• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    EXISTENCE OF SOULUTIONS FOR THE FRACTIONAL (p,q)-LAOLACIAN PROBLEMS INVOLVING A CRITICL SOBLEV EXPNN*

    2021-01-07 06:42:34FanfanCHEN帆帆YangTANG楊陽
    關(guān)鍵詞:楊陽

    Fanfan CHEN(陳 帆帆) Yang TANG(楊陽)?

    School of science,Jiangnan University,Wuxi 214122,China E-mail:1459172465@qq.com;yynjnu@126.com

    which can be found in[6]on a bounded domain??RN.On the other hand,the eigenvalue problem for a Laplacian type equation withp=2 was investigated in[12].For the critical(p,q)-Laplacian proble m

    wheref(u)is a superlinear continuous function with subcritical growth.It was proved in[1]that problem(1.3)has a nontrivial solution forλsufficiently large.

    Recently,equations for a fractional(p,q)-Laplacian operator with critical Sobolev-Hardy exponents have attracted more attention.For instance,authors of[2](see also[21])studied the following problem:

    Here,they proved infinitely many weak solutions of the problem(1.5)for some range ofλ,θ.Also,forV(x)≡1,λ=0,and assuming certain other conditions onN,q,r,they proved the existence ofcat?(?)nonnegative solutions by using the Lusternik-Schnirelmann category theory.Later in[16],Goel et al.studied the nonlinear doubly nonlocal equation

    We generalize the results of[10]to the fractional(p,q)-Laplacian case.Using the auxiliary functions in[18],we obtain some new asymptotic estimates(see Lemma 2.5).This is also the novelty of this article.The main results of the article are the following:

    Theorem 1.1Let 1

    (i)N(p?1)/(N?ps1)≤q<(N?ps1)p/N;

    (ii)N(p?1)/(N?s1)

    (iii)(1?s1/N)p2s1+ps1

    (iv)(p?1)p2s1/(N?ps1)

    which implies thatu?=0.Thus,u=u+is a critical point ofI(u)and a nonnegative solution of problem(Pμ,λ).

    This article is organized as follows:in Section 2,we give some notations and preliminaries.Section 3 and Section 4 deal with the proof of Theorem 1.1 and Theorem 1.2,respectively.

    2 Preliminaries

    In this section,we give some valuable results which will be useful in the study of our problem.Fors∈(0,1),we define

    Denoting byp′=p/(p?1)the H¨older conjugate ofp,|uj(x)?uj(y)|p?2(uj(x)?uj(y))/|x?y|(N+s1p)/p′is bounded inLp′(R2N),and converges to|u(x)?u(y)|p?2(u(x)?u(y))/|x?y|(N+s1p)/p′a.e.in R2N,and(v(x)?v(y))/|x?y|(N+s1p)/p∈Lp(R2N).Therefore,the first integral in(2.4)converges to

    4 Proof of Theorem 1.2

    猜你喜歡
    楊陽
    Analytical formula describing the non-saturating linear magnetoresistance in inhomogeneous conductors
    我愛洋甘菊
    職場高手請(qǐng)“兼職”,請(qǐng)出大驚喜
    Accuracy improvement of quantitative analysis of calorific value of coal by combining support vector machine and partial least square methods in laserinduced breakdown spectroscopy
    A potential germ cell-speci fic marker in Japanese flounder,Paralichthys olivaceus: identi fication and characterization of lymphocyte antigen 75 (Ly75/CD205)*
    數(shù)學(xué)的簡潔美
    東方教育(2017年17期)2017-10-31 19:10:58
    A Historical View on Translation Theory and Practice in China
    我是你的王子
    Experimental study of breach process of landslide dams by overtopping and its initiation mechanisms*
    有朋自遠(yuǎn)方來
    三月三(2015年5期)2015-05-05 16:34:22
    汝城县| 莎车县| 霍山县| 富锦市| 耒阳市| 临邑县| 夏津县| 易门县| 中卫市| 英吉沙县| 苏州市| 桑日县| 行唐县| 淅川县| 宁波市| 社会| 云安县| 宿松县| 武功县| 唐河县| 民权县| 如皋市| 长兴县| 沐川县| 慈利县| 黎城县| 嘉义市| 锡林郭勒盟| 台中市| 南雄市| 吉安市| 错那县| 岳普湖县| 大安市| 武夷山市| 肃北| 鄱阳县| 永兴县| 扬中市| 马龙县| 遂川县|