# On the Elliptic Problems Involving Multisingular Inverse Square Potentials and Concave-Convex Nonlinearities

Authors: Tsing-San Hsu
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

A semilinear elliptic problem(Eλ) with concave-convex nonlinearities and multiple
Hardy-type terms is considered. By means of a variational method, we establish the existence
and multiplicity of positive solutions for problem (Eλ).


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## Body

## 1. Introduction and Main Results

In this paper, we consider the following semilinear elliptic problem:(Eλ)-Δu-∑i=1kμi|x-ai|2u=Q(x)|u|2*-2u+λ|u|q-2u,x∈Ω,u=0,x∈∂Ω,
where Ω⊂ℝN(N≥3) is a smooth bounded domain such that the different points ai∈Ω,i=1,2,…,k,k≥2,0≤μi<μ¯≜((N-2)/2)2,λ>0,1≤q<2,2*≜2N/(N-2) is the critical Sobolev exponent, and Q(x) is a positive bounded function on Ω¯.Problem (Eλ) is related to the well-known Hardy inequality (see [1, 2]):(1.1)∫Ω|u|2|x-a|2dx≤1μ¯∫Ω|∇u|2dx,∀u∈H01(Ω),a∈Ω.In this paper, for∑i=1kμi∈[0,μ¯), we use H≜H01(Ω) to denote the completion of C0∞(Ω) with respect to the norm (1.2)‖u‖=‖u‖H=(∫Ω(|∇u|2-∑i=1kμiu2|x-ai|2)dx)1/2.
By (1.1), this norm is equivalent to the usual norm (∫Ω|∇u|2dx)1/2.The functionu∈H is said to be solution of problem (Eλ) if u satisfies (1.3)∫Ω(∇u∇v-∑i=1kμi|x-ai|2uv-Q(x)|u|2*-2uv-λ|u|q-2uv)dx=0,∀v∈H,
and, by the standard elliptic regularity argument, we have that u∈C2(Ω∖{a1,a2,…,ak})∩C1(Ω¯∖{a1,a2,…,ak}).The energy functional corresponding to problem (Eλ) is defined as follows:(1.4)Jλ(u)≜12∫Ω(|∇u|2-∑i=1kμiu2|x-ai|2)dx-12*∫Ω|u|2*dx-λq∫Ω|u|qdx,
then Jλ(u) is well defined on H and belongs to C1(H,ℝ). The solutions of problem (Eλ) are then the critical points of the functional Jλ.It should be mentioned that, for0∈Ω,λ>0,1≤q<2,0≤μ<μ¯,0≤s<2 and 2*(s)=2(N-s)/(N-2) is the critical Sobolev-Hardy exponent. Note that 2*(0)=2*, the following semilinear elliptic problem:(1.5)-Δu-μ|x|2u=Q(x)|u|2*(s)-2|x|su+λ|u|q-2u,x∈Ω,u=0,x∈∂Ω,
had been extensively studied, and the existence and multiplicity results of positive solutions had been obtained; see [3–7] and references therein.For the casek≥2, our problem (Eλ) can be regarded as a perturbation problem of the following semilinear elliptic problem:(1.6)-Δu-∑i=1kμi|x-ai|2u=Q(x)|u|2*-2u,x∈Ω,u=0,x∈∂Ω.
In [8], by using Morse iteration, the authors studied the asymptotic behavior of solutions for problem (1.6); by critical point theory, the authors also proved the existence of nontrivial solutions to problem (1.6). On the other hand, the authors in [9] also studied problem (1.6); they discussed the corresponding Rayleigh quotient and gave both sufficient and necessary conditions on masses and location of singularities for the minimum to be achieved. In [9], both the case of the whole ℝN and bounded domains are taken into account.To proceed, we make some motivations of the present paper. In [6], the authors studied more general problem than problem (1.5) with μ∈[0,μ¯),s=0, and they proved that there exists Λ>0 such that problem (1.5) has at least two positive solutions for all λ∈(0,Λ). A natural question is whether the above results remain true for problem (Eλ) with multisingular inverse square potentials. In recent work [10], the author studied problem (1.1) with Q(x)≡1 on Ω¯ and showed that there exists Λ>0 such that problem (1.1) has at least two positive solutions for all λ∈(0,Λ). In this paper, we continue the study of [10] by considering the more general function Q(x) instead of Q(x)≡1 and extend the results of [10] to the more general function Q(x).For0≤μi<μ¯ and ai∈Ω,i=1,2,…,k, we can define the best constant (1.7)Sμi≜inf⁡u∈H∖{0}∫Ω(|∇u|2-μi(u2/|x-ai|2))dx(∫Ω|u|2*dx)2/2*,
and from [11], we get that Sμi is independent of Ω. For 0≤μ<μ¯,0≤μi<μ¯, setting (1.8)β≜μ¯-μ,γ≜μ¯+β,γ′≜μ¯-β,βi≜μ¯-μi,γi≜μ¯+βi,γi′≜μ¯-βi,
the authors in [1, 2] proved that Sμi is attained in ℝN by the function (1.9)Uμi(x-ai)=(22*βi2)1/(2*-2)|x-ai|γi′(1+|x-ai|(2*-2)βi)2/(2*-2),
and, moreover, for all ε>0,Vμi,εai(x)≜ε(2-N)/2Uμi((x-ai)/ε) solve the problem (1.10)-Δu-μi|x-ai|2u=|u|2*-2uinRN∖{ai}
and satisfy (1.11)∫RN(|∇Vμi,εai|2-μi|Vμi,εai|2|x-ai|2)dx=∫RN|Vμi,εai|2*dx=SμiN/2.
Note that Sμ is a decreasing function of μ for μ∈[0,μ¯) and(1.12)Uμiai(x)=1(|x-ai|γk/μ¯+|x-ai|γk′/μ¯)μ¯
also attains Sμi for i=1,2,…,k.Now we recall the following standard definition.Assume thatX is a Banach space and X-1 is the dual space of X. The functional I∈C1(X,ℝ) is said to satisfy the Palais-Smale condition at level c ((PS)c in short), if every sequence {un}⊂X satisfying I(un)→c and I′(un)→0 in X-1 has a convergent subsequence.In this paper, we will takeI=Jλ and X=H. To proceed, we need the following assumptions:(ℋ1)
there exists anl∈{1,2,…,k} such that(1.13)SμlN/2Q(al)(2-N)/2=min⁡{SμiN/2Q(ai)(2-N)/2,i=1,2,…,k},(ℋ2)
Q(x) is a positive bounded function on Ω¯, and there exists an x0∈Ω such that Q(x0) is a strict local maximum. Furthermore, there exists τ>(μ¯-μlN)/μ¯ such that(1.14)Q(x0)=QM=max⁡Ω¯Q(x),Q(x)-Q(x0)=o(|x-x0|τ)asx⟶x0,Q(x)-Q(al)=o(|x-al|τ)asx⟶al,(ℋ3)
0≤μi<μ¯ for every i=1,2,…,k and ∑i=1kμi<μ¯.We define the following constants:(1.15)S≜inf⁡u∈H01(Ω)∖{0}∫Ω(|∇u|2-∑i=1kμi(u2/|x-ai|2))dx(∫Ω|u|2*dx)2/2*,(1.16)Λ0≜(2-q(2*-q)QM)(2-q)/(2*-2)(2*-22*-q)|Ω|-((2*-q)/2*)S(2*(2-q))/(2(2*-2))+q/2.The main result of this paper is the following theorem.Theorem 1.1.
Assume that conditions(ℋ1)–(ℋ3) hold; then one has the following.  (i)
Ifλ∈(0,Λ0), then problem (Eλ) has at least one positive solution.(ii)
Ifλ∈(0,(q/2)Λ0), then problem (Eλ) has at least two positive solutions.This paper is organized as follows. In Section2, we give some properties of Nehari manifold. In Sections 3 and 4, we complete proofs of Theorem 1.1. At the end of this section, we explain some notations employed in this paper. Lp(Ω,|x-ai|t) denotes the usual weighted Lp(Ω) space with the weight |x-ai|t.|Ω| is the Lebesgue measure of Ω. Br(x) is a ball centered at x with radius r. O(εt) denotes |O(εt)|/εt≤C, and on(1) denotes on(1)→0 as n→∞. C, Ci will denote various positive constants and omit dx in the integration for convenience.


## 2. Nehari Manifold

In this section, we will give some properties of Nehari manifold. As the energy functionalJλ is not bounded below on H, it is useful to consider the functional on the Nehari manifold(2.1)Mλ={u∈H∖{0}:〈Jλ′(u),u〉=0}.
Thus, u∈ℳλ if and only if(2.2)〈Jλ′(u),u〉=‖u‖2-∫ΩQ(x)|u|2*-λ∫Ω|u|q=0.
Note that ℳλ contains every nonzero solution of problem (Eλ). Moreover, we have the following results.Lemma 2.1.
The energy functionalJλ is coercive and bounded below on ℳλ.Proof.
Ifu∈ℳλ, then by (1.15), (2.2), and Hölder inequality,
(2.3)Jλ(u)=2*-222*‖u‖2-λ(2*-q2*q)∫Ω|u|q≥1N‖u‖2-λ(2*-q2*q)|Ω|(2*-q)/2*S-q/2‖u‖q.
Thus, Jλ is coercive and bounded below on ℳλ.The Nehari manifold is closely linked to the behavior of the function of the formφu : t→Jλ(tu) for t>0. Such maps are known as fibering maps and were introduced by Drábek and Pohozaev in [12] and are also discussed by Brown and Zhang [13]. If u∈H, we have (2.4)φu(t)=t22‖u‖2-t2*2*∫ΩQ(x)|u|2*-λtqq∫Ω|u|q,φu′(t)=t‖u‖2-t2*-1∫ΩQ(x)|u|2*-λtq-1∫Ω|u|q,φu′′(t)=‖u‖2-(2*-1)t2*-2∫ΩQ(x)|u|2*-λ(q-1)tq-2∫Ω|u|q.
It is easy to see that (2.5)tφu′(t)=‖tu‖2-∫ΩQ(x)|tu|2*-λ∫Ω|tu|q,
and so, for u∈H∖{0} and t>0,φu′(t)=0 if and only if tu∈ℳλ, that is, the critical points of φu correspond to the points on the Nehari manifold. In particular, φu′(1)=0 if and only if u∈ℳλ. Thus, it is natural to split ℳλ into three parts corresponding to local minima, local maxima, and points of inflection. Accordingly, we define (2.6)Mλ+={u∈Mλ:φu′′(1)>0},Mλ0={u∈Mλ:φu′′(1)=0},Mλ-={u∈Mλ:φu′′(1)<0}
and note that, if u∈ℳλ, that is, φu′(1)=0, then(2.7)φu′′(1)=(2-q)‖u‖2-(2*-q)∫ΩQ(x)|u|2*(2.8)=λ(2-2*)‖u‖2-(q-2*)∫Ω|u|q.We now derive some basic properties ofℳλ+,ℳλ0, and ℳλ-.Lemma 2.2.
Assume thatu0 is a local minimizer for Jλ on ℳλ and u0∉ℳλ0. Then Jλ′(u0)=0 in H-1.Proof.
Our proof is almost the same as that in Brown-Zhang [13, Theorem 2.3] (or see Binding et al. [14]).Moreover, we have the following result.Lemma 2.3.
Ifλ∈(0,Λ0), then ℳλ0=∅, where Λ0 is the same as in (1.16).Proof.
Suppose the contrary. Then there existsλ∈(0,Λ0) such that ℳλ0≠∅. Then, for u∈ℳλ0 by (1.15) and (2.7), we have that
(2.9)2-q2*-q‖u‖2=∫ΩQ(x)|u|2*≤QMS-2*/2‖u‖2*,
and so
(2.10)‖u‖≥(2-q(2*-q)QM)1/(2*-2)S2*/(2(2*-2)).
Similarly, using (1.15), (2.8), and Hölder inequality, we have that
(2.11)‖u‖2=λ2*-q2*-2∫Ω|u|q≤λ2*-q2*-2|Ω|(2*-q)/2*S-q/2‖u‖q,
which implies that
(2.12)‖u‖≤(λ2*-q2*-2|Ω|(2*-q)/2*)1/(2-q)S-q/(2(2-q)).
Hence, we must have
(2.13)λ≥(2-q(2*-q)QM)(2-q)/(2*-2)(2*-22*-q)|Ω|-(2*-q)/2*S(2*(2-q))/(2(2*-2))+(q/2)=Λ0,
which is a contradiction. This completes the proof.In order to get a better understanding of the Nehari manifold and fibering maps, we consider the functionψu : ℝ+→ℝ defined by (2.14)ψu(t)=t2-q‖u‖2-t2*-q∫ΩQ(x)|u|2*fort>0.
Clearly tu∈ℳλ if and only if ψu(t)=λ∫Ω|u|q. Moreover,(2.15)ψu′(t)=(2-q)t1-q‖u‖2-(2*-q)t2*-q-1∫ΩQ(x)|u|2*fort>0,
and so it is easy to see that, if tu∈ℳλ, then tq-1ψu′(t)=φu′′(t). Hence, tu∈ℳλ+ (or tu∈ℳλ-) if and only if ψu′(t)>0 (or ψu′(t)<0).Foru∈H∖{0}, by (2.15), ψu has a unique critical point at t=tmax(u), where (2.16)tmax⁡(u)=((2-q)‖u‖2(2*-q)∫ΩQ(x)|u|2*)1/(2*-2)>0,
and clearly ψu is strictly increasing on (0,tmax(u)) and strictly decreasing on (tmax(u),∞) with limt→∞ψu(t)=-∞. Moreover, if λ∈(0,Λ0), then (2.17)ψu(tmax⁡(u))=[(2-q2*-q)(2-q)/(2*-2)-(2-q2*-q)(2*-q)/(2*-2)]‖u‖(2(2*-q))/(2*-2)(∫ΩQ(x)|u|2*)(2-q)/(2*-2)=‖u‖q(2*-22*-q)(2-q2*-q)(2-q)/(2*-2)(‖u‖2*∫ΩQ(x)|u|2*)(2-q)/(2*-2)≥‖u‖q(2*-22*-q)(2-q(2*-q)QM)(2-q)/(2*-2)S(2*(2-q))/(2(2*-2))>λ|Ω|(2*-q)/2*S-q/2‖u‖q≥λ∫Ω|u|q.
Therefore, we have the following lemma.Lemma 2.4.
Letλ∈(0,Λ0). For each u∈H∖{0}, one has the following:  (i)
there exist unique0<t+=t+(u)<tmax(u)<t-=t-(u) such that t+u∈ℳλ+,t-u∈ℳλ-,φu is decreasing on (0,t+), increasing on (t+,t-) and decreasing on (t-,∞)(2.18)Jλ(t+u)=inf⁡0≤t≤tmax⁡(u)Jλ(tu),Jλ(t-u)=sup⁡t≥t+Jλ(tu),(ii)
ℳλ-={u∈H∖{0} : (1/∥u∥)t-(u/∥u∥)=1},(iii)
there exists a continuous bijection betweenU={u∈H∖{0} : ∥u∥=1} and ℳλ-. In particular, t- is a continuous function for u∈H∖{0}.Proof.
For the proof see Wu [15, Lemma 2.6].


## 3. Existence of Ground State

First, we remark that it follows from Lemma2.3 that(3.1)Mλ=Mλ+∪Mλ-
for all λ∈(0,Λ0). Furthermore, by Lemma 2.4 it follows that ℳλ+ and ℳλ- are nonempty, and by Lemma 2.1 we may define(3.2)αλ=inf⁡u∈MλJλ(u),αλ+=inf⁡u∈Mλ+Jλ(u),αλ-=inf⁡u∈Mλ-Jλ(u).
Then we get the following result.Theorem 3.1.
One has the following.(i)
Ifλ∈(0,Λ0), then one has αλ+<0.(ii)
Ifλ∈(0,(q/2)Λ0), then αλ->d0 for some d0>0.In particular, for each λ∈(0,(q/2)Λ0), one has αλ+=αλ.Proof.
(i) Let u∈ℳλ+. By (2.7),
(3.3)2-q2*-q‖u‖2>∫ΩQ(x)|u|2*,
and so
(3.4)Jλ(u)=(12-1q)‖u‖2+(1q-12*)∫ΩQ(x)|u|2*<[(12-1q)+(1q-12*)(2-q2*-q)]‖u‖2=-(2*-2)(2-q)22*q‖u‖2<0.
Therefore, αλ+<0.
(ii) Let u∈ℳλ-. By (2.7),
(3.5)2-q2*-q‖u‖2<∫ΩQ(x)|u|2*.
Moreover, by (1.15), we have that
(3.6)∫ΩQ(x)|u|2*≤QMS-2*/2‖u‖2*.
This implies that
(3.7)‖u‖>(2-q(2*-q)QM)1/(2*-2)SN/4∀u∈Mλ-.
By (2.3) and (3.7), we have that
(3.8)Jλ(u)≥‖u‖q[1N‖u‖2-q-λ(2*-q2*q)S-q/2|Ω|(2*-q)/2*]>(2-q(2*-q)QM)q/(2*-2)×SNq/4[1N(2-q(2*-q)QM)(2-q)/(2*-2)S((2-q)N)/4-λ(2*-q2*q)S-q/2|Ω|(2*-q)/2*].
Thus, if λ∈(0,(q/2)Λ0), then
(3.9)Jλ(u)>d0∀u∈Mλ-,
for some positive constant d0. This completes the proof.Remark 3.2.
(i) If λ∈(0,Λ0), then by (1.15), (2.8), and Hölder inequality, for each u∈ℳλ+, we have that
(3.10)‖u‖2<λ2*-q2*-2∫Ω|u|q≤λ2*-q2*-2S-q/2|Ω|(2*-q)/2*‖u‖q≤Λ02*-q2*-2S-q/2|Ω|(2*-q)/2*‖u‖q,
and so
(3.11)‖u‖<(Λ02*-q2*-2S-q/2|Ω|(2*-q)/2*)1/(2-q)∀u∈Mλ+.
(ii) If λ∈(0,(q/2)Λ0), then by Lemma 2.4  (i) and Theorem 3.1  (ii), for each u∈ℳλ- we have that
(3.12)Jλ(u)=sup⁡t≥0Jλ(tu).Now, we use the Ekeland variational principle [16] to get the following results.Proposition 3.3.
(i) Ifλ∈(0,Λ0), then there exists a (PS)αλ sequence {un}⊂ℳλ in H for Jλ.
(ii) Ifλ∈(0,(q/2)Λ0), then there exists a (PS)αλ- sequence {un}⊂ℳλ- in H for Jλ.Proof.
The proof is almost the same as that in Wu [17, Proposition 9].Now, we establish the existence of a local minimum forJλ on ℳλ+.Theorem 3.4.
Assume that condition(ℋ) holds. If λ∈(0,Λ0), then Jλ has a minimizer uλ in ℳλ+ and it satisfies the following:(i)
Jλ(uλ)=αλ=αλ+,(ii)
uλ is a positive solution of problem (Eλ),(iii)
∥uλ∥→0 as λ→0+.Proof.
By Proposition3.3  (i), there is a minimizing sequence {un} for Jλ on ℳλ such that
(3.13)Jλ(un)=αλ+on(1),J′(un)=on(1)inH-1(Ω).
Since Jλ is coercive on ℳλ (see Lemma 2.1), we get that {un} is bounded in H. Going if necessary to a subsequence, we can assume that there exists uλ∈H such that
(3.14)un⇀uλweaklyinH,un⟶uλalmosteverywhereinΩ,un⟶uλstronglyinLs(Ω)∀1≤s<2*.
Thus, we have that
(3.15)λ∫Ω|un|q=λ∫Ω|uλ|q+on(1)asn⟶∞.
First, we claim that uλ is a nonzero solution of problem (Eλ). By (3.13) and (3.14), it is easy to see that uλ is a solution of problem (Eλ). From un∈ℳλ and (2.2), we deduce that
(3.16)λ∫Ω|un|q=q(2*-2)2(2*-q)‖un‖2-2*q2*-qJλ(un).
Let n→∞ in (3.16); by (3.13), (3.15), and αλ<0, we get
(3.17)λ∫Ω|uλ|q≥-2*q2*-qαλ>0.
Thus, uλ∈ℳλ is a nonzero solution of problem (Eλ). Now we prove that un→uλ strongly in H and Jλ(uλ)=αλ. By (3.16), if u∈ℳλ, then
(3.18)Jλ(u)=1N‖u‖2-λ2*-q2*q∫Ω|u|q.
In order to prove that Jλ(uλ)=αλ, it suffices to recall that un,uλ∈ℳλ, by (3.18) and applying Fatou’s lemma to get
(3.19)αλ≤Jλ(uλ)=1N‖uλ‖2-λ2*-q2*q∫Ω|uλ|q≤liminf⁡n→∞(1N‖un‖2-λ2*-q2*q∫Ω|un|q)≤liminf⁡n→∞Jλ(un)=αλ.
This implies that Jλ(uλ)=αλ and limn→∞∥un∥2=∥uλ∥2. Let vn=un-uλ; then Brézis-Lieb’s lemma [18] implies that
(3.20)‖vn‖2=‖un‖2-‖uλ‖2+on(1).
Therefore, un→uλ strongly in H. Moreover, we have uλ∈ℳλ+. On the contrary, if uλ∈ℳλ-, then, by Lemma 2.4, there are unique t0+ and t0- such that t0+uλ∈ℳλ+ and t0-uλ∈ℳλ-. In particular, we have t0+<t0-=1. Since
(3.21)ddtJλ(t0+uλ)=0,d2dt2Jλ(t0+uλ)>0,
there exists t0+<t¯≤t0- such that Jλ(t0+uλ)<Jλ(t¯uλ). By Lemma 2.4  (i),
(3.22)Jλ(t0+uλ)<Jλ(t¯uλ)≤Jλ(t0-uλ)=Jλ(uλ),
which is a contradiction. Since Jλ(uλ)=Jλ(|uλ|) and |uλ|∈ℳλ+, by Lemma 2.2, we may assume that uλ is a nonzero nonnegative solution of problem (Eλ). By Harnack inequality [19], we deduce that uλ>0 in Ω. Finally, by (3.10), we have that
(3.23)‖uλ‖2-q<λ2*-q2*-2|Ω|(2*-q)/2*S-q/2,
and so ∥uλ∥→0 as λ→0+.


## 4. Proof of Theorem1.1

In this section, we will establish the existence of the second positive solution of problem (Eλ) by proving that Jλ attains a local minimum on ℳλ-.Lemma 4.1.
If{un}⊂H is a (PS)c sequence for Jλ, then {un} is bounded in H.Proof.
The argument is similar to that of [10, Lemma 4.1], and here we omit the details.We recall that(4.1)Sμi≜inf⁡u∈H∖{0}∫Ω(|∇u|2-μi(u2/|x-ai|2))dx(∫Ω|u|2*dx)2/2*.Lemma 4.2.
Assume that conditions(ℋ1)–(ℋ3) holds. If {un}⊂H is a (PS)c sequence for Jλ with
(4.2)0<c<c*≜1Nmin⁡{SμlN/2Q(al)(N-2)/2,S0N/2QM(N-2)/2},
then there exists a subsequence of {un} converging weakly to a nonzero solution of problem (Eλ).Proof.
Let{un}⊂H be a (PS)c sequence for Jλ with c∈(0,c*). We know from Lemma 4.1 that {un} is bounded in H, and then there exists a subsequence of {un} (still denoted by {un}) and u0∈H such that
(4.3)un⇀u0weaklyinH,un⇀u0weaklyinL2(Ω,|x-ai|-2)for1≤i≤k,un⇀u0weaklyinL2*(Ω),un⟶u0almosteverywhereinΩ,un⟶u0stronglyinLs(Ω)∀1≤s<2*.
It is easy to see that Jλ′(u0)=0 and
(4.4)λ∫Ω|un|q=λ∫Ω|u0|q+on(1).
Next we verify thatu0≢0. Arguing by contradiction, we assume that u0≡0. By the concentration compactness principle (see [20, 21]), there exist a subsequence, still denoted by {un}, at most countable set 𝒥, a set of different points {xj}j∈𝒥⊂Ω∖{a1,a2,…,ak}, nonnegative real numbers μxj̃,νxj̃,j∈𝒥, and nonnegative real numbers μaĩ,γaĩ,νaĩ(1≤i≤k) such that
(4.5)|∇un|2⇀dμ̃≥|∇u0|2+∑j∈Jμxj̃δxj+∑i=1kμaĩδai,un2|x-ai|2⇀dγ̃=u02|x-ai|2+γaĩδai,|un|2*⇀dν̃=|u0|2*+∑j∈Jνxj̃δxj+∑i=1kνaĩδai,
where δx is the Dirac mass at x. By the Sobolev-Hardy inequalities, we infer that
(4.6)Sμiνaĩ2/2*≤μaĩ-μiγaĩ,1≤i≤k.
We claim that 𝒥 is finite and, for any j∈𝒥, either
(4.7)νxj̃=0orQ(xj)νxj̃≥S0N/2QM(N-2)/N.
In fact, letε>0 be small enough such that ai∉Bε(xj) for all 1≤i≤k and Bε(xi)∩Bε(xj)=⌀ for i≠j,i,j∈𝒥. Let ϕεj be a smooth cut-off function centered at xj such that 0≤ϕεj≤1,ϕεj=1 for |x-xj|≤ε/2,ϕεj=0 for |x-xj|≥ε and |∇ϕεj|≤4/ε. Then
(4.8)lim⁡ε→0lim⁡n→∞∫Ω|∇un|2ϕεj=lim⁡ε→0∫Ωϕεjdμ̃≥lim⁡ε→0(∫Ω|∇u0|2ϕεj+μxj̃)=μxj̃,lim⁡ε→0lim⁡n→∞∫Ωun2|x-ai|2ϕεj=lim⁡ε→0∫Ωϕεjdγ̃=lim⁡ε→0∫Ωu02|x-ai|2ϕεj=0,lim⁡ε→0lim⁡n→∞∫ΩQ(x)|un|2*ϕεj=lim⁡ε→0∫ΩQ(x)ϕεjdν̃=lim⁡ε→0(∫ΩQ(x)|u0|2*ϕεj+Q(xj)νxj̃)=Q(xj)νxj̃,lim⁡ε→0lim⁡n→∞∫Ωun∇un∇ϕεj=0.
Thus we have that
(4.9)0=lim⁡ε→0lim⁡n→∞〈Jλ′(un),unϕεj〉≥μxj̃-Q(xj)νxj̃.
By the Sobolev inequality, S0νxj̃2/2*≤μxj̃ for j∈𝒥; hence we deduce that
(4.10)νxj̃=0orQ(xj)νxj̃≥S0N/2QM(N-2)/2,
which implies that 𝒥 is finite.
Now we consider the possibility of concentraction at pointsai(1≤i≤k). For ε>0 be small enough such that xj∉Bε(ai) for all j∈𝒥 and Bε(ai)∩Bε(aj)=⌀ for i≠j and 1≤i,j≤k. Let φεi be a smooth cut-off function centered at ai such that 0≤φεi≤1,φεi=1 for |x-ai|≤ε/2,φεi=0 for |x-ai|≥ε and |∇φεi|≤4/ε. Then
(4.11)lim⁡ε→0lim⁡n→∞∫Ω|∇un|2φεi=lim⁡ε→0∫Ωφεidμ̃≥lim⁡ε→0(∫Ω|∇u0|2φεi+μaĩ)=μaĩ,lim⁡ε→0lim⁡n→∞∫Ωun2|x-ai|2φεi=lim⁡ε→0∫Ωφεidγ̃=lim⁡ε→0(∫Ωu02|x-ai|2φεi+γaĩ)=γaĩ,lim⁡ε→0lim⁡n→∞∫ΩQ(x)|un|2*φεi=lim⁡ε→0∫ΩQ(x)φεidν̃=lim⁡ε→0(∫ΩQ(x)|u0|2*φεi+Q(ai)νaĩ)=Q(ai)νaĩ,lim⁡ε→0lim⁡n→∞∫Ωun2|x-aj|2φεi=0forj≠i,lim⁡ε→0lim⁡n→∞∫Ωun∇un∇φεi=0.
Thus we have that
(4.12)0=lim⁡ε→0lim⁡n→∞〈Jλ′(un),unφεi〉≥μaĩ-μiγaĩ-Q(ai)νaĩ.
From (4.6) and (4.12) we derive that
(4.13)Sμiνaĩ2/2*≤Q(ai)νaĩ,
and then either νaĩ=0 or νaĩ≥(Sμi/Q(ai))N/2 for all 1≤i≤k.
On the other hand, from the above arguments and (4.4), we conclude that
(4.14)c=lim⁡n→∞(Jλ(un)-12〈Jλ′(un),un〉)=1Nlim⁡n→∞∫ΩQ(x)|un|2*+λ(12-1q)∫Ω|u0|q=1N(∫ΩQ(x)|u0|2*+∑j∈JQ(xj)νxj̃+∑i=1kQ(ai)νaĩ)+λ(12-1q)∫Ω|u0|q=1N(∑j∈JQ(xj)νxj̃+∑i=1kQ(ai)νaĩ)+Jλ(u0).
Ifνaĩ=νxj̃=0 for all i∈{1,2,…,k} and j∈𝒥, then c=0 which contradicts the assumption that c>0. On the other hand, if there exists an i∈{1,2,…,k} such that νaĩ≠0 or there exists a j∈𝒥 with νxj̃≠0, then we infer that
(4.15)c≥1Nmin⁡{Sμ1N/2Q(a1)(N-2)/2,Sμ2N/2Q(a2)(N-2)/2,…,SμkN/2Q(ak)(N-2)/2,S0N/2QM(N-2)/2}=1Nmin⁡{SμlN/2Q(al)(N-2)/2,S0N/2QM(N-2)/2}=c*,
which also contradicts the assumption that c<c*. Therefore u0 is a nonzero solution of problem (Eλ).Lemma 4.3.
Assume that conditions(ℋ1)–(ℋ3) hold. Then for any λ>0, there exist vλ∈H01(Ω) such that
(4.16)sup⁡t≥0Jλ(tvλ)<c*.
In particular,αλ-<c* for all λ∈(0,Λ0) where Λ0 is the same as in (1.16).Proof.
From(ℋ2), we know that there exist ρ0>0,τ>(μ¯-μlN)/μ¯ such that B2ρ0(al)⊂Ω, B2ρ0(x0)⊂Ω,
(4.17)Q(x)=Q(al)+o(|x-al|τ)∀x∈B2ρ0(al),Q(x)=QM+o(|x-x0|τ)∀x∈B2ρ0(x0).
To prove this lemma, we need to distinguish the following two cases:
(4.18)case I:SμlN/2Q(al)(N-2)/2<S0N/2QM(N-2)/2,case II:SμlN/2Q(al)(N-2)/2≥S0N/2QM(N-2)/2.
We first study Case I. The definition ofc* implies that
(4.19)c*=SμlN/2NQ(al)(N-2)/2.
Motivated by some ideas of selecting cut-off functions in [22], we take such cut-off function ηal(x) that satisfies ηal(x)∈C0∞(B2δ0(al)),ηal(x)=1 for |x-al|<δ0,ηal(x)=0 for |x-al|>2δ0,0≤ηal≤1 and |∇ηal|≤C where 0<δ0<min{(1/2)|ai-aj|,i,j=1,2,…,k,i≠j},δ0≤ρ0, and B2δ0(al)⊂Ω. For ε>0, let
(4.20)uμl,εal(x)=ε(N-2)/4ηal(x)[ε|x-al|γl′/μ¯+|x-al|γl/μ¯]μ¯,
where μ¯=((N-2)/2)2,γl′=μ¯-μ¯-μl, and γl=μ¯+μ¯-μl.
We define the following functions on the interval[0,+∞):
(4.21)g(t)≜Jλ(tuμl,εal)=t22∫Ω(|∇uμl,εal|2-μl(uμl,εal)2|x-al|2)-t2*2*∫ΩQ(x)|uμl,εal|2*-t22∑i≠l,i=1kμi∫Ω(uμl,εal)2|x-ai|2-λtqq∫Ω|uμl,εal|q≤t22∫Ω(|∇uμl,εal|2-μl(uμl,εal)2|x-al|2)-t2*2*∫ΩQ(x)|uμl,εal|2*-λtqq∫Ω|uμl,εal|q,g¯(t)≜t22∫Ω(|∇uμl,εal|2-μl(uμl,εal)2|x-al|2)-t2*2*∫ΩQ(x)|uμl,εal|2*.
From Hsu and Lin [6, Lemma 5.3] and after a detailed calculation, we have the following estimates:
(4.22)(∫ΩQ(x)|uμl,εal|2*)2/2*=(∫RNQ(al)|Uμlal|2*)2/2*+O(εN/2),∫Ω(|∇uμl,εal|2-μl(uμl,εal)2|x-al|2)=∫RN(|∇Uμlal|2-μl(Uμlal)2|x-al|2)+O(ε(N-2)/2),(4.23)sup⁡t≥0g¯(t)=SμlN/2NQ(al)(N-2)/2+O(ε(N-2)/2),
where Uμlal is defined as in (1.12).
Using the definitions ofg(t),uμl,εal, we get
(4.24)g(t)≤t22∫Ω(|∇uμl,εal|2-μl(uμl,εal)2|x-al|2),∀t≥0,∀λ>0.
Combining this with (4.22), let ε∈(0,1); then there exists t0∈(0,1) independent of ε such that
(4.25)sup⁡0≤t≤t0g(t)<SμlN/2NQ(al)(N-2)/2,∀λ>0,∀ε∈(0,1).
Using the definitions of g(t) and uμl,εal and by (4.23), we have that
(4.26)sup⁡t≥t0g(t)=sup⁡t≥t0(g¯(t)-tqqλ∫Ω|uμl,εal|q)≤SμlN/2NQ(al)(N-2)/2+O(ε(N-2)/2)-λt0qq∫Bδ0(al)|uμl,εal|q.
Let0<ε≤δ0(γl-γl′)/μ¯; then we have that
(4.27)∫Bδ0(al)|uμl,εal|q=∫Bδ0(al)ε(q(N-2))/4[ε|x-al|γl′/μ¯+|x-al|γl/μ¯]μ¯q≥∫Bδ0(al)ε(q(N-2))/4((2δ0γl/μ¯)μ¯q=C1(N,q,μl,δ0)ε(q(N-2))/4.
Combining with (4.26) and (4.27), for all ε∈(0,δ0(γl-γl′)/μ¯), we get
(4.28)sup⁡t≥t0g(t)≤SμlN/2NQ(al)(N-2)/2+O(ε(N-2)/2)-t0qqC1λε(q(N-2))/4.
Hence, for any λ>0, we can choose small positive constant ελ<min{1,δ0(γl-γl′)/μ¯} such that
(4.29)O(ελ(N-2)/2)-t0qqC1λελ(q(N-2))/4<0.
From (4.25), (4.28), and (4.29), we can deduce that, for any λ>0, there exists ελ>0 such that
(4.30)sup⁡t≥0Jλ(tuμl,ελal)<SμlN/2NQ(al)(N-2)/2.
From Lemma2.4  (i), the definition of αλ-, and (4.30), we can deduce that, for any λ∈(0,Λ0), there exists tελ>0 such that tελuελ∈𝒩λ- and
(4.31)αλ-≤Jλ(tελuμl,ελal)≤sup⁡t≥0Jλ(tuμl,ελal)<SμlN/2NQ(al)(N-2)/2.
Hence Case I is verified.
Next, we investigate Case II. In this case we have that(4.32)c*=S0N/2NQM(N-2)/2=S0N/2NQ(x0)(N-2)/2≤SμlN/2NQ(al)(N-2)/2,
where x0 is the maximum point of Q(x) defined as in (ℋ2).
Ifx0=ai for some i∈{1,2,…,k}, from the fact that Sμi<S0, we obtain
(4.33)c*=S0N/2NQ(ai)(N-2)/2>SμiN/2NQ(ai)(N-2)/2≥SμlN/2NQ(al)(N-2)/2,
which is impossible. Hence x0≠ai for any i∈{1,2,…,k}.
Forε>0, let
(4.34)u0,εx0(x)=ε(N-2)/4ηx0(x)(ε+|x-x0|2)(N-2)/2,
where ηx0(x) is a cut-off function that satisfies ηx0(x)∈C0∞(B2δ0(x0)),ηx0(x)=1 for |x-x0|<δ0,ηx0(x)=0 for |x-x0|>2δ0,0≤ηx0≤1 and |∇ηx0|≤C where 0<δ0<(1/2)min{|x0-a1|,|x0-a2|,…,|x0-ak|,2ρ0} and B2δ0(x0)⊂Ω. Consider the functions defined on the interval [0,+∞):
(4.35)h¯(t)≜t22∫Ω|∇u0,εx0|2-t2*2*∫ΩQ(x)|u0,εx0|2*,h(t)≜Jλ(tu0,εx0)=h¯(t)-t22∑i=1kμi∫Ω(u0,εx0)2|x-ai|2-λtqq∫Ω|u0,εx0|q.
By the same argument as in Case I, we can deduce that
(4.36)sup⁡t≥0h¯(t)=S0N/2NQ(x0)(N-2)/2+O(ε(N-2)/2),∫Ω|u0,εx0|q≥C2(N,q,δ0)εq(N-2)/4∀ε∈(0,δ02),
and, for any λ>0, there exists 0<ελ<min{1,δ02} such that
(4.37)sup⁡t≥0Jλ(tu0,ελx0)<sup⁡t≥0(h¯(t)-λtqq∫Ω|u0,ελx0|q)<S0N/2NQ(x0)(N-2)/2.
From Lemma2.4 (i), the definition of αλ-, and (4.37), we can deduce that, for any λ∈(0,Λ0), there exists tελ>0 such that tελuελ∈𝒩λ- and
(4.38)αλ-≤Jλ(tελu0,ελx0)≤sup⁡t≥0Jλ(tu0,ελx0)<S0N/2NQ(x0)(N-2)/2.
Hence Case II is proved. From Case I and II we conclude Lemma4.3.Now, we establish the existence of a local minimum ofJλ on ℳλ-.Theorem 4.4.
Assume that condition(ℋ) holds. If λ∈(0,(q/2)Λ0), then Jλ has a minimizer Uλ in ℳλ-, and it satisfies the following:(i)
Jλ(Uλ)=αλ-,(ii)
Uλ is a positive solution of problem (Eλ).Proof.
Ifλ∈(0,(q/2)Λ0), then, by Theorem 3.1 (ii), Proposition 3.3 (ii), and Lemma 4.3, there exists a (PS)αλ- sequence {un}⊂ℳλ- in H for Jλ with αλ-∈(0,c*). From Lemma 4.2, there exist a subsequence still denoted by {un} and a nonzero solution Uλ∈H of problem (Eλ) such that un⇀Uλ weakly in H. Now we prove that un→Uλ strongly in H and Jλ(Uλ)=αλ-. By (3.18), if u∈ℳλ, then
(4.39)Jλ(u)=1N‖u‖2-λ2*-q2*q∫Ω|u|q.
First, we prove that Uλ∈ℳλ-. On the contrary, if Uλ∈ℳλ+, then by, the definition of
(4.40)Mλ-={u∈Mλ:φu′′(1)<0}
and Lemma 2.3, we have ∥Uλ∥2<liminfn→∞∥un∥2. By Lemma 2.4 (i), there exists a unique tλ- such that tλ-Uλ∈ℳλ-. Since un∈ℳλ-, by (3.12) and (4.39), we have Jλ(un)≥Jλ(tun) for all t≥0 and
(4.41)αλ-≤Jλ(tλ-Uλ)<liminf⁡n→∞Jλ(tλ-un)≤liminf⁡n→∞Jλ(un)=αλ-,
and this is a contradiction.
In order to prove thatJλ(Uλ)=αλ-, it suffices to recall that un,Uλ∈ℳλ- for all n, by (4.39) and applying Fatou's lemma to get
(4.42)αλ-≤Jλ(Uλ)=1N‖Uλ‖2-λ2*-q2*q∫Ω|Uλ|q≤liminf⁡n→∞(1N‖un‖2-λ2*-q2*q∫Ω|un|q)≤liminf⁡n→∞Jλ(un)=αλ-.
This implies that Jλ(Uλ)=αλ- and limn→∞∥un∥2=∥Uλ∥2. Let vn=un-Uλ; then Brézis-Lieb’s lemma [18] implies that
(4.43)‖vn‖2=‖un‖2-‖Uλ‖2+on(1).
Therefore, un→Uλ strongly in H.
SinceJλ(Uλ)=Jλ(|Uλ|)=αλ- and |Uλ|∈ℳλ-, by Lemma 2.2, we may assume that Uλ is a nonzero nonnegative solution of problem (Eλ). Finally, by the Harnack inequality [19], we deduce that Uλ>0 in Ω.Now, we complete the proof ofTheorem1.1. By Theorems 3.4 and 4.4, we obtain that problem (Eλ) has two positive solutions uλ and Uλ such that uλ∈ℳλ+,Uλ∈ℳλ-. Since ℳλ+∩ℳλ-=∅, this implies that uλ and Uλ are distinct. This completes the proof of Theorem 1.1.



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*Source: 408307-2011-03-16.xml*