# Multiplicity of Positive Solutions for Weighted Quasilinear Elliptic Equations Involving Critical Hardy-Sobolev Exponents and Concave-Convex Nonlinearities

Authors: Tsing-San Hsu; Huei-Li Lin
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

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.


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## Body

## 1. Introduction and Main Results

LetΩ 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:
(1.1)-div⁡(|x|-ap|∇u|p-2∇u)-μ|u|p-2u|x|p(a+1)=|u|p*(a,b)-2u|x|bp*(a,b)+λ|u|q-2u|x|dp*(a,d)inΩ,u=0on⁡∂Ω,
where λ>0, 1<p<N, 0≤μ<μ̅, μ̅≜((N-p)/p-a)p, 0≤a<(N-p)/p, a≤b,d<a+1, 1≤q<p, p*(a,d)≜Np/(N-p(a+1-d)) is the critical Sobolev-Hardy exponent. Note that p*(0,0)=p*≜Np/(N-p) is the critical Sobolev exponent.In this paper,W≜Wa1,p(Ω) denotes the space obtained as the completion of C0∞(Ω) with respect to the norm (∫Ω|x|-ap|∇u|pdx)1/p. The energy functional of (1.1) is defined on W by
(1.2)Jλ(u)=1p∫Ω(|x|-ap|∇u|p-μ|u|p|x|p(a+1))dx-1p*(a,b)∫Ω|u|p*(a,b)|x|bp*(a,b)dx-λq∫Ω|u|q|x|dp*(a,d)dx.
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]:
(1.3)(∫RN|u|p*(a,b)|x|bp*(a,b)dx)p/p*(a,b)≤C∫RN|x|-ap|∇u|pdx,∀u∈C0∞(RN),
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]:
(1.4)∫RN|u|p|u|p(a+1)dx≤1μ̅∫RN|x|-ap|∇u|pdx,∀u∈C0∞(RN),
where μ̅=((N-p)/p-a)p is the best Hardy constant.In the spaceW, we employ the following norm if μ<μ̅:
(1.5)‖u‖=‖u‖W≜(∫Ω(|x|-ap|∇u|p-μ|u|p|x|p(a+1))dx)1/p.
By (1.4) it is equivalent to the usual norm (∫Ω|x|-ap|∇u|pdx)1/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<μ̅:
(1.6)Sμ,a,b(Ω)=inf⁡u∈W∖{0}‖u‖p(∫Ω|u|p*(a,b)/|x|bp*(a,b)dx)p/p*(a,b).
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.Whena=0, we set s=dp*(0,d) and t=bp*(0,b), then (1.1) is equivalent to the following quasilinear elliptic equations:
(1.7)-div⁡(|∇u|p-2∇u)-μ|u|p-2u|x|p=|u|p*(t)-2u|x|t+λ|u|q-2u|x|sinΩ,u=0on⁡∂Ω,
where λ>0, 1<p<N, 0≤μ<μ̅=((N-p)/p)p, 0≤s,t<p, 1≤q<p and p*(t)=p(N-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 Sobolev-Hardy critical exponents (the case that t≠0) (see [17–22] and their references therein). In [23, 24], the authors deal with doubly-critical exponents.Whenp≠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.For0≤a<(N-p)/p, a≤b<a+1, and 0≤μ<μ̅, consider the following limiting problem:
(1.8)-div⁡(|x|-ap|∇u|p-2∇u)-μ|u|p-2u|x|p(a+1)=|u|p*(a,b)-2u|x|bp*(a,b)inRN∖{0},u>0inRN∖{0},u∈Wa1,p(RN),
where Wa1,p(ℝN) is the space obtained as the completion of C0∞(ℝN) with respect to the norm (∫ℝN|x|-ap|∇u|pdx)1/p. From [5, Lemma  2.2], we know (1.8) has a unique ground state solution Up,μ satisfying
(1.9)Up,μ(1)=(p*(a,b)(μ̅-μ)p)1/(p*(a,b)-p)
and all ground states must be of the form Ũε(x)=ε-(N-p)/p-aUp,μ(x/ε) for some ε>0, that is,
(1.10)Sμ,a,b=inf⁡u∈Wa1,p(RN)∖{0}∫RN(|∇u|p-μ|u|p/|x|p(a+1))dx(∫Ω|u|p*(a,b)/|x|bp*(a,b)dx)p/p*(a,b)
is achieved by Ũε. Moreover, Up,μ is radially symmetric and possesses the following properties:
(1.11)lim⁡r→0rα(μ)Up,μ(r)=c1>0,lim⁡r→0rα(μ)+1|Up,μ′(r)|=c1α(μ)≥0,lim⁡r→+∞rβ(μ)Up,μ(r)=c2>0,lim⁡r→+∞rβ(μ)+1|Up,μ′(r)|=c2β(μ)>0,
where ci(i=1,2) are positive constants and α(μ), β(μ) are the zeros of the function
(1.12)f(τ)=(p-1)τp-(N-p(a+1))τp-1+μ,τ≥0,0≤μ<μ̅,
which satisfy 0≤α(μ)<(N-p(a+1))/p<β(μ)<(N-p(a+1))/(p-1). Furthermore, there exist the positive constants c3=c3(μ,p,a,b) and c4=c4(μ,p,a,b) such that
(1.13)c3≤Up,μ(x)(|x|α(μ)/δ+|x|β(μ)/δ)δ≤c4,δ≜N-p(a+1)p.Throughout this paper, letR0 be the positive constant such that Ω⊂B(0;R0), where B(0;R0)={x∈ℝN:|x|<R0}. By Hölder and Sobolev-Hardy inequalities, for all u∈W, we obtain
(1.14)∫Ω|u|q|x|dp*(a,d)≤(∫B(0;R0)|x|-dp*(a,d))(p*(a,d)-q)/p*(a,d)(∫Ω|u|p*(a,d)|x|dp*(a,d))q/p*(a,d)≤(NωN∫0R0r-dp*(a,d)+N-1dr)(p*(a,d)-q)/p*(a,d)(Sμ,a,d)-(q/p)‖u‖q≤(NωNR0N-dp*(a,d)N-dp*(a,d))(p*(a,d)-q)/p*(a,d)(Sμ,a,d)-(q/p)‖u‖q.
Set
(1.15)Λ0=(p-qp*(a,b)-q)(p-q)/(p*(a,b)-p)(p*(a,b)-pp*(a,b)-q)(NωNR0N-dp*(a,d)N-dp*(a,d))-((p*(a,d)-q)/p*(a,d))×(Sμ,a,d)q/p(Sμ,a,b)p*(a,b)(p-q)/p(p*(a,b)-p),
where ωN=2πN/2/NΓ(N/2) is the volume of the unit ball in ℝN.Furthermore, from0≤a<(N-p)/p and a≤d<a+1, we can deduce that
(1.16)0<p(N-pp-a)+pd=N-p(a+1-d)<N,
which implies
(1.17)p*(a,d)=pNN-p(a+1-d)>p,N-dp*(a,d)=NpN-p(a+1-d)(N-pp-a)>0.
Combining these with 1≤q<p, we get Λ0>0.We are now ready to state our main results.Theorem 1.1.
Assume thatN≥3,  0≤μ<μ̅,  0≤a<(N-p)/p,  a≤b,d<a+1,  and 1≤q<p<N. Then one has the following results.(i)
Ifλ∈(0,Λ0), then (1.1) has at least one positive solution in W.(ii)
Ifλ∈(0,(q/p)Λ0), 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 Sections2 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 B(0;R) as a ball centered at the origin with radius R, and ωN=2πN/2/NΓ(N/2) is the volume of the unit ball B(0;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. O(εt) denotes |O(εt)/εt|≤C, and o(εt) means |o(εt)/εt|→0 as ε→0. By o(1) we always mean it is a generic infinitesimal value.


## 2. Nehari Manifold

Since the functionalJλ is not bounded from below on W, we will work on Nehari manifold. For λ>0 we define
(2.1)Nλ={u∈W∖{0}:〈Jλ′(u),u〉=0}.
We recall that any nonzero solutions of (1.1) belong to 𝒩λ. Moreover, by definition, we have that u∈𝒩λ if and only if
(2.2)‖u‖≠0,‖u‖p-1p*(a,b)∫Ω|u|p*(a,b)|x|bp*(a,b)-λq∫Ω|u|q|x|dp*(a,d)=0.
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.
Ifu∈𝒩λ, then by (1.14) and (2.2), we get
(2.3)Jλ(u)=p*(a,b)-pp*(a,b)p‖u‖p-λ(p*(a,b)-qp*(a,b)q)∫Ω|u|q|x|dp*(a,d)(2.4)≥p*(a,b)-pp*(a,b)p‖u‖p-λ(p*(a,b)-qp*(a,b)q)(NωNR0N-dp*(a,d)N-dp*(a,d))(p*(a,d)-q)/p*(a,d)×(Sμ,a,d)-q/p‖u‖q.
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.Defineψλ:W→ℝ, by ψλ(u)=〈Jλ′(u),u〉, that is,
(2.5)ψλ(u)=‖u‖p-∫Ω|u|p*(a,b)|x|bp*(a,b)-λ∫Ω|u|q|x|dp*(a,d).
Note that ψλ is of class C1 with
(2.6)〈ψλ′(u),u〉=p‖u‖p-p*(a,b)∫Ω|u|p*(a,b)|x|bp*(a,b)-qλ∫Ω|u|q|x|dp*(a,d).
Furthermore, if u∈𝒩λ, then by (2.2), we have that
(2.7)〈ψλ′(u),u〉=(p-q)‖u‖p-(p*(a,b)-q)∫Ω|u|p*(a,b)|x|bp*(a,b)(2.8)=(p-p*(a,b))‖u‖p-(q-p*(a,b))λ∫Ω|u|q|x|dp*(a,d).
Now we split 𝒩λ into three sets:
(2.9)Nλ+={u∈Nλ:〈ψλ′(u),u〉>0},Nλ0={u∈Nλ:〈ψλ′(u),u〉=0},Nλ-={u∈Nλ:〈ψλ′(u),u〉<0}.The following result shows that minimizers on𝒩λ are the “usual” critical points for Jλ.Lemma 2.2.
Supposeu0 is a local minimizer of Jλ on 𝒩λ and u0∉𝒩λ0. Then, Jλ′(u0)=0 in W-1.Proof.
See [30, Lemma  2.2].Motivated by the above result, we will get conditions for𝒩λ0=∅.Lemma 2.3.
𝒩λ0=∅ for all λ∈(0,Λ0).Proof.
We argue by contradiction. Suppose that there exists aλ∈(0,Λ0) such that 𝒩λ0≠∅. Let u∈𝒩λ0 be arbitrary, then by (2.2), (2.7), and (2.8), we have that
(2.10)0<‖u‖p=p*(a,b)-qp-q∫Ω|u|p*(a,b)|x|bp*(a,b),0<‖u‖p=λp*(a,b)-qp*(a,b)-p∫Ω|u|q|x|dp*(a,d).
By (1.14), (2.10), and Sobolev-Hardy inequality, we get
(2.11)‖u‖≥(p-qp*(a,b)-q)1/(p*(a,b)-p)(Sμ,a,b)p*(a,b)/p(p*(a,b)-p),‖u‖≤(λp*(a,b)-qp*(a,b)-p)1/(p-q)(NωNR0N-dp*(a,d)N-dp*(a,d))(p*(a,d)-q)/p*(a,d)(p-q)(Sμ,a,d)-(q/p(p-q)).
Hence we must have
(2.12)λ≥(p-qp*(a,b)-q)(p-q)/(p*(a,b)-p)(p*(a,b)-pp*(a,b)-q)(NωNR0N-dp*(a,d)N-dp*(a,d))-((p*(a,d)-q)/p*(a,d))×(Sμ,a,d)q/p(Sμ,a,b)p*(a,b)(p-q)/p(p*(a,b)-p)=Λ0,
which is a contradiction.For eachu∈W∖{0}, let
(2.13)τmax⁡=((p-q)‖u‖p(p*(a,b)-q)∫Ω|u|p*(a,b)/|x|bp*(a,b))1/(p*(a,b)-p).Lemma 2.4.
Ifλ∈(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∈𝒩λ- and a unique τ+=τ+(u)>0 such that τ+u∈𝒩λ+. Moreover, τ+<τmax<τ- and
(2.14)Jλ(τ+u)=inf⁡0≤τ≤τmax⁡Jλ(τu),Jλ(τ-u)=sup⁡τ≥τmax⁡Jλ(τu).Proof.
The proof is similar to that of [29, Lemma  2.7] and is omitted.We remark that by Lemma2.3 we have, 𝒩λ=𝒩λ+∪𝒩λ- for all λ∈(0,Λ0). Furthermore, by Lemma 2.4 it follows that 𝒩λ+ and 𝒩λ- are non-empty and by Lemma 2.1 we may define
(2.15)αλ=inf⁡u∈NλJλ(u),αλ+=inf⁡u∈Nλ+Jλ(u),αλ-=inf⁡u∈Nλ-Jλ(u).Theorem 2.5.
(i)
Ifλ∈(0,Λ0), then one has αλ≤αλ+<0.(ii)
Ifλ∈(0,(q/p)Λ0), then αλ->d0 for some positive constant d0.
In particular, for eachλ∈(0,(q/p)Λ0), one has αλ=αλ+<0<αλ-.Proof.
(i) Letu∈𝒩λ+. By (2.7),
(2.16)p-qp*(a,b)-q‖u‖p>∫Ω|u|p*(a,b)|x|bp*(a,b),
and so also using (2.2),
(2.17)Jλ(u)=(1p-1q)‖u‖p+(1q-1p*(a,b))∫Ω|u|p*(a,b)|x|bp*(a,b)<[(1p-1q)+(1q-1p*(a,b))(p-qp*(a,b)-q)]‖u‖p=-(p-q)(p*(a,b)-p)pqp*(a,b))‖u‖p<0.
Therefore, from the definition of αλ and αλ+, we can deduce that αλ≤αλ+<0.
(ii) Letu∈𝒩λ-. By (2.7),
(2.18)p-qp*(a,b)-q‖u‖p<∫Ω|u|p*(a,b)|x|bp*(a,b).
Moreover, by Sobolev-Hardy inequality,
(2.19)∫Ω|u|p*(a,b)|x|bp*(a,b)≤(Sμ,a,b)-(p*(a,b)/p)‖u‖p*(a,b).
This implies
(2.20)‖u‖>(p-qp*(a,b)-q)1/(p*(a,b)-p)(Sμ,a,b)p*(a,b)/p(p*(a,b)-p)∀u∈Nλ-.
By (2.4) and (2.20), we have
(2.21)Jλ(u)≥‖u‖q[(NωNR0N-dp*(a,d)N-dp*(a,d))(p*(a,d)-q)/p*(a,d)p*(a,b)-pp*(a,b)p‖u‖p-q-λ(p*(a,b)-qp*(a,b)q)×(NωNR0N-dp*(a,d)N-dp*(a,d))(p*(a,d)-q)/p*(a,d)(Sμ,a,d)-(q/p)]>(p-qp*(a,b)-q)q/(p*(a,b)-p)(Sμ,a,b)qp*(a,b)/p(p*(a,b)-p)×[(NωNR0N-dp*(a,d)N-dp*(a,d))(p*(a,d)-q)/p*(a,d)p*(a,b)-pp*(a,b)p(p-qp*(a,b)-q)(p-q)/(p*(a,b)-p)(Sμ,a,b)p*(a,b)(p-q)/p(p*(a,b)-p)-λ(p*(a,b)-qp*(a,b)q)(NωNR0N-dp*(a,d)N-dp*(a,d))(p*(a,d)-q)/p*(a,d)(Sμ,a,d)-(q/p)]=(qpΛ0-λ)(p-qp*(a,b)-q)q/(p*(a,b)-p)(Sμ,a,b)qp*(a,b)/p(p*(a,b)-p)(p*(a,b)-qp*(a,b)q)×(NωNR0N-dp*(a,d)N-dp*(a,d))(p*(a,d)-q)/p*(a,d)(Sμ,a,d)-(q/p).
Thus, if λ∈(0,(q/p)Λ0), then
(2.22)Jλ(u)>d0∀u∈Nλ-,
for some positive constant d0.Remark 2.6.
Ifλ∈(0,(q/p)Λ0), then by Lemma 2.4 and Theorem 2.5, for each u∈W∖{0}, we can easily deduce that
(2.23)τ-u∈Nλ-,Jλ(τ-u)=sup⁡τ≥0Jλ(τu)≥αλ->0.


## 3. Proof of the Main Results

First, we define the Palais-Smale (simply by (PS)) sequences, (PS)-values, and (PS)-conditions inW for Jλ as follows.Definition 3.1.
(i) Forc∈ℝ, a sequence {un} is a (PS)c-sequence in W for Jλ if Jλ(un)=c+o(1) and Jλ′(un)=o(1) strongly in W-1 as n→∞.
(ii)c∈ℝ is a (PS)-value in W for Jλ if there exists a (PS)c-sequence in W for Jλ.
(iii)Jλ satisfies the (PS)c-condition in W if any (PS)c-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) Ifλ∈(0,Λ0), then Jλ has a (PS)αλ-sequence {un}⊂𝒩λ.
(ii) Ifλ∈(0,(q/p)Λ0), then Jλ has a (PS)αλ--sequence {un}⊂𝒩λ-.Proof.
The proof is similar to [29, Proposition  3.3] and the details are omitted.Now, we establish the existence of a local minimum forJλ on 𝒩λ.Theorem 3.3.
Assume thatN≥3,  0≤μ<μ̅,  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+.Proof.
By Proposition3.2(i), there exists a minimizing sequence {un}⊂𝒩λ such that
(3.1)Jλ(un)=αλ+o(1),Jλ′(un)=o(1)inW-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→∞(3.2)un⇀uλweaklyinW,un⟶uλstronglyinLq(Ω,|x|-dp*(a,d))for1≤q<p,un⟶uλa.e.inΩ,∇un⟶∇uλa.e.inΩ,un|x|a+1⇀uλ|x|a+1weaklyinLp(Ω),∫Ω|un|p*(a,b)-2un|x|bp*(a,b)v⟶∫Ω|uλ|p*(a,b)-2uλ|x|bp*(a,b)v,∀v∈W.
Consequently, passing to the limit in 〈Jλ′(un),v〉, by (3.1) and (3.2), as n→∞, we have
(3.3)∫Ω(|∇uλ|p-2∇uλ∇v|x|ap-μ|uλ|p-2uλv|x|p(a+1))-∫Ω|uλ|p*(a,b)-2uλv|x|bp*(a,b)-λ∫Ω|uλ|q-2uλv|x|dp*(a,d)=0,
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
(3.4)λ∫Ω|un|q|x|dp*(a,d)=q(p*(a,b)-p)p(p*(a,b)-q)‖un‖p-p*(a,b)qp*(a,b)-qJλ(un).
Let n→∞ in (3.4), by (3.1) and (3.2), and since αλ<0 by (i) of Theorem 2.5, we get
(3.5)λ∫Ω|uλ|q|x|dp*(a,d)≥-p*(a,b)qp*(a,b)-qαλ>0.
Thus uλ≢0, and since Jλ′(uλ)=0, it follows that uλ∈𝒩λ and, in particular, Jλ(uλ)≥αλ.
Next, we will show, up to a subsequence, thatun→uλ strongly in W and Jλ(uλ)=αλ. From the fact un,u∈𝒩λ, (2.3) and the Fatou’s lemma, it follows that
(3.6)αλ≤Jλ(uλ)=p*(a,b)-ppp*(a,b)‖uλ‖p-λ(p*(a,b)-qp*(a,b)q)∫Ω|uλ|q|x|dp*(a,d)≤lim⁡inf⁡n→∞[p*(a,b)-pp*(a,b)p‖un‖p-λ(p*(a,b)-qp*(a,b)q)∫Ω|un|q|x|dp*(a,d)]=lim⁡inf⁡n→∞Jλ(un)=αλ,
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∈𝒩λ-, by Lemma 2.4, there exist unique τλ+ and τλ- such that τλ+uλ∈𝒩λ+, τλ-uλ∈𝒩λ- and τλ+<τλ-=1. Since
(3.7)ddτJλ(τλ+uλ)=0,d2dτ2Jλ(τλ+uλ)>0,
there exists τ̅∈(τλ+,τλ-) such that Jλ(τλ+uλ)<Jλ(τ̅uλ). By Lemma 2.4 we get that
(3.8)Jλ(τλ+uλ)<Jλ(τ̅uλ)≤Jλ(τλ-uλ)=Jλ(uλ),
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
(3.9)‖uλ‖p-q<λ(p*(a,b)-qp*(a,b)-p)(NωNR0N-dp*(a,d)N-dp*(a,d))(p*(a,d)-q)/p*(a,d)(Sμ,a,d)-(q/p),
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 (PS)αλ--condition.Lemma 3.4.
Let{un} be a bounded sequence in W. If {un} is a (PS)c-sequence for Jλ with
(3.10)c∈(0,p*(a,b)-pp*(a,b)p(Sμ,a,b)p*(a,b)/(p*(a,b)-p)),
then there exists a subsequence of {un} converging weakly to a nonzero solution of (1.1).Proof.
Let{un}⊂W be a (PS)c-sequence for Jλ with c∈(0,((p*(a,b)-p)/p*(a,b)p)(Sμ,a,b)p*(a,b)/(p*(a,b)-p)). Since {un} is bounded in W and the embedding 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→∞(3.11)un⇀u0weaklyinW,un⇀u0weaklyinLp*(a,b)(Ω,|x|-bp*(a,b))un⟶u0stronglyinLq(Ω,|x|-dp*(a,d))for1≤q<p,un⟶u0a.e.inΩ.
Using the same argument in Theorem 3.3, we deduce that Jλ′(u0)=0 and
(3.12)λ∫Ω|un|q|x|dp*(a,d)=λ∫Ω|u0|q|x|dp*(a,d)+o(1).
Next we verify thatu0≢0. Arguing by contradiction, we assume u0≡0. Set
(3.13)l=lim⁡n→∞∫Ω|un|p*(a,b)|x|bp*(a,b).
Since Jλ′(u0)=0 and {un} is bounded in W, then by (3.12), we can deduce that
(3.14)0=〈lim⁡n→∞Jλ′(un),un〉=lim⁡n→∞(‖un‖p-∫Ω|un|p*(a,b)|x|bp*(a,b))=lim⁡n→∞‖un‖p-l,
that is,
(3.15)lim⁡n→∞‖un‖p=l.
Ifl=0, then by (3.12)–(3.15), we get
(3.16)c=lim⁡n→∞Jλ(un)=lim⁡n→∞(1p‖un‖p-∫Ω|un|p*(a,b)|x|bp*(a,b)-λ∫Ω|un|q|x|dp*(a,d))=0,
which contradicts c>0. Thus we conclude that l>0. Furthermore, the Sobolev-Hardy inequality implies that
(3.17)‖un‖p≥Sμ,a,b(∫Ω|un|p*(a,b)|x|bp*(a,b))p/p*(a,b).
Then as n→∞, we have l=limn→∞∥un∥p≥Sμ,a,blp/p*(a,b), which implies that
(3.18)l≥(Sμ,a,b)p*(a,b)/(p*(a,b)-p).
Hence, from (3.12)–(3.18) we get
(3.19)c=lim⁡n→∞Jλ(un)=1plim⁡n→∞‖un‖p-1p*(a,b)lim⁡n→∞∫Ω|un|p*(a,b)|x|bp*(a,b)-λqlim⁡n→∞∫Ω|un|q|x|dp*(a,d)=(1p-1p*(a,b))l≥p*(a,b)-pp*(a,b)p(Sμ,a,b)p*(a,b)/(p*(a,b)-p).
This contradicts the definition of c. Therefore, u0 is a nontrivial solution of (1.1).Lemma 3.5.
IfN≥3,  0≤μ<μ̅,  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
(3.20)sup⁡τ≥0Jλ(τvλ)<p*(a,b)-pp*(a,b)p(Sμ,a,b)p*(a,b)/(p*(a,b)-p).
In particular,αλ-<((p*(a,b)-p)/p*(a,b)p)(Sμ,a,b)p*(a,b)/(p*(a,b)-p) for all λ∈(0,Λ0).Proof.
LetUp,μ be a ground state solution of (1.8), ρ>0 small enough such that B(0;ρ)⊂Ω, η∈C0∞(Ω), 0≤η(x)≤1, η(x)=1 for |x|<ρ/2,η(x)=0 for |x|≥ρ. Set Ũε(x)=ε-((N-p)/p)-aUp,μ(x/ε) and uε(x)=η(x)Ũε(x),ε>0. Then, following the same lines as in [5], we get the following estimates as ε→0:
(3.21)‖uε‖p=(Sμ,a,b)p*(a,b)/(p*(a,b)-p)+O(εβ(μ)p+p(a+1)-N),(3.22)∫Ω|uε|p*(a,b)|x|bp*(a,b)=(Sμ,a,b)p*(a,b)/(p*(a,b)-p)+O(ε(β(μ)+b)p*(a,b)-N),(3.23)∫Ω|uε|q|x|dp*(a,d)≥{CεN-dp*(a,d)-qδ,N-dp*(a,d)β(μ)<q<p*(a,d),Cεq(β(μ)-δ)|ln⁡ε|,q=N-dp*(a,d)β(μ),Cεq(β(μ)-δ),1≤q<N-dp*(a,d)β(μ),
where b(μ) is given in the introduction satisfying δ=(N-p(a+1))/p<b(μ)<(N-p(a+1))/(p-1).
Now we consider the following functions:(3.24)g(τ)=Jλ(τuε)=τpp‖uε‖p-τp*(a,b)p*(a,b)∫Ω|uε|p*(a,b)|x|bp*(a,b)-λτqq∫Ω|uε|q|x|dp*(a,d),g¯(τ)=τpp‖uε‖p-τp*(a,b)p*(a,b)∫Ω|uε|p*(a,b)|x|bp*(a,b).
Using the definitions ofg and uε, we get
(3.25)g(τ)=Jλ(τuε)≤τpp‖uε‖p,∀τ≥0,λ>0.
Combining this with (3.21), let ε∈(0,1), then there exists τ0∈(0,1) independent of ε such that
(3.26)sup⁡0≤τ≤τ0g(τ)<p*(a,b)-pp*(a,b)p(Sμ,a,b)p*(a,b)/(p*(a,b)-p),∀λ>0,∀ε∈(0,1).
On the other hand, by the fact for B1,B2>0(3.27)max⁡τ≥0(τppB1-τp*(a,b)p*(a,b)B2)=p*(a,b)-pp*(a,b)p(B1)p*(a,b)/(p*(a,b)-p)(B2)-(p/(p*(a,b)-p)),
and by (3.21) and (3.22), we can get that
(3.28)max⁡τ≥0g¯(τ)=p*(a,b)-pp*(a,b)p‖uε‖p*(a,b)/(p*(a,b)-p)(∫Ω|uε|p*(a,b)|x|bp*(a,b))-(p/(p*(a,b)-p))=p*(a,b)-pp*(a,b)p((Sμ,a,b)p*(a,b)/(p*(a,b)-p)+O(εβ(μ)p+p(a+1)-N)p*(a,b)/(p*(a,b)-p)×((Sμ,a,b)p*(a,b)/(p*(a,b)-p)+O(ε(β(μ)+b)p*(a,b)-N))-(p/(p*(a,b)-p))=p*(a,b)-pp*(a,b)p(Sμ,a,b)p*(a,b)/(p*(a,b)-p)+O(εβ(μ)p+p(a+1)-N).
Hence as λ>0, 1≤q<p, by (3.28) we have that
(3.29)sup⁡τ≥τ0g(τ)=sup⁡τ≥τ0(g¯(τ)-λτqq∫Ω|uε|q|x|dp*(a,d))≤p*(a,b)-pp*(a,b)p(Sμ,a,b)p*(a,b)/(p*(a,b)-p)+O(εβ(μ)p+p(a+1)-N)-λτ0qq∫Ω|uε|q|x|dp*(a,d).
(i) If1≤q<(N-dp*(a,d))/β(μ), then by (3.23) we have that
(3.30)∫Ω|uε|q|x|dp*(a,d)≥Cεq(β(μ)-δ)
and since b(μ)>δ=(N-p(a+1))/p, then
(3.31)β(μ)p+p(a+1)-N=p(β(μ)-δ)>q(β(μ)-δ).
Combining this with (3.26) and (3.29), for any λ>0, we can choose ελ small enough such that
(3.32)sup⁡τ≥0Jλ(τuελ)<p*(a,b)-pp*(a,b)p(Sμ,a,b)p*(a,b)/(p*(a,b)-p).
(ii) If(N-dp*(a,d))/β(μ)≤q<p, then by (3.23) and b(μ)>δ=(N-p(a+1))/p we have that
(3.33)∫Ω|uε|q|x|dp*(a,d)≥{CεN-dp*(a,d)-qδ,N-dp*(a,d)β(μ)<q<p*(a,d),Cεq(β(μ)-δ)|ln⁡ε|,q=N-dp*(a,d)β(μ),N-dp*(a,d)-qδ≤qβ(μ)-qδ<p(β(μ)-δ)=β(μ)p+p(a+1)-N.
Combining this with (3.26) and (3.29), for any λ>0, we can choose ελ small enough such that
(3.34)sup⁡τ≥0Jλ(τuελ)<p*(a,b)-pp*(a,b)p(Sμ,a,b)p*(a,b)/(p*(a,b)-p).
From (i) and (ii), (3.20) holds by taking vλ=uελ.
From Lemma2.4, the definition of αλ-, and (3.20), for any λ∈(0,Λ0), we obtain that there exists τλ->0 such that τλ-vλ∈𝒩λ- and
(3.35)αλ-≤Jλ(τλ-vλ)≤sup⁡t≥0Jλ(τvλ)<p*(a,b)-pp*(a,b)p(Sμ,a,b)p*(a,b)/(p*(a,b)-p).
The proof is thus complete.Now, we establish the existence of a local minimum ofJλ on 𝒩λ-.Theorem 3.6.
Assume thatN≥3,  0≤μ<μ̅,  0≤a<(N-p)/p,  a≤b,d<a+1 and 1≤q<p<N. If λ∈(0,(q/p)Λ0), then there exists Uλ∈𝒩λ- such that(i)
Jλ(Uλ)=αλ-,(ii)
Uλ is a positive solution of (1.1).Proof.
Ifλ∈(0,(q/p)Λ0), then by Theorem 2.5 (ii), Proposition 3.2 (ii), and Lemma 3.5, there exists a (PS)αλ--sequence {un}⊂𝒩λ- in W for Jλ with αλ-∈(0,((p*(a,b)-p)/p*(a,b)p)(Sμ,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 thatUλ∈𝒩λ-. Arguing by contradiction, we assume Uλ∈𝒩λ+. Since 𝒩λ- is closed in W, we have ∥Uλ∥<liminfn→∞∥un∥. Thus, by Lemma 2.4, there exists a unique τλ- such that τλ-Uλ∈𝒩λ-. If u∈𝒩λ, then it is easy to see that
(3.36)Jλ(u)=p*(a,b)-pp*(a,b)p‖u‖p-λ(p*(a,b)-qp*(a,b)q)∫Ω|u|q|x|dp*(a,d).
From Remark 2.6, un∈𝒩λ-, ∥Uλ∥<liminfn→∞∥un∥, and (3.36), we can deduce that
(3.37)αλ-≤Jλ(τλ-Uλ)<lim⁡n→∞Jλ(τλ-un)≤lim⁡n→∞Jλ(un)=αλ-.
This is a contradiction. Thus, Uλ∈𝒩λ-.
Next, by the same argument as that in Theorem3.3, we get that un→Uλ strongly in W and Jλ(Uλ)=αλ->0 for all λ∈(0,(q/p)Λ0). 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). Finally, by [5, Lemma  2.3], we obtain that Uλ is a positive solution of (1.1).Now, we complete the proof of Theorem1.1. The part (i) of Theorem 1.1 immediately follows from Theorem 3.3. When 0<λ<(q/p)Λ0<Λ0, by Theorems 3.3 and 3.6, we obtain (1.1) has two positive solutions uλ and Uλ such that uλ∈𝒩λ+, Uλ∈𝒩λ-. Since 𝒩λ+∩𝒩λ-=∅, this implies that uλ and Uλ are distinct. This completes the proof of Theorem 1.1.



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*Source: 579481-2012-03-22.xml*