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 therein. In4 , Wu has proved that there existsλ0 >0 such thatPμ,λ,f,gadmits at least two
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 > 0 such 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.H01Ω denotes the standard Sobolev space, whose norm · is
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 · 2D
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. OnH01Ω, we use the norm
u2μ |∇u|2− μ
|x|2u2
dx. 2.1
Thanks to the Hardy inequality, the norm · μis equivalent to the usual norm · ofH01Ω.
H01Ωwith 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|2
u2 dx
RN|u|2∗dx2/2∗ . 2.2 From7,8 ,Sμis independent ofΩ⊂RNin the sense that if
SμΩ inf
u∈H01Ω\{0}
Ω
|∇u|2− μ/|x|2
u2 dx
Ω|u|2∗dx2/2∗ , 2.3 thenSμΩ SμRN Sμ.
Letμ N−2/22,γ1 μ−
μ−μ,γ2
μ
μ−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 a convergent subsequence.
3. Nehari Manifold
Associated withPμ, we consider the energy functionalJλinH, for eachu∈Has follows:
Jλu 1
2u2μ−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 u2μ−λ 2∗−q
2∗q f|u|qdx 3.4
≥ 1
Nu2μ−λ 2∗−q
2∗q
S−q/2μ |Ω|
2∗−q
/2∗uqμf
∞. 3.5
Thus,Jλis coercive and bounded below onNλ. Define
ψλu
Jλu, u
. 3.6
Then foru∈ Nλ,
ψλu, u
2u2μ−λq f|u|qdx−2∗ g|u|2∗dx 2−qu2μ−
2∗−q g|u|2∗dx λ
2∗−q f|u|qdx− 2∗−2
u2μ.
3.7
Similar to the method used in Tarantello13 , we splitNλinto three parts:
Nλ
u∈ Nλ:
ψλu, u
>0 , N0λ
u∈ Nλ:
ψλu, u 0
, 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 0 in 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∈ N0λ, we have
u2μ 2∗−q
2−q g|u|2∗dx, u2μλ2∗−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|∞S2μ∗/2
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∈NλJλu, α−λ inf
u∈N−λJλu. 3.12
Then we get the following result.
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∗−qu2μ> 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 u2μ<0.
3.14
Therefore, from the definitions ofαλ,αλ, we can deduce thatαλ≤αλ<0.
iiLetu∈ N−λ. By3.7
2−q
2∗−qu2μ< g|u|2∗dx. 3.15
Moreover, byg1and the Sobolev embedding theorem, g|u|2∗dx≤S−2μ ∗/2u2μ∗g
∞. 3.16
This implies
uμ>
2−q 2∗−q
|g|∞
1/2∗−2
SN/4μ ∀u∈ N−λ. 3.17
By3.5in the proof ofLemma 3.1
Jλu≥ uqμ
1
Nu2−qμ −λS−q/2μ
2∗−q
2∗q |Ω|2∗−q/2∗|f|∞
>
2−q 2∗−qg
∞
q/
2∗−2 SqN/4μ
1
NS2−qN/4μ
2−q 2∗−qg
∞
2−q/2∗−2
−λS−q/2μ 2∗−q
2∗q |Ω|2∗−q/2∗f∞ .
3.18
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∈Hwith
g|u|2∗dx >0, we write
tmax
2−qu2μ 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) if
f|u|qdx≤0, then there exists a uniquet− > tmaxsuch thatt−u∈ N−λand Jλ
t−u sup
t≥0Jλtu, 3.21
iiif
f|u|qdx >0, then there exist unique 0< t< tmax< t−such thattu∈ Nλ,t−u∈ N−λ and
Jλ
tu inf
0≤t≤tmax
Jλtu, Jλ
t−u sup
t≥0Jλtu. 3.22
Proof. The proof is almost the same as that in Brown-Wu 16, 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,
iiuλis a positive solution ofPμ, iiiJλuλ → 0 asλ → 0.
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 q 2∗−2 2
2∗−qun2
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
Nu2μ−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 Nuλ2
μ− 2∗−q
2∗q λ fuλqdx
≤lim inf
n→ ∞
1 Nun2
μ−2∗−q
2∗q λ funqdx
≤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
vn2
μun2
μ−uλ2
μon1. 4.7
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
dtJλt0uλ 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−q/2μ |Ω|2∗−q/2∗uλq
μf
∞. 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∗dxu2μ−λ fx|u|qdx. 5.1
Consequently,
Jλu 1
2 − 1 2∗
u2μ−
1 q− 1
2∗
λ fx|u|qdx. 5.2
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
Nu2μ−2∗−q
2∗q f∞|u|q2∗|Ω|2∗−q/2∗λ
≥ 1
Nu2μ−2∗−q 2∗q f
∞S−q/2μ uqμ|Ω|2∗−q/2∗λ
≥ 1
Nu2μ− 1
Nu2μ−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−2/2∞ SN/2μ −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
μun2
μ− u2μon1, 5.6
gxvn2∗dx gxun2∗dx− gx|u|2∗dxon1. 5.7
SinceJλun con1,Jλun on1and5.5–5.7, we can deduce that 1
2vn2
μ− 1
2∗ gxvn2∗dxc−Jλu on1, 5.8 vn2
μ− gxvn2∗dxon1. 5.9
Hence, we may assume that vn2
μ−→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/N∞ SμlN−2/N. Eitherl 0 orl ≥ |g|−N−2/2∞ SN/2μ . Ifl 0, this completes the proof.
Assume thatl≥ |g|−N−2/2∞ SN/2μ , from Lemmas5.1,5.8, and5.10, we get
c≥1 2 − 1
2∗
lJλu≥ 1
Ng−N−2/2
∞ SN/2μ −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∗ >0 such that forλ∈0,Λ∗, one has
sup
t≥0Jλtv< 1
Ng−N−2/2
∞ SN/2μ −C0λ2/2−q, 5.12
whereC0is the positive constant given inLemma 5.1.
In particular,α−λ<1/N|g|−N−2/2∞ SN/2μ −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
2u2μ− 1
2∗ g|u|2∗dx, χB0,2ρ0
⎧⎨
⎩
1 ifx∈B 0,2ρ0
, 0 ifx /∈B
0,2ρ0 ,
Qu u2μ
gχB0,2ρ01/2∗
u2
2∗
.
5.13
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/NSN/2μ OεN−2/2.
On that purpose, we need to establish the following estimatesasε → 0:
gχB0,2ρ01/2∗
uε2
2∗ ε−N−2/2|U|2L2∗RNOε, 5.16 uε2
μ ε−N−2/2
RN
|∇U|2− μ
|x|2U2
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
gχB0,2ρ01/2∗ uε2∗
2∗
B0,2δ0gxuε2∗dx RN
η2∗xgx
!ε|x|γ1/√
μ|x|γ2/√
μ"Ndx.
5.18
On the other hand, it is clear that
RN
1 ε|x|γ1/√
μ|x|γ2/√
μNdxε−N/2
RN
1
!|y|γ1/√
μ|y|γ2/√
μ"Ndy ε−N/2|U|2L∗2∗RN.
5.19
Combining the equalities above, we have ε−N/2|U|2L∗2∗RN− |
gχB0,2ρ01/2∗ uε|22∗∗
RN\B0,δ0
1−η2∗xgx ε|x|γ1/√
μ|x|γ2/√
μNdx
B0,δ0
1−gx ε|x|γ1/√
μ|x|γ2/√
μNdx, 5.20
hence
0≤ε−N/2|U|2L∗2∗RN−gχB0,2ρ01/2∗ uε2∗
2∗
≤ RN\B0,δ0
1
ε|x|γ1/√
μ|x|γ2/√
μNdx
B0,δ0
o
|x|β ε|x|γ1/√
μ|x|γ2/√
μNdx,
≤ RN\B0,δ0
1
|x|γ2N/√
μdx
B0,δ0
o
|x|β
|x|γ2N/√
μdx, NωN
∞ δ0
rN−1 rγ2N/√
μdr δ0
0
o rβ
rN−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−
gχB0,2ρ01/2∗ uε2∗
2∗|U|−2L2∗∗RNεN/2≤C1|U|−2L2∗∗RNεN/2, 5.22 that is,
1−C1|U|−2L2∗∗RNεN/2≤
gχB0,2ρ01/2∗ uε2∗
2∗|U|−2L2∗∗RNεN/2≤1. 5.23 Now, letεbe small enough such thatC1|U|−22∗∗εN/2<1, then from5.23we can deduce that
1−C1|U|−2L2∗∗RNεN/2≤
1−C1|U|−2L2∗∗RNεN/2#2/2∗
≤gχB0,2ρ01/2∗ uε2
2∗|U|−2L2∗RNεN−2/2≤1,
5.24
which yields that
|U|2L2∗RNε−N−2/2−C1|U|2−2L2∗R∗ Nε≤gχB0,2ρ01/2∗
uε2
2∗≤ |U|2L2∗RNε−N−2/2, 5.25
equivalently, equality5.16is valid.
Set|U|2μ
RN|∇U|2 −μ/|x|2U2dx. Combining with5.16and5.17, we obtain that
Q uε
ε−N−2/2|U|2μO1 ε−N−2/2|U|2L2∗RNOε |U|2μO
εN−2/2
|U|2L2∗RNO εN/2.
5.26
Hence
Quε−Sμ |U|2μO
εN−2/2
|U|2L2∗RNO
εN/2− |U|2μ
|U|2L2∗RN
|U|2L2∗RNO
εN−2/2
− |U|2μOεN/2 |U|2L2∗RNO
εN/2#
|U|2L2∗RN
O
εN−2/2 .
5.27
Using the fact
maxt≥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≥0I tuε
1 N
Q uεN/2
. 5.29
From5.27, we conclude that supt≥0Ituε≤1/NSN/2μ OεN−2/2.
Step 2. Let ε λ4/2−qN−2. We claim that there exists Λ∗ > 0 such that supt≥0Jλtuε <
1/NSN/2μ −C0λ2/2−qfor allλ∈0,Λ∗. Letδ1>0 be such that
1
NSN/2μ −C0λ2/2−q>0, ∀λ∈ 0, δ1
. 5.30
Using the definitions ofJλ, uεand byf2,g2, we get
Jλ tuε
≤ t2 2uε2
μ, ∀t≥0, λ >0, 5.31
which implies that there existst0 ∈0,1satisfying sup
0≤t≤t0
Jλ
tuε
< 1
NSN/2μ −C0λ2/2−q, ∀λ∈ 0, δ1
. 5.32
Using the definitions ofJλ, uε, and by the results inStep 1andf2, we have supt≥t0
Jλ
tuε
sup
t≥t0
I
tuε
−tq
qλ fxuεqdx
≤ 1
NSN/2μ O
εN−2/2
−tq0 qβ0λ
B0,δ0
uεqdx.
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
NSN/2μ O
λ2/2−q
−tq0
qβ0C1λ. 5.35
Hence, we can chooseδ2>0 such that O
λ2/2−q
−tq0
qβ0C1λ <−C0λ2/2−q λ∈ 0, δ2
. 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≥0Jλ
tuε
< 1
NSN/2μ −C0λ2/2−q. 5.37
Step 3. Prove thatα−λ<1/NSN/2μ −C0λ2/2−qfor allλ∈0,Λ∗. Byf2,g2, and the definition ofuε, we have
fxuεqdx >0, gxuε2∗dx >0. 5.38
