theorem 3.3. When 0 < λ < q/pΛ0 < Λ0 , by Theorems 3.3 and 3.6, we obtain
1.1 has two positive solutions uλ and Uλ such that uλ ∈ Nλ , Uλ ∈ N−λ . Since Nλ ∩ N−λ ∅,
this implies that uλ and Uλ are distinct. This completes the proof of Theorem 1.1.
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