Volume 2009, Article ID 584203,17pages doi:10.1155/2009/584203
Research Article
Multiple Positive Solutions for
Singular Elliptic Equations with Concave-Convex
Nonlinearities and Sign-Changing Weights
Tsing-San Hsu and Huei-Li Lin
Center for General Education, Chang Gung University, Kwei-Shan, Tao-Yuan 333, Taiwan
Correspondence should be addressed to Tsing-San Hsu,[email protected]
Received 5 December 2008; Accepted 11 March 2009
Recommended by Pavel Drabek
We study existence and multiplicity of positive solutions for the following Dirichlet equations: −Δu−μ/|x|2uλfx|u|q−2ugx|u|2∗−2uinΩ,u0 on∂Ω, where 0∈Ω⊂RNN≥3is a
bounded domain with smooth boundary∂Ω,λ >0, 0≤ μ < μ N−22/4, 2∗ 2N/N−2, 1≤q <2, andf, gare continuous functions onΩwhich are somewhere positive but which may change sign onΩ.
Copyrightq2009 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.
1. Introduction and Main Results
In this paper, we study the existence and multiplicity of positive solutions for the following singular elliptic equation:
−Δu− μ
|x|2uλfx|u|
q−2ugx|u|p−2u inΩ,
u0 on∂Ω,
Pμ,λ,f,g
where 0∈Ω⊂RNN ≥3is a bounded domain with smooth boundary∂Ω,λ > 0, 0≤μ <
μ N−22/4,μis the best constant in the Hardy inequality, 1≤q <2< p, andf, g:Ω → R
are continuous functions which are somewhere positive but which may change sign onΩ.
We will assume in this paper thatpis a critical Sobolev exponent, that is,p2∗2N/N−2.
Whenμ 0 and weight functions fx ≡ gx ≡ 1 onΩ,Pμ,λf,ghas been studied
extensively for 2 < p ≤ 2∗ and variousq > 1. See, for example, 1–3 and the references
solutions for allλ∈0, λ0with 1≤q <2, a subcritical exponentp∈2,2∗,gx≡1 onΩand
fis a continuous function which change sign inΩ. In a recent work5 , Hsu-Lin have showed
the existence and multiplicity of positive solutions ofPμ,λ,f,gwith a critical exponentp2∗
and sign-changing weight functionsf, g.
To proceed, we make some motivations of the present paper. In 6 , Chen studied
Pμ,λ,f,g assuming that 0 ≤ μ < μ −1, 1 ≤ q < 2, p2∗ and fx ≡ gx ≡ 1 on Ω. He
proved that there existsΛ>0 such thatPμ,λ,f,ghas at least two positive solutions inH01Ω
for anyλ ∈ 0,Λ. But we do not see any multiplicity results aboutPμ,λ,f,gin the case of
the critical exponentp2∗and the weight functionsf, gsign-changing. In the present paper,
we continue the study of5 by considering the general caseμ ∈ 0, μ. We will extend the
results of6 to the more general case withμ ∈ 0, μand the weight functionsf, g which
may change sign onΩ. Our assumptions are
f1f ∈CΩandf max{f,0}/≡0 inΩ,
g1g ∈CΩandg max{g,0}/≡0 inΩ.
Set
Λ1
2−q
2∗−qg∞
2−q/2∗−2 2∗−2
2∗−qf∞
|Ω|q−2∗/2∗SN/2−N/4qq/2
μ >0, 1.1
where|Ω|is the Lebesgue measure ofΩ, andSμis the best Sobolev constantsee2.2. Now,
we state the first main result about the existence of positive solution ofPμ,λ,f,g.
Theorem 1.1. Assumef1and g1hold. Ifλ ∈ 0,Λ1, thenPμ,λ,f,g(simply written asPμ
from now on) has at least one positive solution inH01Ω.
In order to get the second positive solution of Pμ, we need some additional
assumptions aboutfandg. We assume the following conditions onfandg:
f2there existβ0andρ0 >0 such thatB0,2ρ0⊂Ωandfx≥β0for allx∈B0,2ρ0;
g2|g|∞ g0 maxx∈Ωgx,gx > 0 for all x ∈ B0,2ρ0and there existsβ ∈
μ−μN/
μ,
μ−μN1/
μsuch that
gx g0 o|x|β as x−→0. 1.2
Theorem 1.2. Assume thatf1-f2andg1-g2hold. Then there existsΛ2 > 0such that for
λ∈0,Λ2,Pμhas at least two positive solutions inH01Ω.
This paper is organized as follows. In Sections2and3, we give some preliminaries and
some properties of Nehari manifold. In Sections4and5, we complete proofs of Theorems1.1
and1.2.
2. Preliminaries
Throughout this paper,f1 andg1will be assumed. The dual space of a Banach space
Ewill be denoted by E−1.H1
induced by the standard inner product. We denote the norm inL2Ωby| · |2and the norm
inL2RNby| · |
L2RN.D1,2RN {u ∈ L2 ∗
RN :∇u ∈L2RN}with usual norm · 2
D
RN|∇·|2dx.|Ω|is the Lebesgue measure ofΩ.Bx, ris a ball centered atxwith radiusr.Oεt
denotes|Oεt|/εt≤C,oεtdenotes|oεt|/εt → 0 asε → 0, andon1denoteson1 → 0
asn → ∞. All integrals are taken overΩunless stated otherwise.C,Ciwill denote various
positive constants, the exact values of which are not important. OnH1
0Ω, we use the norm
u2μ |∇u|2− μ
|x|2u 2
dx. 2.1
Thanks to the Hardy inequality, the norm · μis equivalent to the usual norm · ofH01Ω.
H1
0Ωwith the norm · μis simply denoted byH. For allμ∈0, μ, we define the constant
Sμ inf u∈D1,2RN\{0}
RN
|∇u|2−μ/|x|2u2dx
RN|u|2 ∗
dx2/2∗ . 2.2
From7,8 ,Sμis independent ofΩ⊂RNin the sense that if
SμΩ inf
u∈H1 0Ω\{0}
Ω
|∇u|2−μ/|x|2u2dx
Ω|u|2∗dx
2/2∗ , 2.3
thenSμΩ SμRN S
μ.
Letμ N−2/22,γ1
μ−μ−μ,γ2
μμ−μ; Catrina and Wang 9 ,
Terracini10 proved thatSμis attained by the function
Ux 1
|x|γ1/ √
μ|x|γ2/ √
μ
√
μ. 2.4
Moreover, forε >0,Uεx ε−N−2/24Nμ−μ/N−2 N−2/4Ux/εsatisfies
−Δu− μ
|x|2u|u|
2∗−2u inRN\ {0},
u−→0 as|x| −→ ∞.
2.5
From11, Theorem B, all the positive solutions of problem2.5must have the form ofUε.
Moreover,UεattainsSμ.
We end these preliminaries by the following definition.
Definition 2.1. Letc∈R,Ebe a Banach space andI ∈C1E,R.
i{un}is aPSc-sequence inEforIifIun con1andIun on1strongly in
E−1asn → ∞.
iiWe say thatIsatisfies thePSc-condition if anyPSc-sequence{un}inEforIhas
3. Nehari Manifold
Associated withPμ, we consider the energy functionalJλinH, for eachu∈Has follows:
Jλu 1 2u
2
μ−
λ q f|u|
qdx− 1
2∗ g|u|
2∗dx. 3.1
It is well known thatJλis ofC1inH, and the solutions ofPμare the critical points of the
energy functionalJλsee Rabinowitz12 .
As the energy functionalJλ is not bounded below onH, it is useful to consider the
functional Nehari manifold
Nλ
u∈H\ {0}:Jλu, u0. 3.2
Thus,u∈ Nλif and only if
Jλu, uu2μ−λ f|u|qdx− g|u|2∗dx0. 3.3
Note that Nλ contains every nonzero solution of Pμ. Moreover, we have the following
results.
Lemma 3.1. The energy functionalJλis coercive and bounded below onNλ.
Proof. If u ∈ Nλ, then by f1, 3.3, the H ¨older inequality and the Sobolev embedding
theorem
Jλu 2∗−2 2∗2 u
2
μ−λ 2∗−q
2∗q f|u|
qdx 3.4
≥ 1
Nu
2
μ−λ 2∗−q
2∗q
S−μq/2|Ω|
2∗−q/2∗uq
μf∞. 3.5
Thus,Jλis coercive and bounded below onNλ.
Define
ψλu Jλu, u. 3.6
Then foru∈ Nλ,
ψλu, u2u2μ−λq f|u|qdx−2∗ g|u|2∗dx
2−qu2μ−2∗−q g|u|2∗dx
λ2∗−q f|u|qdx−2∗−2u2μ.
Similar to the method used in Tarantello13 , we splitNλinto three parts:
N
λ
u∈ Nλ:
ψλu, u>0,
N0
λ
u∈ Nλ:
ψλu, u0,
N− λ
u∈ Nλ:
ψλu, u<0.
3.8
Then, we have the following results.
Lemma 3.2. Assume thatuλis a local minimizer forJλonNλ anduλ/∈ N0λ. ThenJλuλ 0in
H−1Ω.
Proof. Our proof is almost the same as that in Brown-Zhang14, Theorem 2.3 or see
Binding-Dr´abek-Huang15 .
Lemma 3.3. Ifλ∈0,Λ1, thenN0λ∅, whereΛ1is the same as in1.1.
Proof. Suppose otherwise, that is there existsλ∈0,Λ1such thatN0λ/∅. Then by3.7, for
u∈ N0
λ, we have
u2
μ
2∗−q 2−q g|u|
2∗dx,
u2
μλ
2∗−q 2∗−2 f|u|
qdx.
3.9
Moreover, byf1,g1, the H ¨older inequality, and the Sobolev embedding theorem, we have
uμ≥
2−q
2∗−q|g|∞S
2∗/2
μ
1/2∗−2
,
uμ≤
λ2
∗−q
2∗−2Sμ
−q/2|Ω|2∗−q/2∗f ∞
1/2−q
.
3.10
This implies
λ≥
2−q
2∗−qg∞
2−q/2∗−2 2∗−2
2∗−qf∞
|Ω|q−2∗/2∗SN/2−N/4qq/2
μ Λ1, 3.11
which is a contradiction. Thus, we can conclude that ifλ∈0,Λ1, we haveN0λ∅.
ByLemma 3.3, we writeNλNλ∪ N−λand define
αλ inf u∈Nλ
Jλu, αλ inf
u∈N λ
Jλu, α−λ inf
u∈N−λJλu. 3.12
Lemma 3.4. iIfλ∈0,Λ1, then one hasαλ≤αλ<0.
ii If λ ∈ 0,q/2Λ1, then α−λ > d0 for some positive constant d0 depending on
λ, μ, q, N, Sμ, |f|∞,|g|∞and|Ω|.
Proof. iLetu∈ Nλ. By3.7
2−q 2∗−qu
2
μ> g|u|2 ∗
dx, 3.13
and so
Jλu
1
2 −
1 q
u2μ
1 q−
1
2∗ g|u|
2∗dx
<
1
2 −
1 q
1 q−
1 2∗
2−q
2∗−q
u2μ
−2−q
qN u
2
μ<0.
3.14
Therefore, from the definitions ofαλ,αλ, we can deduce thatαλ≤αλ<0. iiLetu∈ N−λ. By3.7
2−q 2∗−qu
2
μ< g|u|2 ∗
dx. 3.15
Moreover, byg1and the Sobolev embedding theorem,
g|u|2∗dx≤Sμ−2∗/2u2 ∗
μg∞. 3.16
This implies
uμ>
2−q
2∗−q|g|∞
1/2∗−2
SN/μ 4 ∀u∈ N−λ. 3.17
By3.5in the proof ofLemma 3.1
Jλu≥ uqμ
1 Nu
2−q
μ −λS−μq/2
2∗−q 2∗q |Ω|
2∗−q/2∗|f|
∞
>
2−q
2∗−qg∞
q/2∗−2
SqN/μ 4
1 NS
2−qN/4
μ
2−q
2∗−qg∞
2−q/2∗−2
−λS−μq/2
2∗−q 2∗q |Ω|
2∗−q/2∗f ∞
.
Thus, ifλ∈0,q/2Λ1, then
Jλu> d0 ∀u∈ N−λ, 3.19
for some positive constantd0d0λ, q, N, Sμ,|f|∞,|g|∞,|Ω|. This completes the proof.
For eachu∈Hwithg|u|2∗dx >0, we write
tmax
2−qu2
μ
2∗−q g|u|2∗dx
1/2∗−2
>0. 3.20
Then the following lemma holds.
Lemma 3.5. Letλ∈0,Λ1. For eachu∈Hwith
g|u|2∗dx >0, one has the following: (i) iff|u|qdx≤0, then there exists a uniquet− > t
maxsuch thatt−u∈ N−λand
Jλ
t−u sup
t≥0
Jλtu, 3.21
iiiff|u|qdx >0, then there exist unique0< t< tmax< t−such thattu∈ N
λ,t−u∈ N−λ
and
Jλ
tu inf
0≤t≤tmax
Jλtu, Jλ
t−usup
t≥0
Jλtu. 3.22
Proof. The proof is almost the same as that in Brown-Wu16, Lemma 2.6 , and is omitted here.
4. Proof of
Theorem 1.1
First, we will use the idea of Tarantello13 to get the following results.
Proposition 4.1. iIfλ∈0,Λ1, then there exists aPSαλ-sequence{un} ⊂ NλinHforJλ.
iiIfλ∈0,q/2Λ1, then there exists aPSα−λ-sequence{un} ⊂ N−λinHforJλ.
Proof. The proof is almost the same as that in Wu 4, Proposition 9 or see Hsu-Lin 5,
Proposition 3.3 .
Now, we establish the existence of a local minimum forJλonNλ.
Theorem 4.2. Ifλ∈0,Λ1, thenJλhas a minimizeruλinNλand it satisfies
iJλuλ αλαλ,
Proof. ByProposition 4.1i, there exists a minimizing sequence{un}forJλonNλsuch that
Jλ
un
αλon1, Jλ
un
on1 inH−1. 4.1
Since Jλ is coercive on Nλ see Lemma 3.1, we get that {un} is bounded in H. Going if
necessary to a subsequence, we can assume that there existsuλ∈Hsuch that
un uλ weakly inH,
un−→uλ almost every where in Ω,
un−→uλ strongly inLsΩ ∀1≤s<2∗.
4.2
First, we claim thatuλis a nontrivial solution ofPμ. By4.1and4.2, it is easy to see that
uλis a solution ofPμ. Fromun∈ Nλand3.4, we deduce that
λ funqdx
q2∗−2 22∗−qun
2
μ−
2∗q 2∗−qJλ
un
. 4.3
Letn → ∞in4.3, by4.1,4.2, andαλ<0, we get
λ fuλqdx≥ −
2∗q
2∗−qαλ>0. 4.4
Thus,uλ∈ Nλis a nontrivial solution ofPμ. Now we prove thatun → uλstrongly inHand
Jλuλ αλ. By4.3, ifu∈ Nλ, then
Jλu 1 Nu
2
μ−
2∗−q 2∗q λ f|u|
qdx. 4.5
In order to prove thatJλuλ αλ, it suffices to recall thatuλ ∈ Nλ, by4.5and applying
Fatou’s lemma to get
αλ≤Jλ
uλ
1
Nuλ
2
μ−
2∗−q
2∗q λ fuλ
q
dx
≤lim inf
n→ ∞
1 Nun
2
μ−
2∗−q
2∗q λ fun
q
dx
≤lim inf
n→ ∞ Jλ
un
αλ.
4.6
This implies thatJλuλ αλand limn→ ∞un2μ uλ2μ.Letvnun−uλ, then by Br´ezis-Lieb
lemma17 implies that
Therefore, un → uλ strongly in H. Moreover, we have uλ ∈ Nλ. On the contrary, ifuλ ∈
N−
λ, then byLemma 3.5, there are uniquet0 andt−0 such thatt0uλ ∈ Nλ andt−0uλ ∈ N−λ. In
particular, we havet0 < t−0 1. Since
d dtJλt
0uλ 0,
d2
dt2Jλ
t0uλ
>0, 4.8
there existst0 < t≤t−0such thatJλt0uλ< Jλtuλ. ByLemma 3.5,
Jλ
t0uλ
< Jλ
tuλ
≤Jλ
t−0uλ
Jλ
uλ
, 4.9
which is a contradiction. SinceJλuλ Jλ|uλ|and|uλ| ∈ Nλ, byLemma 3.2we may assume
thatuλis a nontrivial nonnegative solution ofPμ. Standard arguments implies thatuλis a
positive solution ofPμ. Moreover, byLemma 3.4iand3.5, we have
0> αλ>−λ
2∗−q
2∗q
S−μq/2|Ω|2 ∗−q/2∗
uλqμf∞. 4.10
This implies thatJλuλ → 0 asλ → 0.
Now, we begin the proof ofTheorem 1.1: ByTheorem 4.2, we obtainPμhas a positive
solutionuλ.
5. Proof of
Theorem 1.2
Next, we will establish the existence of the second positive solution ofPμby proving that
Jλ satisfies thePSα−
λ-condition.
Lemma 5.1. Assume thatf1and g1hold. If{un}is aPSc-sequence forJλ withun uin
H, thenJλu 0, and there exists a constantC0depending onq, N, Sμ,|f|∞and|Ω|, such that
Jλu≥ −C0λ2/2−q.
Proof. If{un}is aPSc-sequence forJλ withun uinH, it is easy to see thatJλu 0. This
implies thatJλu, u0, and
gx|u|2∗dxuμ2−λ fx|u|qdx. 5.1
Consequently,
Jλu
1
2 −
1 2∗
u2μ−
1 q−
1 2∗
Using the H ¨older inequality, the Young inequality, and the Sobolev embedding theorem, we have
Jλu
1
2 −
1 2∗
u2μ−1 q−
1 2∗
λ fx|u|qdx
≥ 1
Nu
2
μ−
2∗−q 2∗q f
∞|u|q2∗|Ω|2 ∗−q/2∗
λ
≥ 1
Nu
2
μ−
2∗−q 2∗q f
∞S−q/
2
μ uqμ|Ω|2 ∗−q/2∗
λ
≥ 1
Nu
2
μ−
1 Nu
2
μ−C0λ2/2−q−C0λ2/2−q,
5.3
whereC0is a positive constant depending onq, N, Sμ,|f|∞,and|Ω|.
Lemma 5.2. Assume thatf1andg1hold. Then the functionalJλsatisfies thePSc-condition
for allc ∈ −∞,1/N|g|∞−N−2/2SN/μ 2−C0λ2/2−qwhere C0 is the positive constant given in
Lemma 5.1.
Proof. Let {un} ⊂ H be aPSc-sequence which satisfiesJλun con1and Jλun
on1. Using standard arguments it follows that {un}is bounded inH. Thus, there exists a
subsequence still denoted by{un}and a functionu∈Hsuch that
un u weakly inH,
un−→u strongly in LsΩ ∀1≤s<2∗,
un−→u a.e. onΩ.
5.4
Byf1,g1, andLemma 5.1, we have thatJλu 0 and
λ fxunqdxλ fx|u|qdxon1, 5.5
Let vn un −u. Then by g is continuous onΩ, Br´ezis-Lieb lemma see17 , and
Vitali’s theorem, we obtain
vn2
μun
2
μ− u2μon1, 5.6
gxvn2 ∗
dx gxun2 ∗
SinceJλun con1,Jλun on1and5.5–5.7, we can deduce that
1 2vn
2
μ−
1
2∗ gxvn
2∗
dxc−Jλu on1, 5.8
vn2μ− gxvn2 ∗
dxon1. 5.9
Hence, we may assume that
vn2μ−→l, gx|vn|2 ∗
dx−→l. 5.10
By the Sobolev inequality, we havevn2μ ≥ Sμ|vn|22∗, combining with5.10, we get thatl ≥
|g|−∞N−2/NSμlN−2/N. Eitherl 0 orl ≥ |g|∞−N−2/2SN/μ 2. Ifl 0, this completes the proof.
Assume thatl≥ |g|∞−N−2/2SμN/2, from Lemmas5.1,5.8, and5.10, we get
c≥
1
2 −
1 2∗
lJλu≥ 1 Ng
−N−2/2
∞ SN/μ 2−C0λ2/2−q, 5.11
which is a contradiction. Therefore,l0 and we conclude thatun → uinH.
Lemma 5.3. Assume thatf1-f2andg1-g2hold. Then there existv∈HandΛ∗ >0such that forλ∈0,Λ∗, one has
sup
t≥0
Jλtv< 1 Ng
−N−2/2
∞ SN/μ 2−C0λ2/2−q, 5.12
whereC0is the positive constant given inLemma 5.1.
In particular,α−λ<1/N|g|∞−N−2/2SμN/2−C0λ2/2−qfor allλ∈0,Λ∗.
Proof. Without loss of generality, we can assume that|g|∞ 1. In fact, if|g|∞/1, we may
consider new coefficientsg∗x gx/|g|∞whose maximum equals to 1.
For convenience, we introduce the following notations:
Iu 1 2u
2
μ−
1 2∗ g|u|
2∗dx,
χB0,2ρ0
⎧ ⎨ ⎩
1 ifx∈B0,2ρ0
,
0 ifx /∈B0,2ρ0
,
Qu u
2
μ gχB0,2ρ0
1/2∗
u22∗
.
Fromg2, we know that there exists 0< δ0≤ρ0such that for allx∈B0,2δ0,
gx g0 o|x|β for someβ∈
⎛ ⎜ ⎝
μ−μN
μ ,
μ−μN1
μ
⎞ ⎟
⎠. 5.14
Motivated by some ideas of selecting cut-offfunctions in18 , we take such cut-offfunction
ηxthat satisfiesηx∈C∞0 B0,2δ0,ηx 1 for|x|< δ0, ηx 0 for|x|>2δ0,0≤η≤1
and|∇η| ≤C. Forε >0, let
uεx ηx
!
ε|x|γ1/ √
μ|x|γ2/ √
μ"
√
μ, 5.15
whereμ∈0, μ,μ N−2/22,γ1
μ−
μ−μ, andγ2
μ
μ−μ.
Step 1. Show that supt≥0Ituε≤1/NSμN/2OεN−2/2.
On that purpose, we need to establish the following estimatesasε → 0:
gχB0,2ρ0
1/2∗
uε
2
2∗ ε
−N−2/2|U|2
L2∗RNOε, 5.16
uε2μ ε−N−2/2
RN
|∇U|2− μ
|x|2U 2
dxO1, 5.17
whereUis defined as in2.4, andωN 2πN/2/NΓN/2is the volume of the unit
ballB0,1inRN. We only show that equality5.16is valid, proofs of5.17are very similar
to18 . Byg2and the definition ofuε, we get that
gχB0,2ρ0
1/2∗
uε
2∗
2∗ B0,2δ 0
gxuε2 ∗
dx
RN
η2∗xgx
!
ε|x|γ1/ √
μ|x|γ2/ √
μ"N
dx.
5.18
On the other hand, it is clear that
RN
1
ε|x|γ1/ √
μ|x|γ2/ √
μN
dxε−N/2 RN
1
!
|y|γ1/ √
μ|y|γ2/ √
μ"N
dy
ε−N/2|U|L2∗2∗RN.
Combining the equalities above, we have
ε−N/2|U|2∗
L2∗RN− |
gχB0,2ρ0
1/2∗u
ε|2 ∗
2∗
RN\B0,δ 0
1−η2∗xgx
ε|x|γ1/ √
μ|x|γ2/ √
μN
dx
B0,δ0
1−gx
ε|x|γ1/ √
μ|x|γ2/ √
μN
dx, 5.20
hence
0≤ε−N/2|U|2L∗2∗RN−gχB0,2ρ0
1/2∗
uε2 ∗
2∗
≤
RN\B0,δ 0
1
ε|x|γ1/ √
μ|x|γ2/ √
μN
dx
B0,δ0
o|x|β
ε|x|γ1/ √
μ|x|γ2/ √
μN
dx,
≤
RN\B0,δ0
1
|x|γ2N/ √
μdx B0,δ
0
o|x|β
|x|γ2N/ √
μdx,
NωN ∞
δ0 rN−1
rγ2N/ √
μdr δ0
0
orβrN−1
rγ2N/ √ μ dr, ωN μ
μ−μ δ−
√
μ−μ/√μ #
N
0
o1δβ−
√
μ−μ/√μN
0
β−μ−μ/μ#N
≤C1Const.,
5.21
which leads to
0≤1−gχB0,2ρ0
1/2∗
uε
2∗
2∗|U|
−2∗
L2∗RNεN/2≤C1|U|−2 ∗
L2∗RNεN/2, 5.22
that is,
1−C1|U|−L22∗∗RNε
N/2≤gχ
B0,2ρ0
1/2∗
uε
2∗
2∗|U|
−2∗
L2∗RNε
N/2≤1. 5.23
Now, letεbe small enough such thatC1|U|−2∗
2∗ εN/2<1, then from5.23we can deduce that
1−C1|U|L−22∗∗RNεN/2≤
1−C1|U|L−22∗∗RNεN/2
#2/2∗
≤gχB0,2ρ0
1/2∗
uε22∗|U|−L22∗RNεN−2/2≤1,
5.24
which yields that
|U|2L2∗RNε−N−
2/2−C1|U|2−2∗
L2∗RNε≤gχB0,2ρ0
1/2∗
uε22∗≤ |U|L22∗RNε−N−
equivalently, equality5.16is valid. Set|U|2
μ
RN|∇U|2 −μ/|x|2U2dx. Combining with5.16and5.17, we obtain
that
Quε
ε−N−
2/2|U|2
μO1
ε−N−2/2|U|2
L2∗RNOε
|U|
2
μO
εN−2/2 |U|2
L2∗RNO
εN/2.
5.26
Hence
Quε−Sμ
|U|2μOεN−2/2
|U|2
L2∗RNO
εN/2− |U|2μ
|U|2
L2∗RN
|U|
2
L2∗RNO
εN−2/2− |U|2
μOεN/2
|U|2
L2∗RNO
εN/2#|U|2
L2∗RN
OεN−2/2.
5.27
Using the fact
max
t≥0
t2
2a− t2∗
2∗b
1/N
a b2/2∗
N/2
for any a, b >0, 5.28
we can deduce that
sup
t≥0
Ituε
1
N
Quε N/2
. 5.29
From5.27, we conclude that supt≥0Ituε≤1/NSN/μ 2OεN−2/2.
Step 2. Let ε λ4/2−qN−2. We claim that there exists Λ∗ > 0 such that sup
t≥0Jλtuε <
1/NSN/μ 2−C0λ2/2−qfor allλ∈0,Λ∗.
Letδ1>0 be such that
1 NS
N/2
μ −C0λ2/2−q>0, ∀λ∈
0, δ1
. 5.30
Using the definitions ofJλ, uεand byf2,g2, we get
Jλ
tuε
≤ t2
2uε
2
which implies that there existst0 ∈0,1satisfying
sup
0≤t≤t0 Jλ tuε < 1 NS N/2
μ −C0λ2/2−q, ∀λ∈
0, δ1
. 5.32
Using the definitions ofJλ, uε, and by the results inStep 1andf2, we have
sup
t≥t0 Jλ
tuε
sup
t≥t0
Ituε
−tq
qλ fxuε
q dx ≤ 1 NS N/2
μ O
εN−2/2−t
q
0
qβ0λ B0,δ0uε
q
dx.
5.33
Let 0< ε≤δγ2−γ1/ √
μ
0 , we have
B0,δ0
uεqdx B0,δ0
1
ε|x|γ1/ √
μ|x|γ2/ √
μ
√
μqdx
≥
B0,δ0
1
2δγ2/
√
μ
0
√μqdx
C1
N, q, μ, δ0
.
5.34
Combining with5.33and5.34, for allελ4/2−qN−2 ∈0, δγ2−γ1/ √
μ
0 , we get
sup
t≥t0 Jλ tuε ≤ 1 NS N/2
μ O
λ2/2−q−t q
0
qβ0C1λ. 5.35
Hence, we can chooseδ2>0 such that
Oλ2/2−q−t
q
0
qβ0C1λ <−C0λ
2/2−q λ∈0, δ
2
. 5.36
If we setΛ∗min{δ1, δ2−q √
μ−μ
0 , δ2}>0, then forλ∈0,Λ∗andελ4/2−qN−2, we have
sup
t≥0
Jλ tuε < 1 NS N/2
μ −C0λ2/2−q. 5.37
Step 3. Prove thatα−λ<1/NSN/μ 2−C0λ2/2−qfor allλ∈0,Λ∗.
Byf2,g2, and the definition ofuε, we have
fxuεqdx >0, gxuε2 ∗
Combining this withLemma 3.5, from the definition ofα−λand the results inStep 2, we obtain that there existstε>0 such thattεuε∈ N−λand
α−λ≤Jλ
tεuε
≤sup
t≥0
Jλ
tuε
< 1 NS
N/2
μ −C0λ2/2−q 5.39
for allλ∈0,Λ∗.
Now, we establish the existence of a local minimum ofJλonN−λ.
Theorem 5.4. There existsΛ2 >0such that forλ∈0,Λ2the functionalJλhas a minimizerUλin
N−
λand satisfies
iJλUλ α−λ,
iiUλis a positive solution ofPμinH,
whereΛ2 min{Λ∗,q/2Λ1},Λ∗is defined as inLemma 5.3, andΛ1is defined as in1.1.
Proof. By Proposition 4.1ii, there exists a PSα−
λ-sequence {un} ⊂ N
−
λ in H for Jλ for all
λ∈0,q/2Λ1. From Lemmas5.2,5.3and3.4ii, forλ∈0,Λ∗,JλsatisfiesPSα−λ-condition
andα−λ > 0. SinceJλ is coercive onNλ seeLemma 3.1, we get that{un}is bounded inH.
Therefore, there exist a subsequence still denoted by{un}andUλ∈ N−λsuch thatun → Uλ
strongly inHandJλUλ α−λ > 0 for allλ∈ 0,Λ2. Finally, by using the same arguments
as in the proof ofTheorem 4.2, for allλ ∈ 0,Λ2, we have thatUλ is a positive solution of
Pμ.
Now, we complete the proof ofTheorem 1.2: By Theorems4.2and5.4, we obtainPμ
has two positive solutionsuλandUλsuch thatuλ ∈ Nλ,Uλ ∈ N−λ. SinceNλ∩ N−λ ∅, this
implies thatuλandUλare distinct.
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