1 Introduction and Main Result

In this article,we study the following concave-convex elliptic equations involving critical Sobolev exponent

where Ω⊂RN(N≥3)is an open bounded domain with smooth boundary,1＜q＜2,λ＞0.is the critical Sobolev exponent for the embedding ofinto Lp(Ω)for everywhereis a Sobolev space equipped with the normThe coefficient function f ∈ Lr(Ω)is nonzero and nonnegative,whereAnd g ∈ C(Ω)is a positive function.

More precisely,we say that a functionis called a weak solution of problem(1.1),if for allthere holds

where u+=max{u,0}.

It is well known that the pioneer work is Brézis and Nirenberg[4],that is,the existence of positive solutions of semilinear elliptic equations involving critical exponent is related to the dimension of space.After that,semilinear elliptic problems with critical exponent were extensively considered(see[1,2,5,7–30,33,35–37,39–42]).Particularly,Ambrosetti,Brézis,and Cerami[2]studied the following problem

where 0＜q＜1＜ p≤ 2∗−1.They obtained the classic results by the sub-supersolution method and variational method,that is,there exists λ0 ＞ 0 such that problem(1.3)has at least two positive solutions for λ ∈ (0,λ0),a positive solution for λ = λ0,and no positive solution for λ ＞ λ0.After that,many authors considered the concave-convex-type elliptic problems(see[1,9,14,20,21,24–28,35,40]).Particularly,Korman in[22]considered the exact number of positive solutions for problem(1.3)in the unit ball B⊂RN(N≥3).By the bifurcation theory,it is obtained that there exists a critical number＞0 such that problem(1.3)has two positive solutions for λ ∈(0,),exactly one positive solution for λ=,and no positive solution for λ＞.As any positive solution to problem(1.3)in the unit ball B is radial,Tang in[35]proved this result of[22]by an ordinary differential equation method.In[1],Ambrosetti,Azorer,and Peral studied problem(1.3)in RN(N≥3),that is,

近年来，食品安全事件频发，可用于课堂教学的案例数不胜数。无论是三聚氰胺毒奶粉事件还是地沟油事件，以及后来发生的染色馒头、瘦肉精和毒胶囊事件，都在充分说明着将案例教学的重点放在食品添加剂的卫生安全问题、食物中毒及其预防、食品卫生安全监督管理等几个应用性、实践性较强的章节。

Recently,Lin considered problem(1.1)under the following conditions in[28].

(h1)

(h2)There exist k points a1,a2,···,akin Ω such that

电力企业要设置专项资金，投入到安全工器具的管理当中去，建立相应的工器具存放室并配备具有干燥功能的存放柜，使安全工器具的存放有一个良好的环境，同时在人员的配置上要合理，配置的人员要有相关专业的资格证书，对安全工器具的功能、存放特点等方面要十分的了解，专业知识强，技能高。

and for some σ ＞ N such that g(x)−g(ai)=O(|x− ai|σ)as x → aiuniformly in i.(h3)Choosing r0＞0 such thatwhereThere exists a positive number d0such that f(x)≥d0＞0 for any

For the convenience of the reader,we repeat the main results of[28]as follows.

Theorem 1.1 Assume thatand(h1)–(h3)hold,then problem(1.1)has k+1 positive solutions for λ＞0 small enough.Moreover,one of the solutions is a ground state solution.

Moreover,Cao and Chabrowski in[7]considered problem(1.1)with q=1 and obtained the multiplicity of positive solutions.Very recently,we studied problem(1.1)with 2≤ q＜ 2∗and obtained some existence and multiplicity results by the variational method;see[27].

In this article,an interesting question is whether there exist k+1 positive solutions for problem(1.1)without constraining conditions of＜q＜2 and N ＞4.We are interested in the existence of k+1 positive solutions for problem(1.1)with 1＜q＜2 and N≥3.Wefind the reason of restricting ＜q＜2 and N ＞4 in[28]is that the author obtained k positive solutions from the prolongation of the origin.Now,we try to get rid of the constraining conditions of＜q＜2 and N＞4 via obtaining k positive solutions from the prolongation of the first solution.In fact,the first solution is a positive ground state solution of problem(1.1).

Lemma 2.2 Iλis coercive and bounded from below on N.

As well known,the function

is an extremal function for the minimization problem(1.4),that is,it is a positive solution of the following problem

Moreover,

For all u ∈ (Ω),associated with problem(1.1),we define the energy functional Iλ by

采用复式断面作为设计断面时，第二台阶的河道岸墙高度不得大于2.5 m；超过2.5 m时应设置第三台阶。

From the result in[31],one knows that the functional Iλ is of class C1on(Ω).As well known that there exists a one to one correspondence between the nonnegative solutions of problem(1.1)and the critical points of Iλ on (Ω).

In this article,assume that the coefficient functions f,g satisfy the following conditions:

(f)f ∈ Lr(Ω)with f ≥ 0 and f/≡ 0,where

(g)g is continuous on Ω and g＞ 0.

(h′2)There exist k points a1,a2,···,akin Ω satisfying

and moreover,as x→aiuniformly in i∈N+and 1≤i≤k.

Notice that u is a weak solution of problem(1.1),then u satisfies the following equation

So,if such a solution exists,then it must lie in Nehari manifold N,which is defined by

To obtain the multiplicity of positive solutions,we split N=N+∪N0∪N −with N+,N0,N−defined as follows:

This article is organized as follows.In Section 2,we give some preliminaries.We give the proof of existence of a positive ground state solution of problem(1.1)in Section 3.In Section 4,we give the proof of Theorem 1.2.

Here is our main result.

According to Theorem 2.3 in[6],we have the following lemma.

Remark 1.3 To our best knowledge,our results are up to date.Our results complement the case of 1＜q≤ in[28],and we also obtain the same results but without restricting N ＞4.It is worth noticing that the growth condition inthat is,as x→aiuniformly in i∈N+and 1≤i≤k,which is more general than the corresponding condition of(h2)in[28].Moreover,we get rid of the condition(h3)which is important for estimating the level value of functional Iλin Lemma 4.2 of[28].

When λ ∈ (0,Λ),we will prove that N ± /= Ø and N0={0}in Section 2.

where 0＜ q＜ 1,h ∈ L1(RN)∩L∞(RN).When h ≥ 0,h/≡ 0,and λ ＞ 0 small enough,they obtained two positive solutions.Moreover,[9,14]and[20]considered the multiplicity of positive solutions for concave-convex p-Laplacian problems with critical Sobolev exponent.

2 Some Preliminary Results

In this section,we give some lemmas about properties of the functional Iλon N,which are valuable preparation for the proof of our main result.

为方便计算，偏差和偏差变化率的模糊集合采用三角隶属函数，由于输出量的变化范围较小，为了取得更佳的精确度，选用正态函数作为输出的隶属度函数。输入输出变量的语言赋值表见表1、表2。

Lemma 2.1 For eachsuch thatwe have the followings:

(i)Assume thatthen there exists a unique t− =t−(u)＞ tmaxsuch that t−u∈N − and

(ii)Assume thatthen there exist uniquesuch that t+u ∈ N+and t−u ∈ N − for all λ ∈ (0,Λ),and

Furthermore,N0={0}and N−is a closed set for all 0＜λ＜Λ.

Proof According to(g),there existssuch thatFor eachsuch thatFor all t≥0,we define k:R+ →R by

Clearly,we obtain k(0)=0 and k(t)→−∞ as t→∞.Because

then k′(tmax)=0,k′(t)＞ 0 for 0 ＜ t＜ tmax,and k′(t)＜ 0 for t＞ tmax,where

本研究的不足之处是未分析HSP27表达对骨髓瘤患者接受BTZ治疗后疗效和预后的影响，这是后续研究的内容之一。总之，本研究揭示了HSP27通过负反馈机制介导骨髓瘤对BTZ耐药，深化了对骨髓瘤耐药机制的认识，为靶向HSP家族蛋白治疗MM提供了新的治疗靶点和体外实验依据。

Thus,k(t)achieves its maximum at tmax.Moreover,by the Sobolev embedding theorem,one has

Case(i):There is a unique t− ＞ tmax,such that

陈校长不假思索地说：“此次校庆能受到社会各界的广泛支持。许多领导、校友，兄弟学校的代表以及同学家长，纷纷表示校庆活动既隆重又简朴，师生节目呈现出本校良好的精神风貌和才华。这意味着学校为此次校庆所付出的努力得到了大家的认可，令人十分感动！”

Then,we have

and

which implies that t−u∈N −.It follows that

Consequently,one has

Case(ii):By the Hölder and Sobolev inequalities,(1.4),and(2.3),for all 0＜λ＜Λ,we obtain

马兰属植物的化学成分和药理作用及应用研究进展…………………………………………………… 崔 昊等（8）：1149

there exist unique t+and t−such that

for 0＜t+＜tmax＜t−.We have t+u∈N+,t−u∈N − and

for each t∈ [t+,t−],and Iλ(t+u)≤ Iλ(tu)for each t∈ [0,t+].Thus,

Next,we prove that N0={0}for all 0＜λ＜Λ.By contradiction,suppose that there exists u0∈N0with u0/=0.Obviously,u0∈N,it follows that

and

总而言之,肾小球肾炎患者接受优质护理的临床效果比较优秀,患者的总体护理评分比较高,因此在临床中应该进行推广使用。

Consequently,

问：正如您所说，您构建的数学能力模型被广泛地应用到许多研究中，包括PISA测试也是基于您的数学能力模型进行了课程设计．作为数学教育领域内一个重要话题，学生的能力测评也备受关注，中国的教育研究者对于PISA测试也一直很关注，那么您是如何看待国际学生学业水平测试在数学教育领域中的作用呢？

Then,according to(2.3),(2.4),(2.6),and(2.7),for all 0＜λ＜Λ,one has

从表1可知，华东师范大学的季浏老师是发文量最多的作者，达到了17篇，其中有2篇独立完成，15篇合作完成，研究的主题主要是体育课程、绩效评价等。其次是华东师范大学的汪晓赞老师，发文10篇，全是合作完成，她研究的主题是体育课程标准、课程评价等。排名第3的是北京体育大学的姚蕾老师发表7篇，独立完成4篇，研究的主题涉及学校体育、体育教学评价等。由此可知，这些核心作者发文通常是以与他人合作的方式，研究的主题也涉及广泛，但同时涉及了体育教学评价这一研究主题。表1中排名前10的核心作者中有5位作者是属于体育类师范院校，其中华东师范大学有3位作者，因此华东师范大学为体育教学评价领域的研究做出了突出的贡献。

which is a contradiction.Thus,N0={0}for all 0＜λ＜Λ.

Finally,we prove that N−is a closed set for all 0＜λ＜Λ.Assume that{un}⊂N−such that un→u in as n→+∞,then we need prove that u∈N −.As un∈N −,from the definition of N−,one has

Consequently,as un→u in as n→+∞,it follows from(2.8)that

Consequently,for all u∈N,it follows from(2.9)that

which contradicts u=0.Thus,u∈N −for all λ ∈(0,Λ).Therefore,N −is a closed set infor all λ ∈ (0,Λ).This completes the proof of Lemma 2.1.

1．1 一般资料 对2006－2011年在如东县接受儿童系统管理的27 662例3个月内小婴儿听力筛查资料进行调查分析。

We denote by|·|pthe usual Lp-norm.Let S be the best Sobolev constant and Λ be a constant,respectively,

Proof By the Hölder and Sobolev inequalities,one has

thus u∈N − ∪N0.If u∈N0,because N0={0}for all λ ∈(0,Λ),one has u=0.However,from(2.8),byand(1.4),one has

because 1＜ q＜ 2＜ 2∗,which implies that Iλ is coercive and bounded from below on N.Then,the proof of Lemma 2.2 is completed.

For 0＜λ＜Λ,by Lemma 2.1,one has N=N+∪N −∪{0}.Consequently,from Lemma 2.2,the following definitions are well defined

Lemma 2.3 (i) α ≤ α+ ＜0 for all λ ∈(0,Λ);

(ii)there exists a positive constant c0depending on λ,N,S,and|f|r,such that α− ≥ c0 ＞ 0 for all

伴CNS炎性脱髓鞘的原发性干燥综合征8例临床分析 … ……………………… 李晖，李昌茂，张斌，等 134

Proof (i)For all u∈N+,we have

consequently,because of 1＜q＜2 and u/=0,it follows that

which implies that α+ ＜0.Thus,α ≤ α+ ＜0.

(ii)For u ∈ N −,byand the Sobolev embedding theorem,one has

this implies that

农业机械的价格高昂但使用效率却较低。一般来说，农业机械的成本不低，但在具体使用过程中却容易发生各种问题：一是，农机具可靠性较差、质量不过关，在具体使用过程中易发生故障、造成损失；二是，农业机械与玉米种植的需求不完全匹配，大型机械在较小的玉米田地中无法有效运行，再者，不同种植模式下所种植的玉米植株行间距差别较大，而一些机械在收割行间距大于80cm，小于40cm的玉米植株时确实力有不逮；三是，操作和使用者的技术不过硬，无法充分发挥机械设备的功能。上述原因导致部分农户对玉米全程机械化失去信心。

Consequently,from(2.10)and(2.11),we obtain

for allThis implies that there exists a constant c0=c0(λ,N,S,|f|r)＞ 0 such that α− ≥ c0＞ 0 for allThus,the proof of Lemma 2.3 is completed.

Theorem 1.2 Assume that 1 ＜ q ＜ 2,N ≥ 3,and f,g satisfy(f),(g)andthen there exists Λ∗ ＞ 0 such that for λ ∈ (0,Λ∗),problem(1.1)has at least k+1 positive solutions.Moreover,there exists one solution uλ which is a ground state solution with Iλ(uλ) → 0 and‖uλ‖ → 0 as λ → 0+.

Lemma 2.4 Suppose that u0is a local minimizer of Iλon N and 0＜ λ ＜ Λ,then

Proof Suppose that u0is a local minimizer of Iλon N,then u0is a solution of the optimization problem

where

Furthermore,by the theory of Lagrange multipliers,there exists θ∈R such thatAs u0∈N,we get

As 0＜ λ ＜ Λ,from Lemma 2.3,one has u0/∈N0.Consequently,θ=0 and inThis completes the proof of Lemma 2.4.

3 Existence of a Positive Ground State Solution

According to[28]or[38],we show the existence of a(PS)α-sequence and a(PS)α−-sequence in

Lemma 3.1 (i)For all λ ∈ (0,Λ),there exists a(PS)α-sequence{un} ⊂ N in for Iλ.

(ii)For all then we have α− ＞ 0,and there exists a(PS)α−-sequence infor Iλ.

The proof of Lemma 3.1 is similar to the proof of Proposition 9 in[38];the reader can refer to[38].Now,we have the following proposition.

Proposition 3.2 Let λ ∈ (0,Λ),then there exists uλ ∈ N+such that

(i)Iλ(uλ)= α = α+;

(ii)uλ is a positive ground state solution of problem(1.1).Moreover,one has Iλ(uλ)→ 0 and‖uλ‖ → 0 as λ → 0+.

Proof By Lemma 3.1,there exists a minimizing sequence{un}⊂N such that

From Lemma 2.1,then Iλ is coercive on N.It follows that{un}is bounded onGoing if necessary to a subsequence,still denoted by{un},there exists such that

From(3.1),we haveConsequently,combining with(3.2),it follows that

which implies that uλ is a solution of problem(1.1).Particularly,choosing ϕ =uλ in(3.3),one has uλ∈N.Because{un}⊂N,one has

Letting n→∞in(3.4),with(3.1),(3.2),and the fact α＜0,we obtain

Therefore,uλ∈N is a nontrivial solution of problem(1.1).

Next,we prove that un → uλ strongly in as n → ∞ and Iλ(uλ)= α.By Vitali’s theorem(see[32]pp:133),we claim that

Indeed,we only need to prove thatis equi-absolutely-continuous.Noting that{un}is bounded,by the Sobolev embedding theorem,so there exists a constant C1 ＞ 0 such that|un|2∗ ≤ C1 ＜ ∞.By the Hölder inequality,for every ε＞ 0,setting δ＞ 0,when E⊂Ω with mesE＜δ,we have

where the last inequality is from the absolutely-continuity ofRThus,our claim is proved to be true.Because of uλ∈N,by the Fatou Lemma,it follows from(3.5)that

which implies that Iλ(uλ)= α andCombining with(3.2),un ⇀ uλ as n → ∞ in it shows that un → uλ strongly in Moreover,we can claim that uλ ∈ N+.Indeed,if uλ ∈ N −,by Lemma 2.1,there exist uniqueandsuch thatBecause of

there existssuch thatAccording Lemma 2.1,one obtains

which is contradiction.Thus,according to Lemma 2.3(i),Iλ(uλ)= α,and uλ ∈ N+,one has Iλ(uλ)= α = α+.

Finally,we prove that uλ is a positive solution of problem(1.1).In fact,we choosein(3.3),one hasthis implies that =0,so uλ ≥ 0.Consequently,as uλ0,by Lemma 2.4 and the strong maximum principle,then uλis a positive solution of problem(1.1).Therefore,uλis a positive ground state solution of problem(1.1).Moreover,from(2.10)and Lemma 2.4,we obtain

which implies that Iλ(uλ)= α → 0 as λ → 0+.As uλ ∈ N+,from(2.9),one has

which implies.Therefore, ‖uλ‖ → 0 as λ → 0+.Then,the proof of Proposition 3.2 is completed.

4 Proof of Theorem 1.2

In this part,we want to obtain k positive solutions of problem(1.1).To get rid of the constraining conditions of＜q＜2 and N ＞4 in[28],we try to obtain k positive solutions from the prolongation of the first solution uλin N −.Before proving Theorem 1.2,we give some important lemmas.

Lemma 4.1 For allIλ satisfies the(PS)c-condition in

ProofLet{un}⊂be a(PS)c-sequence satisfying Iλ(un)=c+o(1)and=o(1).We claim that{un}is bounded in In fact,for n large enough,one has

which implies that{un}is bounded in So,our claim is true.Therefore,‖un‖ is bounded.Set vn=un − u∗.By Brézis-Lieb’s Lemma,we have

consequently,from(3.5),one has

and

Becausewith(4.1)–(4.3),we obtain

Now,we can assume that

as n→∞.Applying the Sobolev inequality,one obtains

Then,l≥Sl22∗,which implies that either l=0 or l≥ SN2.If l≥ SN2,by(4.4)and(4.5),we have

which is contradicts the definition of c.Therefore,l=0 and un→ u∗strongly inThis completes the proof of Lemma 4.1.

We consider the following critical exponent problem

Associated with(4.6),we consider the energy functional I∞ inthat is

Now,following the methods of[5],let ηi∈be a radially symmetric function such that 0 ≤ ηi≤ 1,|∇ηi|≤ C and for 1 ≤ i≤ k,we define

and

where U(x)is defined as(1.6).

Lemma 4.2 Suppose that(f),(g),andhold,then

where 1≤i≤k.

Proof According to[4]or[16],we can easy to obtain the following classical results

Moreover,one has

where Cl＞0(l=2,3,4,5,6,7)are positive constants.As uλis a positive solution of problem(1.1),one has

Moreover,by a standard method,we get uλ ∈ C1(Ω,R+)and there exists a positive constant C8(C8independent of x)such that uλ ＜ C8.Now,we give the following two elementary inequalities:

For any m＞2,there exists a positive constant C9=C9(m)such that

where M＞0 is a positive constant.

Then,combining with(4.10),(4.12)and(4.11),we easily obtain

where we can assume that≥1 for all t≥and|x − ai|≤ .We denote

Notice that Φε(0)=0, uniformly for all ε.On the one hand,whenone hasThus,we have

uniformly in i.On the other hand,whenthen and it attains for some tε ＞0.So,there exist two constants such that T0＜tε＜T0.In fact,from uniformly for all ε,we choosethen there exists T0＞0 such that

According to the monotonicity of Φε near t=0,we have tε ＞ T0.Similarly,we can obtainLet

Then,we have

Let=0,one has

such that for all 0 ＜ t＜ tmaxand for all t＞ tmax,so Φε,1(t)attains its maximum at tmax.By(g)and,letting ε→ 0+,one has

In fact,for all ε＞ 0,it follows that

Because ofthat is,for all ε ＞ 0,there exists δ＞ 0 such that

When ε＞ 0 small enough,for δ＞ ε12,it follows from(4.14),(g),andthat

where cN=[N(N−2)]N2,and C,C′＞ 0 are constants.Consequently,one has

which implies that

Then,from the arbitrariness of η,we obtain(4.13).Combining with(4.7)and(4.13),one has

Consequently,from(4.8),it follows that

thus

From the boundness ofwith g ＞ 0,and uλ ∈ C1(Ω),there exist three positive constants C10,C11,and C12such that

and from(4.9),one obtains

Consequently,it follows thatTherefore,

where C13＞0 is a constant and 1≤i≤k.

While for all 0＜t＜,it follows from(g),(4.7),(4.8),and(4.11)that

This completes the proof of Lemma 4.2.

According to[36],we have the following lemma.

Lemma 4.3For any 1≤i≤k,there exists such thatN − for all λ ∈(0,Λ).Moreover,

Proof According to Lemma 2.1,there exists u∈ H10(Ω){0},consequently,there exists unique t−(u)＞ 0 such that t−(u)u ∈ N −.Set

and

Then,Thus,and N+⊂H1.In particular,uλ∈N+⊂H1.By simple computations,then there exists a suitable constant M ＞ 0 such that 0 ＜ t−(u)＜ M for u ∈ with ‖u‖ =1.Letwe claim that

for ε ＞ 0 small.Indeed,from(4.10),(4.8),and(4.9),one has

for ε＞0 small.Define h:there existssuch thatMoreover,according to Lemma 4.2,one has

This completes the proof of Lemma 4.3.

From(g)andwe can choose r0 ＞ 0 such that

andwhereLetSuppose thatfor some ρ0 ＞ 0.According to[3],let φ :be a barycenter map defined by

where χ:RN →RN

From Lemma 4.3,there exists such thatfor any 1≤i≤k.Then,we have the following lemma.

Lemma 4.4For any 1≤i≤k,thenMoreover,there exists ε0＞0 such that for all 0＜ ε＜ ε0,thenfor each 1≤i≤k.

Proof According to the definition of φ,we have

as ε→0+.This implies that there exists ε0＞0 such thatfor any 0＜ ε＜ ε0and each 1≤ i≤k.Then,the proof of Lemma 4.4 is completed.

For each 1≤i≤k,we define

Next,we shall show thatare(PS)-values.So,we give the following lemmas.

Lemma 4.5 For all cIλ satisfies the(PS)c-condition in

Proof Assuming the contrary,we can suppose that there exists a sequence{un}⊂N0 such thatand

Then,I∞ has a(PS)-sequence{un}in(see[24]).It follows from Lemma 2.2 that there exist a subsequence{un}and u0∈ such that un ⇀ u0weakly in H10(Ω).BecauseΩ is a bounded domain,is not achieved.Applying the Palais-Smale Decomposition Lemma(see Theorem 3.1 in Struwe[34]),there exist two sequences{zn}⊂ Ω,{Rn}⊂R+,z0∈Ω,and a positive solution v0∈H1(RN)of the critical problem(1.6)with I∞(v0)=1NSN2such that zn→z0,Rn→∞as n→∞and

Let and as n→∞,one hasΩn→RNas n→∞.Then,

Now,we will show that Because of

which impliesandwhich is a contradiction.This completes the proof of Lemma 4.5.

Lemma 4.6 There exists such that if 0＜λ＜Λ∗and u∈N−withis given in Lemma 4.5),then we have φ(u) ∈

Proof Similar to Lemma 2.1,then there is a unique positive number

such that tuu∈N0.Now,we want to prove that tu＜C14for some constant C14＞0(independent of u and λ).From(2.11),if u ∈ N −,then

For λ ∈ (0,),because Iλ is coercive on N andby(2.10),we obtain‖u‖2＜ C16for some constant C16＞ 0.Next,we claim that|u|2∗＞ C17＞ 0,where C17=If u ∈ N −,that is

consequently,fromone obtains

Thus,tu＜C14for some constant C14＞0.Thus,we have

From the above inequality,we deduce that

Hence,there exists such that tuu∈N0and

for 0＜ λ ＜ Λ∗.Consequently,by Lemma 4.5,it follows that

for any 0＜ λ ＜ Λ∗.The proof of Lemma 4.6 is completed.

From the above lemma,one has

By Lemma 4.3,we have

Lemma 4.7For u∈,then there exist τ＞0 and a differentiable functionalsuch that ζ(0)=1,ζ(v)(u−v)∈for any v∈B(0;τ),and

for any ϕ ∈ (Ω).

ProofThe proof is almost the same as in[10]or[38].For each u∈,define a function Fu:R×→R,given by

Then,Fu(1,0)=〉=0 and

According to the implicit function theorem,there exist τ＞ 0 and a differentiable functionsuch that ζ(0)=1,

and

which is equivalent to

That is,ζ(v)(u−v)∈This completes the proof of Lemma 4.7.

Lemma 4.8 For each 1 ≤ i≤ k,λ ∈ (0,Λ∗),there is ainfor Iλ.

Proof For each 1≤i≤k,by(4.15)and(4.16),we obtain

Then,Letbe a minimizing sequence forApplying Ekeland’s variational principle,there exists a subsequencesuch thatand

From(4.17),we may assume thatfor sufficiently large n.By Lemma 4.7,there exist a＞ 0 and a differentiable functionalsuch that =1,for anyLet vσ=σv with‖v‖=1 andThen,andFrom(4.18)and by the mean value theorem,as σ → 0,we obtain

where t0 ∈ (0,1)is a constant and

as σ →0.Hence,

where o(1)→ 0 as σ → 0.From Lemma 4.7,then there exists a positive constant M0such thatfor all n and i.Then,strongly inas n→∞.This completes the proof of Lemma 4.8.

Proof of Theorem 1.2 By Lemma 4.8,there exists a(PS)βλi -sequence{un}⊂infor Iλ and each 1 ≤ i≤ k.As Iλ satisfies the(PS)β-condition forfrom(4.16),then Iλhas at least k critical points in N − for 0＜ λ ＜ Λ∗.It follows that problem(1.1)has k nonnegative solutions inApplying the strong maximum principle,problem(1.1)has at least k positive solutions.Combining with Proposition 3.2,we complete the proof of Theorem 1.2.

References

[1]Ambrosetti A,Azorero J G,Peral I.Elliptic variational problems in RNwith critical growth.J DiffEqu,2000,168:10–32.

[2]Ambrosetti A,Brézis H,Cerami G.Combined effects of concave and convex nonlinearities in some elliptic problems.J Funct Anal,1994,122:519–543

[3]Bahri A,Li Y Y.On a min-max procedure for the existence of a positive solution for certain scalar field equations in RN.Rev Mat Iberoam,1990,6:1–15

[4]Brézis H,Nirenberg L.Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents.Comm Pure Appl Math,1983,36:437–477

[5]Brézis H,Nirenberg L.A minimization problem with critical exponent and nonzero data//Symmetry in Nature.Scuola Normale Superiore Pisa,1989,I:129–140

[6]Brown K J,Zhang Y P.The Nehari manifold for a semilinear elliptic problem with a sign-changing weight function.J DiffEqu,2003,193:481–499

[7]Cao D M,Chabrowski J.Multiple solutions of nonhomogeneous elliptic equation with critical nonlinearity.DiffInteg Equ,1997,10:797–814

[8]Cao D M,Li G B,Zhou H S.Multiple solution for nonhomogeneous elliptic with critical sobolev exponent.Proc Roy Soc Edinburgh Sect A,1994,124:1177–1191

[9]Cao D M,Peng S J,Yan S S.In finitely many solutions for p-Laplacian equation involving critical Sobolev growth.J Funct Anal,2012,262:2861–2902

[10]Cao D M,Zhou H S.Multiple positive solutions of nonhomogeneous semilinear elliptic equations in RN.Proc Roy Soc Edinburgh Sect A,1996,126:443–463

[11]Cerami G,Zhong X X,Zou W M.On some nonlinear elliptic PDEs with Sobolev-Hardy critical exponents and a Li-Lin open problem.Calc Var Partial DiffEqu,2015,54:1793–1829

[12]Chen Y P,Chen J Q.Multiple positive solutions for a semilinear equation with critical exponent and prescribed singularity.Nonlinear Anal,2016,130:121–137

[13]Clapp M,del Pino M,Musso M.Multiple solutions for a non-homogeneous elliptic equation at the critical exponent.Proc Roy Soc Edinburgh Sect A,2004,134:69–87

[14]Drábek P,Huang Y X.Multiplicity of positive solutions for some quasilinear elliptic equation in RNwith critical Sobolev exponent.J DiffEqu,1997,140:106–132

[15]Du Y H,Guo Z M.Finite Morse index solutions of weighted elliptic equations and the critical exponents.Calc Var Partial DiffEqu,2015,54:3161–3181

[16]Ghoussoub N,Yuan C.Multiple solutions for quasilinear PDEs involving the critical Sobolev and Hardy exponents.Trans Amer Math Soc,2000,352:5703–5743

[17]Guo Z Y.Ground states for a nonlinear elliptic equation involving multiple Hardy-Sobolev rritical exponents.Adv Nonlinear Stud,2016,16:333–344

[18]Hirano N,Kim W S.Multiple existence of solutions for a nonhomogeneous elliptic problem with critical exponent on RN.J DiffEqu,2010,249:1799–1816

[19]Hirano N,Micheletti A M,Pistoia A.Multiple exitence of solutions for a nonhomogeneous elliptic problem with critical exponent on RN.Nonlinear Anal,2006,65:501–513

[20]Huang Y S.Multiple positive solutions of nonhomogeneous equations involving the p-Laplacian.Nonlinear Anal,2001,43:905–922

[21]Hsu T S,Lin H L.Three positive solutions for semilinear elliptic problems involving concave and convex nonlinearities.Proc Roy Soc Edinburgh Sect A,2012,142:115–135

[22]Korman P.On uniqueness of positive solutions for a class of semilinear equations.Discrete Contin Dyn Syst,2002,8:865–871

[23]Lan Y Y,Tand C L.Perturbation methods in semilinear elliptic problems involving critical Hardy-Sobolev exponent.Acta Mathematica Scientia,2014,34B:703–712

[24]Li T X,Wu T F.Multiple positive solutions for a Dirichlet problem involving critical Sobolev exponent.J Math Anal Appl,2010,369:245–257

[25]Liao F F,Yang M H,Tang C L.The existence a second positive solution of a class of elliptic equations with concave and convex nonlinearities with a critical exponent(Chinese).J Southwest Univ,2012,34:83–86

[26]Liao J F,Liu J,Zhang P,Tang C L.Existence of two positive solutions for a class of semilinear elliptic equations with singularity and critical exponent.Ann Pol Math,2016,116:273–292

[27]Liao J F,Liu J,Zhang P,Tang C L.Existence and multiplicity of positive solutions for a class of elliptic equations involving critical Sobolev exponents.RACSAM,2016,110:483–501

[28]Lin H L.Positive solutions for nonhomogeneous elliptic equations involving critical Sobolev exponent.Nonlinear Anal,2012,75:2660–2671

[29]Liu X Q,Liu J Q,Wang Z Q.Quasilinear elliptic equations with critical growth via perturbation method.J DiffEqu,2013,254:102–124

[30]Naito Y,Sato T.Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent.Ann Mat Pura Appl,2012,191:25–51

[31]Rabinowitz P H.Minimax methods in critical point theory with applications to differential equations//Regional Conference Series in Mathematics.American Mathematical Society,1986

[32]Rudin W.Real and complex analysis.New York,London etc:McGraw-Hill,1966

[33]Song Y Y,Wu X P,Tang C L.Multiple positive solutions for Robin problem involving critical weighted Hardy-Sobolev exponents with boundary singularities.J Math Anal Appl,2014,414:211–236

[34]Struwe M.Variational Methods(second edition).Berlin,Heidelberg:Springer-Verlag,1996

[35]Tang M.Exact multiplicity for semilinear elliptic Dirichlet problems involving concave and convex nonlinearities.Proc Roy Soc Edinburgh Sect A,2003,133:705–717

[36]Tarantello G.On nonhomogeneous elliptic equations involving critical Sobolev exponent.Ann Inst H Poincare Anal Non Lineaire,1992,9:281–304

[37]Tarantello G.Multiplicity results for an inhomogeneous Neumann problem with critical exponent.Manuscripta Math,1993,18:57–78

[38]Wu T F.On semilinear elliptic equation involving concave-convex nonlinearities and sign-changing weight function.J Math Anal Appl,2006,318:253–270

[39]Wu T F.On the semilinear elliptic equation involving critical exponent and sign-changing weight function.Commun Pure Appl Anal,2008,7:383–405

[40]Wu T F.Three positive solutions for Dirichlet problems involving critical Sobolev exponent and signchanging weight.J DiffEqu,2010,249:1549–1578

[41]Zhang J,Ma S W.In finitely many sign-changing solutions for the Brézis-Nirenberg problem involving Hardy potential.Acta Mathematica Scientia,2016,36B:527–536

[42]Zhang Z.On ground state solutions for quasilinear elliptic equations with a general nonlinearity in the critical growth.J Math Anal Appl,2013,401:232–241