亚洲免费av电影一区二区三区,日韩爱爱视频,51精品视频一区二区三区,91视频爱爱,日韩欧美在线播放视频,中文字幕少妇AV,亚洲电影中文字幕,久久久久亚洲av成人网址,久久综合视频网站,国产在线不卡免费播放

        ?

        MULTIPLICITY RESULTS FOR A NONLINEAR ELLIPTIC PROBLEM INVOLVING THE FRACTIONAL LAPLACIAN?

        2017-01-21 05:31:17YongqiangXU許勇強(qiáng)

        Yongqiang XU(許勇強(qiáng))

        Department of Mechanical and Electrical Engineering,Xiamen University,Xiamen 361005,China; School of Mathematics and Statistics,Minnan Normal University,Zhangzhou 363000,China

        Zhong TAN(譚忠)

        School of Mathematical Sciences,Xiamen University,Xiamen 361005,China

        Daoheng SUN(孫道恒)

        Department of Mechanical and Electrical Engineering,Xiamen University,Xiamen 361005,China

        MULTIPLICITY RESULTS FOR A NONLINEAR ELLIPTIC PROBLEM INVOLVING THE FRACTIONAL LAPLACIAN?

        Yongqiang XU(許勇強(qiáng))

        Department of Mechanical and Electrical Engineering,Xiamen University,Xiamen 361005,China; School of Mathematics and Statistics,Minnan Normal University,Zhangzhou 363000,China

        E-mail:yqx458@126.com

        Zhong TAN(譚忠)

        School of Mathematical Sciences,Xiamen University,Xiamen 361005,China

        E-mail:ztan85@163.com

        Daoheng SUN(孫道恒)

        Department of Mechanical and Electrical Engineering,Xiamen University,Xiamen 361005,China

        E-mail:sundh@xmu.edu.cn

        In this paper,we consider a class of superlinear elliptic problems involving fractional Laplacian(??)s/2u=λf(u)in a bounded smooth domain with zero Dirichlet boundary condition.We use the method on harmonic extension to study the dependence of the number of sign-changing solutions on the parameter λ.

        fractional Laplacian;existence;asymptotic;Sobolev trace inequality

        2010 MR Subject Classifcation35J99;45E10;45G05

        1 Introduction

        Problems of the type

        for diferent kind of nonlinearities f,were the main subject of investigation in past decades.See for example the list[2,4,5,10,14,16,17].Specially,in 1878,Rabinowitz[14]gave multiplicity results of(1.1)for any positive parameter λ as n=1.But he found that the number of solutions of(1.1)is independent on λ.Under some conditions on f,Costa and Wang[5]proved that the number of signed and sign-changing solutions is dependent on the parameter λ as n≥1.

        Recently,fractional Laplacians attracted much interest in nonlinear analysis.Cafarelli et al.[7,8]studied a free boundary problem.Since the work of Cafarelli and Silvestre[9],whointroduced the s-harmonic extension to defne the fractional Laplacian operator,several results of version of the classical elliptic problems were obtained,one can see[3,6]and their references.

        In this paper,we consider the nonlinear elliptic problem involving the fractional Laplacian power of the Dirichlet Laplacian

        where ??Rn(n≥2)is a bounded domain with smooth boundary??,λ is a positive parameter, s∈(0,2),(??)s/2stands for the fractional Laplacian,and f:R→R satisfes:

        For the defnition of fractional Laplacian operator we follow some idea of[3].In particular, we defne the eigenvalues ρkof(??)s/2as the power s/2 of the eigenvalues λkof(??),i.e., ρk=λs/2kboth with zero Dirichlet boundary data.

        Let N(λ)be the number of sign-changing solutions of(Pλ).Our main result is the following theorem.

        2 Preliminaries

        Denote the half cylinder with base on a bounded smooth domain ? by

        and its lateral boundary by

        Denote H?s/2(?)the dual space of Hs/20(?).(??)s/2is given by

        Associated to problem(Pλ),the corresponding energy functional I1:Hs/20(?)→ R is defned as follows:

        Defnition 2.1We say that u∈Hs/20(?)is a weak solution of(Pλ)if

        So if δ>0 small enough,there exists Cδ>0 such that

        Let δ>0 be small enough such that

        and(2.4)is satisfed.Let βδbe a C∞function satisfying that βδ=1 if|t|≤δ,βδ=0 if |t|≥2δ,and 0≤βδ≤1,for any t∈R.Defne

        and consider the following equation

        Combining(Pλ)with(2.6),through direct calculation,we have the following lemma.

        Lemma 2.1If w is a solution of(Qδ,λ)and

        then u(x)=λ?1/(p?2)w(x),x∈? is a solution of Pλ.

        To treat the nonlocal Qδ,λ,we will study a corresponding extension problem in one more dimension,which allows us to investigate Qδ,λby studying a local problem via classical nonlinear variational methods.

        For any regular function u,the fractional Laplacian(??)s/2acting on u is defned by

        In fact the extension technique is developed originally for the fractional Laplacian defned in the whole space[9],and the corresponding functional spaces are well defned on the homogeneous fractional Sobolev spaceand the weighted Sobolev spaceIf φ is smooth enough,it can be computed by the following singular integral:where P.V.is the principal value and cn,s/2is a normalization constant.And it is obtained, from[9],that formula(2.8)for the fractional Laplacian in the whole space equivalent to that obtained from Fourier transform(i.e.,the fractional Laplacian(??)s/2of a function φ∈S is defned by

        where S denotes the Schwartz space of rapidly decreasing C∞function in Rn,F is the Fourier transform).

        With this extension,we can reformulate our problem(Qδ,λ)as

        Defnition 2.2We say that u∈Hs/20(?)is an energy solution of problem(Qδ,λ)if u=tr?w,wheresatisfes

        The corresponding energy functional is defned by

        In the following,we collect some results of the space

        Lemma 2.2(see[3]) Let n≥s and 2#=2n

        n?s.Then there exists a constant C,depending only n,such that,for all ω∈Hs/20,L(C),

        By H¨older’s inequality,since ? is bounded,the above lemma leads to:

        Lemma 2.3(see[3]) (i) Let 1≤ q≤ 2#for n≥ s.Then,we have that for all

        where C depends only on n,q and the measure of ?.

        Lemma 2.4(see[3])

        3 Proof of Theorem 1.1

        and

        The following lemma is an elliptic regularity result,which is crucial in our proof.That is, we deduce the regularity of bounded weak solutions to the nonlinear problem

        Lemma 3.1Assume n≥2.Let q∈C(R)andfor some constantis a weak solution of the nonlinear problem(Qs),then there exists C=C(p,L,n)>0 such that

        ProofAs before,the precise meaning for(Qs)is that w∈Hs/20,L(C),w(·,0)=u,and w is a weak solution of

        Denote

        By direct computation,we see

        Multiplying(3.3)by ?β,Tand integrating by parts,we obtain

        Combining(3.4)and(3.5),we have

        On the other hand,

        where C1dependent on n and q,and q>p.

        From(3.6)–(3.7),we have

        Let T→∞,we get

        So,we have

        where

        Let l=p(β+1),then

        By Sobolev inequality,we have

        So,we fnish the proof of Lemma 3.1.

        Defne

        and

        By(2.5)and(3.1),we have

        Lemma 3.2Under assumption(F),the functionals I and Iδ,λsatisfy(PS)cconditions.

        ProofWe just prove the case that Iδ,λsatisfes(PS)cconditions.The other case can be obtained similarly.Assume that there exists a(PS)csequence{uk}?Hs/20(?),i.e.,

        By(F),we get

        which implies that wk→w0in Hs/20(?),as k→∞.Using the same method,we can prove that I also satisfes(PS)ccondition.

        Proof of Theorem 1.1In order to employ the method from[12],we defne,on E(=

        which is a closed convex cone.From[1](Theorem 7.38),we know that the Banach spaceis densely embedded inand

        is a closed convex cone in X.Furthermore,P=P?∪P under the topology of X,i.e.,there exist interior points in P.So,as in[12],we may defne a partial order relation in X:u,v∈X,u>v?u?v∈P{0};u?v?u?v∈P?.We also defne W=P∪(?P).

        Defne

        we obtain the similar deformation lemma.

        Lemma 3.3Fix c≥0 and ε∈(0,14].Then,there exists a homeomorphism map η:such that

        ProofFirst,due to(PS)ccondition,we can choose a constant ε>0 such that

        Let

        and

        where

        Then ψ(u)is locally Lipschitz on E,consider

        Since kf(ξ(t,u))k≤1 for all ξ(t,u)the Cauchy problem(3.22)has a unique solution ξ(t,u)continuous on R×E.Follow the argument as in[13],we can obtain that there exist T>0 such that η(u)=ξ(T,u)satisfes the conclusion.

        Denote by 0<λ1<λ2≤λ3≤···all the eigenvalues of??in ? with zero Dirichlet boundary condition and by e1,e2,e3,···the corresponding eigenfunctions,with the explicit meaning that each λiis counted as many times as its multiplicity.

        Denote

        Let ri>0 be such that ri+1>rifor i=1,2,···,ri→∞(i→∞)and

        Let

        and?Bkbe the boundary of Bkin Xk.Defne a sequence{Λk}of functions inductively as

        and for k=2,3,···

        Defne,for k=1,2,···,

        Using the similar method as(Proposition 5.2,[11]),we can obtain that when n≥ 2, there exist non-positive constants C and D such thatfor k∈N,where γ=(s/n)p(p?2)?1.

        and

        Thus,

        So,

        On the other hand,using the similar method from[15],we have

        where M is a constant dependent on p,n and ?,which is a contradiction.

        Thus for n≥2,there exists a sequence{kj}?N such that

        For j=1,2,···,defne

        By(3.23),it is easy to deduce that

        Claim 1

        and by(3.19),

        Claim 2is a critical value of Iδ,λ.

        It contradicts to the defnition of

        If ujis a critical point of Iδ,λandthen

        and

        Combining(3.25)–(3.26),(2.5),(3.1)and Claim 1,we have

        Since

        and dkjis independent of δ and λ,by Lemma 3.1,we know that for any j∈N,there is λj>0 such that for any λ≥λj,

        Then by Lemma 2.1,and using the similar argument as[12],we can prove that when λ≥λjandis a sign-changing solution of(Pλ).Thus

        [1]Adams R A.Sobolev Spaces.New York:Academic Press,1975

        [2]Bahri A,Lions P L.Solutions of superlinear elliptic equations and their Morse indices.Comm Pure Appl Math,1992,45:1205–1215

        [3]Barrios B,Colorado E,de Pablo A,S′anchez U.On some critical problems for the fractinal Laplacian operator.J Difer Equ,2012,252:6133–6162

        [4]Cao D M.Multiple positive solutions of inhomogeneous semilinear elliptic equations unbounded damain in R2.Acta Math Sci,1994,14(2):297–312

        [5]Costa D G.Multiplicity results for a class of superlinear elliptic problems.Amer Math Soc,1993,48: 137–151

        [6]Cabr′e X,Tan J.Positive solutions of nonlinear problems involving the square root of the Laplacian.Adv Math,2010,224:2052–2093

        [7]Cafarelli L,Roquejofre J M,Sire Y.Variational problems for free boundaries for the fractional Laplacian. J Eur Math Soc,2010,12:1151–1179

        [8]Cafarelli L,Salsa S,Silvestre L.Regularity estimates for the fractional Laplacian.Invent Math,2008,171: 425–461

        [9]Cafarelli L,Silvestre L.An extension problem related to the fractional Laplacian.Commun Part Difer Equ,2007,32:1245–1260

        [10]Deng Y B,Li Y,Zhao X J.Multiple solutions for an inhomogeneous semilinear elliptic equation in Rn. Acta Math Sci,2003,23(1):1–15

        [11]Ekeland I,Choussoub N.Selected new aspects of the calculus of variations in the large.Bull Amer Math Soc,2002,39:207–265

        [12]Li S J,Wang Z Q.Ljusternik-Sohnirelman theorey in partially order Hilber space.Trans Amer Math Soc, 2002,354:3207–3227

        [13]Liu Z L,Wang Z-Q.Sign-changing solutions of nonlinear elliptic equations.Front Math China,2008,3: 1–18

        [14]Rabinowitz P H.Some minimax theorems and applications to nonlinear PDE//Nonlinear Analysis.Acad Press,1978:161–177

        [15]Tan J.Positive solutions for non local elliptic problems.Disc Cont Dyn Syst,2013,33:837–859

        [16]Wang Z-Q.On a superlinear elliptic equation.Anal Nonl,1991,8:43–58

        [17]Wang Z-Q.Multiplicity results for a class of superlinear elliptic problems.Amer Math Soc,2004,133: 787–794

        ?Received April 4,2015;revised May 12,2016.This research was supported by China Postdoctoral Science Foundation Funded Project(2016M592088)and National Natural Science Foundation of China-NSAF (11271305).

        人妻少妇偷人精品免费看| 亚洲欧美国产精品久久久| 色中文字幕视频在线观看| av在线播放免费网站| 青草网在线观看| av天堂一区二区三区精品| 日本高清乱码中文字幕| 巨胸喷奶水www视频网站| 国产精品11p| 久久综合激激的五月天| 久久黄色国产精品一区视频| 欧美乱人伦人妻中文字幕| 真实国产老熟女粗口对白| 国产偷国产偷亚洲欧美高清| 国产高清大片一级黄色| 亚洲精品无码不卡在线播he| 亚洲av高清在线观看三区| 日韩有码中文字幕av| 亚洲国产精品一区二区久久恐怖片| 精品少妇一区二区三区免费观 | 欧美日韩在线免费看| 日韩在线手机专区av | 男人和女人做爽爽视频| 国产精品国产三级国av| 在线播放a欧美专区一区| 亚洲中文字幕有综合久久| 国产精品国产三级国产专播下| а天堂中文在线官网| 国产免费一级在线观看| 免费观看视频在线播放| 色播视频在线观看麻豆 | 啊v在线视频| 国产精品美女自在线观看| 午夜久久久久久禁播电影| 福利视频一二三在线观看| 级毛片无码av| 日本一二三四高清在线| 野狼第一精品社区| 久草热这里只有精品在线| 一区二区三区四区黄色av网站 | 久久久久亚洲av无码专区体验|