Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 806397, 15 pages
doi:10.1155/2012/806397
Research Article
Existence of Multiple Solutions for a Singular
Elliptic Problem with Critical Sobolev Exponent
Zonghu Xiu
Science and Information College, Qingdao Agricultural University, Qingdao 266109, China
Correspondence should be addressed to Zonghu Xiu, qingda@163.com
Received 11 October 2012; Revised 1 November 2012; Accepted 5 November 2012
Academic Editor: Jifeng Chu
Copyright q 2012 Zonghu Xiu. 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 consider the existence of multiple solutions of the singular elliptic problem −div|x|−ap
∗
∗
|∇u|p−2 ∇u |u|p−2 u/|x|a1p f|u|r−2 u h|u|s−2 u |x|−bp |u|p −2 u, ux → 0 as |x| → ∞, where
x ∈ RN , 1 < p < N, a < N − p/p, a ≤ b ≤ a 1, r, s > 1, p∗ Np/N − pd, d a 1 − b. By
the variational method and the theory of genus, we prove that the above-mentioned problem has
infinitely many solutions when some conditions are satisfied.
1. Introduction and Main Results
In this paper, we consider the existence of multiple solutions for the singular elliptic problem
|u|p−2 u
∗
∗
− div |x|−ap |∇u|p−2 ∇u a1p f|u|r−2 u h|u|s−2 u |x|−bp |u|p −2 u,
|x|
ux −→ 0
x ∈ RN ,
1.1
as |x| −→ ∞,
where 1 < p < N, a < N − p/p, a ≤ b ≤ a 1, r > 1, p∗ Np/N − pd, d a 1 − b. fx
and hx are nonnegative functions in RN .
In recent years, the existence of multiple solutions on elliptic equations has been
considered by many authors. In 1, Assunção et al. considered the following quasilinear
degenerate elliptic equation:
− div |x|−ap |∇u|p−2 ∇u λ|x|−a1p |u|p−2 u |x|−bq |u|q−2 u f,
1.2
2
Abstract and Applied Analysis
where x ∈ RN , 1 < p < N, q Np/N −pa1−b. When λ 0, f εg, where 0 < ε ≤ ε0 and
q
0 ≤ g ∈ Lb RN ∗ ; the authors proved that problem 1.2 has at least two positive solutions.
Rodrigues in 2 studied the following critical problem on bounded domain Ω ∈ RN :
∗
∗
− div |x|−ap |∇u|p−2 ∇u |x|−bp |u|p −2 u |x|−β f|u|r−2 u,
ux 0,
x ∈ Ω,
1.3
on ∂Ω.
By the variational method on Nehari manifolds 3, 4, the author proved the existence of at
least two positive solutions and the nonexistence of solutions when some certain conditions
are satisfied. When p 2 and a −1, Miotto and Miyagaki in 5 considered the semilinear
Dirichlet problem in infinite strip domains
−Δu u λfx|u|q−1 hx|u|p−1 ,
ux 0,
x ∈ Ω,
on ∂Ω.
1.4
The authors also proved that problem 1.4 has at least two positive solutions by the methods
of Nehari manifold. For other references, we refer to 6–11 and the reference therein. In fact,
motivated by 1, 2, 5, we consider the problem 1.1. Since our problem is singular and is
studied in the whole space RN , the loss of compactness of the Sobolev embedding renders a
variational technique that is more delicate. By the variational method and the theory of genus,
we prove that problem 1.1 has infinitely many solutions when some suitable conditions are
satisfied.
In order to state our result, we introduce some weighted Sobolev spaces. For r, s ≥ 1
and g gx > 0 in RN , we define the spaces Lr RN , g and Ls RN , g as being the set of
Lebesgue measurable functions u : RN → R1 , which satisfy
ur,g
uLr RN ,g RN
gx|u|r dx
1/r
us,g uLs RN ,g s
RN
1/s
gx|u| dx
< ∞,
1.5
< ∞.
Particularly, when gx ≡ 1, we have
ur uLr RN RN
|u|r dx
1/r
< ∞.
1.6
1,p
We denote the completion of C0∞ RN by X Wa RN with the norm of
uX RN
|x|−ap |u|p dx
1/p
,
1.7
where 1 < p < N and a < N − p/p. It is easy to find that X is a reflexive and separable
Banach space with the norm uX .
Abstract and Applied Analysis
3
The following Hardy-Sobolev inequality is due to Caffarelli et al. 12, which is called
Caffarelli-Kohn-Nirenberg inequality. There exist constants S1 , S2 > 0 such that
RN
|x|
−bp∗
RN
|x|
p∗
|u| dx
−a1p
p/p∗
≤ S1
RN
p
|u| dx ≤ S2
RN
∀u ∈ C0∞ RN ,
|x|−ap |∇u|p dx,
1.8
∀u ∈ C0∞ RN ,
|x|−ap |∇u|p dx,
1.9
where p∗ Np/N − pd is called the Sobolev critical exponent.
In the present paper, we make the following assumptions:
RN \ {0} for 1 < r < p, where g1 |x|a1rσ1 , σ1 p/p −
A1 fx ∈ Lσ1 RN , g1 L∞
loc
r;
RN \{0} for p < r < p∗ , where g2 |x|brσ2 , σ2 p∗ /p∗ −r.
A2 fx ∈ Lσ2 RN , g2 L∞
loc
∗
RN \ {0} for p < s < p∗ , where g3 |x|μbp , μ p∗ /p∗ − s.
A3 hx ∈ Lμ RN , g3 L∞
loc
Then, we give some basic definitions.
Definition 1.1. u ∈ X is said to be a weak solution of 1.1 if for any ϕ ∈ C0∞ RN there holds
|x|
RN
−ap
|∇u|
p−2
∇u · ∇ϕ |u|p−2 uϕ
|x|
a1p
r−2
dx RN
f|u|
RN
|x|
uϕ dx −bp∗
|u|
p∗ −2
RN
h|u|s−2 uϕ dx
1.10
uϕ dx.
Let Iu : X → R1 be the energy functional corresponding to problem 1.1, which is
defined as
1
Iu p
RN
−ap
|x|
|u|p
p
|∇u| |x|a1p
1
dx −
r
1
f|u| dx −
s
RN
r
1
h|u| dx − ∗
p
RN
|u|p
s
RN
∗
dx,
∗
|x|bp
1.11
for all u ∈ X. Then the functional I ∈ C1 X, R1 and for all ϕ ∈ X, there holds
I u, ϕ RN
|x|
−ap
|∇u|
p−2
∇u∇ϕ −
RN
hx|u|s−2 uϕ dx −
|u|p−2 uϕ
|x|a1p
∗
RN
|x|−bp |u|p
dx −
∗
−2
RN
fx|u|r−2 uϕdx
1.12
uϕ dx.
It is well known that the weak solutions of problem 1.1 are the critical points of the
functional Iu, see 13. Thus, to prove the existence of weak solutions of 1.1, it is sufficient
to show that Iu admits a sequence of critical points in X.
Our main result in this paper is the following.
4
Abstract and Applied Analysis
Theorem 1.2. Let 1 < p < N, a < N − p/p, a ≤ b ≤ a 1, r > 1, p∗ Np/N − pd, d a 1 − b, max{r, p} < s < p∗ . Assume A1 –A3 are fulfilled. Then problem 1.1 has infinitely
many solutions in X.
2. Preliminary Results
Our proof is based on variational method. One important aspect of applying this method is
to show that the functional Iu satisfies P Sc condition which is introduced in the following
definition.
Definition 2.1. Let c ∈ R1 and X be a Banach space. The functional Iu ∈ C1 X, R satisfies
the P Sc condition if for any {un } ⊂ X such that
I un −→ 0 in X ∗ as n −→ ∞
Iun −→ c,
2.1
contains a convergent subsequence in X.
The following embedding theorem is an extension of the classical Rellich-Kondrachov
compactness theorem, see 14.
Lemma 2.2. Suppose Ω ⊂ RN is an open bounded domain with C1 boundary and 0 ∈ Ω. N ≥
1,p
3, a < N − p/p. Then the embedding W0 Ω, |x|−ap → Lr Ω, |x|−α is continuous if 1 ≤ r ≤
Np/N − p and 0 ≤ α ≤ 1 ar N1 − r/p, and is compact if 1 ≤ r < Np/N − p and
0 ≤ α < 1 ar N1 − r/p.
Now we prove an embedding theorem, which is important in our paper.
Lemma 2.3. Assume A1 -A2 and 1 < r < p∗ . Then the embedding X → Lr RN , f is compact.
Proof. We split our proof into two cases.
i Consider 1 < r < p.
By the Hölder inequality and 1.9 we have that
urLr
RN ,f RN
fx|u|r dx ≤
RN
|u|p |x|−a1p dx
RN
|u|p |x|−a1p dx
r/p
f σ N
L 1 R ,g1 r/p RN
f σ1 |x|a1rσ1 dx
1/σ1
2.2
r/p
≤ S2 urX f Lσ1 RN ,g1 ,
where g1 |x|a1rσ1 , σ1 p/p − r. Then the embedding is continuous. Next, we will prove
that the embedding is compact.
Let BR be a ball center at origin with the radius R > 0. For the convenience, we denote
Lr RN , f by Z, that is, Z Lr RN , f. Assume {un } is a bounded sequence in X. Then {un }
is bounded in XBR . We choose α 0 in Lemma 2.2, then there exist u ∈ ZBR and a
Abstract and Applied Analysis
5
subsequence, still denoted by {un }, such that un − uLr BR → 0 as n → ∞. We want to prove
that
sup
lim
uZBRc R → ∞ u∈X\{0}
uX
0,
2.3
where BRc RN \ BR . In fact, we obtain from 2.2 that
r/p
urZBc ≤ S2 urX f Lσ1 Bc ,g1 .
R
R
2.4
The fact f ∈ Lσ1 RN , g1 shows that
lim
R→∞
f σ1 g1 dx 0.
2.5
1/p 1/r
≤ S2 f Lσ1 Bc ,g1 ,
2.6
BRc
Then 2.4 and 2.5 imply that
uZBRc uX
R
which gives 2.3.
In the following, we will prove that un → u strongly in ZRN .
Since X is a reflexive Banach space and {un } is bounded in X. Then we may assume,
up to a subsequence, that
un u
in X.
2.7
In view of 2.3, we get that for any ε > 0 there exists Rε > 0 large enough such that
un ZBRc ≤ εun X
ε
n 1, 2, . . ..
2.8
On the other hand, due to the compact embedding XBRε → ZRε in Lemma 2.2, we have
that
lim un − uZBRε 0.
2.9
un − uZBRε < ε,
2.10
n→∞
Therefore, there is N0 > 0 such that
6
Abstract and Applied Analysis
for n > N0 . Thus, the inequalities 2.8 and 2.10 show that
un − uZ ≤ un − uZBRε un − uZBRc
ε
≤ un − uZBRε un ZBc uZBc
R
R
ε
ε
2.11
≤ 1 un X uX ε.
This shows that {un } is convergent in Z Lr RN , f.
ii Consider p ≤ r < p∗ .
It follows from 1.8 and the Hölder inequality that
urLr RN ,f
−bp∗
r
RN
r/p
≤ S1
fx|u| dx ≤
RN
RN
|x|−ap |∇u|p dx
|x|
p∗
|u| dx
r/p f |x|
σ2
RN
RN
r/p
≤ S1 urX f Lσ2 RN ,g2 ,
r/p∗ f σ2 |x|brσ2 dx
1/σ2
brσ2
1/σ2
dx
2.12
where g2 |x|brσ2 , σ2 p∗ /p∗ − r. Thus, the fact of f ∈ Lσ2 RN , g2 and 2.12 imply that the
embedding is continuous. Similar to the proof of i we can also prove that the embedding
X → Lr RN , f is compact for p ≤ r < p∗ .
Similarly, we have the following result of compact embedding.
Lemma 2.4. Assume 1 < p < s < p∗ and A3 , then the embedding X → Ls RN , h is compact.
The following concentration compactness principle is a weighted version of the
Concentration Compactness Principle II due to Lions 15–18, see also 19, 20.
Lemma 2.5. Let 1 < p < N, −∞ < a < N −p/p, a ≤ b ≤ a1, p∗ Np/N −pd, d a1−b.
1,p
Suppose that {un } ⊂ Wa RN is a sequence such that
1,p
un u
in Wa
|x|−ap |∇un |p μ
|x|
−bp∗
p∗
|un |
η
un −→ u
RN ,
in M RN ,
in M R
N
2.13
,
a.e. on RN ,
where μ, η are measures supported on Ω and MRN is the space of bounded measures in RN . Then
there are the following results.
Abstract and Applied Analysis
7
1 There exists some at most countable set J, a family {xj ∈ Ω | j ∈ J} of distinct points in
RN , and a family {ηj | j ∈ J} of positive numbers such that
∗
∗
η |x|−bp |u|p ηj δxj ,
2.14
j∈J
where δxj is the Dirac measure at xj .
2 The following equality holds
μ j δx j ,
μ ≥ |x|−ap |∇u|p 2.15
j∈J
for some family {μj > 0 | j ∈ J} satisfying
p/p∗
≤ μj
S 1 ηj
p/p∗
ηj
≤ ∞.
∀j ∈ J,
j∈J
2.16
3 There hold
lim sup
n → ∞
lim sup
n → ∞
−ap
Ω
Ω
|x|
|x|
−bp∗
p
|∇un | dx Ω
dμ μ∞ ,
2.17
p∗
|∇un | dx Ω
dη η∞ ,
where
μ∞ : lim lim sup
R → ∞ n → ∞
η∞ : lim lim sup
R → ∞ n → ∞
Ω
Ω
BRc
|x|−ap |∇un |p dx,
2.18
−bp∗
BRc
|x|
p∗
|∇un | dx.
Lemma 2.6. Let 1 < p < r < s < p∗ . Then Iu satisfies the P Sc condition with c ≤ 1/r −
p∗ /p∗ −p
, where S1 is as in 1.8.
1/p∗ S1
Proof. We will split the proof into three steps.
Step 1. {un } is bounded in X.
Let {un } be a P Sc sequence of Iu in X, that is,
Iun −→ c,
I un −→ 0
in X ∗ as n −→ ∞.
2.19
8
Abstract and Applied Analysis
Then, we have
1
I un , un
r
1
1 1
1
1 1
p
p∗
s
−
−
− ∗ un Lp∗ RN un X un Ls RN ,h p r
r s
r p
1 1
p
−
≥
un X .
p r
1 c un X ≥ Iun −
2.20
Since p > 1, 2.20 shows that {un } is bounded in X.
∗
Step 2. There exists {un } in X such that un → u in Lp RN .
∗
∗
The inequality 1.8 shows that {un } is bounded in Lp RN , |x|−bp . Then the
above argument and the compactness embedding in Lemma 2.2 mean that the following
convergence hold:
un u
un u
1,p
in W0
RN ,
∗
∗
in Lp RN , |x|−bp ,
un −→ u
2.21
a.e. in RN .
It follows from Lemma 2.5 that there exist nonnegative measures μ and η such that
∗
∗
∗
∗
|x|−bp |un |p η |x|−bp |u|p ηj δxj ,
j∈J
|x|−ap |∇un |p ≥ |x|−ap |∇u|p μ j δx j .
2.22
2.23
j∈J
∗
Thus, in order to prove un → u in Lp RN it is sufficient to prove that ηj η∞ 0.
For the proof of ηj 0, we define the functional ψ ∈ C0∞ RN such that
ψ ≡ 1,
in B xj , ε ,
ψ ≡ 0,
c
in B xj , 2ε ,
2
∇ψ ≤ ,
ε
2.24
where xj belongs to the support of dη. It follows from 2.1 that
lim I un , un ψ 0.
n→∞
2.25
Abstract and Applied Analysis
9
Since un X is bounded, we can get from 1.8-1.9, Lemmas 2.3 and 2.5 that
lim
n→∞
−ap
RN
|x|
|∇un |
p−2
∇un ∇ψun dx lim
n→∞
|x|
RN
−bp∗
−
|un | ψ dx −
−ap
RN
p∗
|x|
−→
|x|−a1p |un |p ψ dx
p
|∇un | ψ dx RN
RN
fx|un |r ψ dx
s
RN
hx|un | ψ dx
RN
ψdη −
RN
ψdμ ηj − μj
as ε −→ 0.
2.26
On the other hand,
lim
n→∞
RN
|x|−ap |∇un |p−2 ∇un ∇ψun dx
≤ lim
n→∞
RN
|x|−ap |∇un |p dx
N
∇ψ dx
≤c
p−1/p RN
1/N 1/p
N−p/Np
−aNp/N−p
|x|
B2ε
p
|x|−ap |un |p ∇ψ dx
Np/N−p
|u|
2.27
dx
B2ε
N−p/Np
−aNp/N−p
≤c
|x|
Np/N−p
|u|
−→ 0
dx
ε −→ 0,
B2ε
where B2ε Bxj , 2ε. Then μj ηj ; furthermore, 2.16 implies that μj ηj 0 or ηj >
p∗ /p∗ −p
S1
. We will prove that the later does not hold. Suppose otherwise, there exists some
p∗ /p∗ −p
j0 ∈ J such that ηj0 > S1
. Then 2.19 and Lemma 2.4 show that
1
I un , un
r
∗
∗
1
1
1
p
−
− ∗
|x|−bp |un |p dx
un X r
r p
Ω
1
1
1
p∗ /p∗ −p
− ∗ ηj 0 >
− ∗ S1
,
p
r p
c o1 Iun −
≥
1
p
1
r
2.28
which contradicts the hypothesis of c. Then μj ηj 0.
Similarly, we define the functional ψ1 ∈ C0∞ RN as
ψ1 ≡ 0,
|x| < R,
ψ1 ≡ 1,
|x| > 2R,
∇ψ1 ≤ 2 .
R
2.29
10
Abstract and Applied Analysis
Then, the similar proof as above shows that η∞ μ∞ 0. Thus, we can deduce from 2.22
that
∗
RN
∗
|x|−bp |un |p dx −→
∗
RN
∗
∗
|x|−bp |u|p dx
as n −→ ∞,
2.30
∗
which implies that un → u in Lp RN , |x|−bp .
Step 3. {un } converges strongly in X.
The following inequalities 21 play an important role in our proof:
⎧ ⎨ c |ξ|p−2 ξ − |ζ|p−2 ζ, ξ − ζ
for p ≥ 2,
p/2 |ξ − ζ|p ≤
2−p/2
⎩ c |ξ|p−2 ξ − |ζ|p−2 ζ, ξ − ζ
for 1 < p < 2.
|ξ|p |ζ|p
2.31
Our aim is to prove that {un } is a Cauchy sequence of X. In fact, let ψ un − um in 1.12, it
follows from 2.19 that
|x|−a1p |un |p−2 un − |um |p−2 um un − um dx
Amn RN
I un − I um , un − um
fx |un |r−2 un − |um |r−2 um un − um dx
RN
RN
RN
2.32
hx |un |s−2 un − |um |s−2 um un − um dx
∗
∗
∗
|x|−bp |un |p −2 un − |um |p −2 um un − um dx,
where
Amn RN
|x|−ap |∇un |p−2 ∇un − |∇um |p−2 ∇um · ∇un − um dx.
2.33
Using the inequalities 2.31, we can get by direct computation that
Amn
⎧ ⎪
⎪
|x|−ap |∇un − um |p dx,
⎨c
N
2/p
R
≥
⎪
⎪
⎩c
,
|x|−ap |∇un − um |p dx
p≥2
2.34
1 < p < 2,
RN
with some constant c > 0, independent of n and m.
Then the Hölder inequality together with 1.8 and 2.30 yield that
RN
∗
∗
∗
|x|−bp |un |p −2 un − |um |p −2 um un − um dx −→ 0 as n, m −→ ∞.
2.35
Abstract and Applied Analysis
11
Similarly, we have from the Hölder inequality, Lemmas 2.3 and 2.4 that
RN
fx |un |r−2 un − |um |r−2 um un − um dx −→ 0 as n, m −→ ∞,
RN
hx |un |
s−2
un − |um |
s−2
um un − um dx −→ 0
2.36
as n, m → ∞.
Therefore, the above estimates imply that un − um X → 0 n, m → ∞, that is, {un } is a
Cauchy sequence of X. Then {un } converges strongly in X and we complete the proof.
Similarly, we have the following lemma.
Lemma 2.7. Let 1 < r < p < s < p∗ . Then Iu satisfies the P Sc condition with c ≤ 1/s −
p∗ /p∗ −p
p/p−r
s − r/s − pS2 r/p−r r − ps − r/prsfLσ1 RN ,g , where S1 , S2 are as
1/p∗ S1
1
in 1.8, and 1.9 respectively.
Proof. Step 1. {un } is bounded in X.
Let {un } be a P Sc sequence of Iu in X. Then we have from Lemma 2.3 that
1
I un , un
s
1
1 1
1
1 1
p
p∗
−
−
− ∗ un Lp∗ RN un X −
un rLr RN ,f p s
r s
s p
1 1
1 1 r/p
p
−
−
S2 un rX f Lσ1 RN ,g1 .
≥
un X −
p s
r s
c 1 un X ≥ Iun −
2.37
Since 1 < r < p < s, 2.37 shows that un is bounded in X.
∗
Step 2. There exists {un } in X such that un → u in Lp RN .
p∗ /p∗ −p
by
Similar to the proof of Lemma 2.5, we can get that μj ηj 0 or ηj > S1
p∗ /p∗ −p
applying the functional ψ. Now we prove that there is no j0 ∈ J such that ηj0 > S1
Suppose otherwise, then
1
I u, un
s
1
1 1
1 1
1
p
p∗
−
−
− ∗ un Lp∗ RN un X −
un rLr RN ,f p s
r s
s p
1
1 1
1 1 r/p
1
p
p∗ /p∗ −p
−
−
− ∗ S1
≥
.
S2 un rX f Lσ1 RN ,g1 un X −
p s
r s
s p
.
c o1 Iun −
2.38
Let
qt 1 1 r/p 1 1 p
−
t −
−
S2 f Lσ1 RN ,g1 tr ,
p s
r s
t ≥ 0.
2.39
12
Abstract and Applied Analysis
Then qt has the unique minimum point at
1/p−r
s − r r/p S2 f Lσ1 RN ,g1 ,
s−p
r/p−r r − p s − r p/p−r
s−r
f σ N .
S2
qt0 L 1 R ,g1 s−p
prs
t0 2.40
Then it follows from 2.38 that
c o1 ≥
r/p−r r − p s − r p/p−r
s−r
1
1
p∗ /p∗ −p
f σ N ,
−
S2
S1
L 1 R ,g1 s p∗
s−p
prs
2.41
which contradicts the hypothesis of c.
Step 3. {un } converges strongly in X.
By Lemma 2.4, this result can be similarly obtained by the method in Lemma 2.6, so
we omit the proof.
3. Existence of Infinitely Solutions
In this section, we will use the minimax procedure to prove the existence of infinity many
solutions of problem 1.1. Let A denotes the class of A ⊂ X \ {0} such that A is closed in
X and symmetric with respect to the origin. For A ∈ A, we recall the genus γA which is
defined by
γA : min m ∈ N : ∃φ ∈ CA, Rm \ {0}, φx −φ−x .
3.1
If there is no mapping φ as above for any m ∈ N, then γA ∞, and γ∅ 0. The following
proposition gives some main properties of the genus, see 13, 22.
Proposition 3.1. Let A, B ∈ A. Then
1 if there exists an odd map g ∈ CA, B, then γA ≤ γB,
2 if A ⊂ B, then γA ≤ γB,
3 γA B ≤ γA γB.
4 if S is a sphere centered at the origin in RN , then γS N,
5 if A is compact, then γA < ∞ and there exists δ > 0 such that Nδ A ∈ A and
γNδ A γA, where Nδ A {x ∈ X : x − A ≤ δ}.
Lemma 3.2. Assume (A1 )–(A3 ). Then for any m ∈ N, there exists ε εm > 0 such that
γ{u ∈ X : Iu ≤ −ε} ≥ m.
3.2
Abstract and Applied Analysis
13
Proof. For given m ∈ N , let Xm be a m-dimensional subspace of X. If p < r < s < p∗ , then for
u ∈ Xm we have
Iu 1
1
1
1
1
1
p
p∗
p
uX − urLr RN ,f − usLs RN ,h − ∗ uLp∗ ≤ uX − urLr RN ,f .
p
r
s
p
p
r
3.3
The fact that all the norms on finite dimensional space are equivalent implies that for all
u ∈ Xm
Iu ≤
1
p
uX − curX ,
p
3.4
for some constant c > 0. Then there exist large ρ > 0 and small ε > 0 such that
Iu ≤ −ε,
uXm ρ.
3.5
Denote
Sρ u ∈ Xm : uXm ρ .
3.6
Then Sρ is a sphere centered at the origin with radius of ρ and
Sρ ⊂ {u ∈ X : Iu ≤ −ε} I −ε .
3.7
Therefore, Proposition 3.1 shows that γI −ε ≥ γSρ m.
If r < p < s < p∗ , we have
Iu 1
1
1
1
1
1
p
p∗
p
uX − urLr RN ,f − usLs RN ,h − ∗ uLp∗ ≤ uX − usLs RN ,h .
p
r
s
p
p
s
3.8
Since usLs RN ,h is also a norm and all norms on the finite dimensional space Xm are
equivalent, we have
Iu ≤
1
p
uX − cusX .
p
3.9
Then there exist large σ > 0 and small ε > 0 such that
Iu ≤ −ε,
uXm σ.
3.10
Denote
Sσ u ∈ Xm : uXm σ .
3.11
14
Abstract and Applied Analysis
Then Sσ is a sphere centered at the origin with radius of σ and
Sσ ⊂ {u ∈ X : Iu ≤ −ε} I −ε .
3.12
Therefore, Proposition 3.1 shows that γI −ε ≥ γSσ m.
Let Am {A ∈ A : γA ≥ m}. It is easy to check that Am1 ⊂ Am m 1, 2, . . .. We
define
cm inf supIu.
3.13
c1 ≤ c2 ≤ · · · ≤ cm ≤ · · · .
3.14
A∈Am u∈A
It is not difficult to find that
and cm > −∞ for any m ∈ N since Iu is coercive and bounded below. Furthermore, we
define the set
Kc u ∈ X : Iu c, I u 0 .
3.15
Then, Kc is compact and we have the following important lemma, see 22.
Lemma 3.3. All the cm are critical values of Iu. Moreover, if c cm cm1 · · · cmτ , then
γKc ≥ 1 τ.
Proof of Theorem 1.2. In view of Lemmas 2.6 and 2.7, Iu satisfies the P Sc condition in X.
Furthermore, as the standard argument of 13, 22, 23, Lemma 3.3 gives that Iu has infinity
many critical points with negative values. Thus, problem 1.1 has infinitely many solutions
in X, and we complete the proof.
Acknowledgments
The author would like to express his sincere gratitude to the anonymous reviewers for the
valuable comments and suggestions.
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