摘要本文研究如下带有临界增长的分数阶Kirchhoff方程ε2sM(ε2s-3 ∫∫R3×R3·(|u(x)?u(y)|2)/(|x-y|3+2s)dxdy)(-Δ)su+V(x)u=λW(x)f(u)+K(x)|u|2s*-2u,x ∈ R3,其中M是一个连续正的Kirchhoff函数,λ>0是一个参数,3/4<s<1,2s*:=6/(3-2s)是3维的临界指数,并且V(x),W(x)和K(x)都是正位势函数.在Kirchhoff函数M和位势函数的适当假设下,当ε>0充分小和λ足够大时,我们首先证明了上述问题正基态解的存在性.其次,证明了基态解集中在一个由位势函数所刻画的特定集合中.最后,研究了基态解的衰减估计. | | 服务 | | | 加入引用管理器 | | E-mail Alert | | RSS | 收稿日期: 2020-03-25 | | 基金资助:国家自然科学基金资助项目(11771385)
| 通讯作者:赵富坤,E-mail:fukunzhao@163.comE-mail: fukunzhao@163.com |
[1] Ambrosio V., Isernia T., Concentration phenomena for a fractional schr? dinger-kirchhoff type equation, Math. Meth. Appl. Sci., 41, 2018. [2] Autuori G., Fiscella A., Pucci P., Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Potential Anal., 2015, 125:699-714. [3] Caponi M., Pucci P., Existence theorems for entire solutions of stationary Kirchhoff fractional p-Laplacian equations, Ann. Mat. Pura Appl., 2016, 195(6):2099-2129. [4] Chen G., Zheng Y., Concentration phenomenon for fractional nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 2014, 13(6):2359-2376. [5] Corrêa S., Figueiredo G., On an elliptic equation of p-Kirchhoff type via variational methods, Bull. Austral. Math. Soc., 2006, 74(2):263-277. [6] D'Ancona P., Spagnolo S., Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 1992, 108(2):247-262. [7] Deng Y., Peng S., Shuai W., Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in R3, J. Funct. Anal., 2015, 269(11):3500-3527. [8] Di Nezza E., Palatucci G., Valdinoci E., Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 2012, 136(5):521-573. [9] Ding Y., Liu X., Semi-classical limits of ground states of a nonlinear Dirac equation, J. Differential Equations, 2012, 252:4962-498. [10] Ding Y., Liu X., Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities, Manuscripta Math., 2013, 140(1-2):51-82. [11] Ding Y., Wei J., Multiplicity of semiclassical solutions to nonlinear Schrödinger equations, J. Fixed Point Theory Appl., 2017, 19(1):987-1010. [12] Dipierro M., Medina S., Valdinoci E., Fractional elliptic problems with critical growth in the whole of Rn, volume 15 of Appunti. Scuola Normale Superiore di Pisa (Nuova Serie)[Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], Edizioni della Normale, Pisa, 2017. [13] Felmer P., Quaas A., Tan J., Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 2012, 142(6):1237-1262. [14] Figueiredo G., Ikoma N., Santos Júnior J., Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Ration. Mech. Anal. 2014, 213(3):931-979. [15] Figueiredo G., Santos Júnior J., Multiplicity and concentration behavior of positive solutions for a Schrödinger-Kirchhoff type problem via penalization method, ESAIM Control Optim. Calc. Var. 2014, 20(2):389-415. [16] Fiscella A., Valdinoci E., A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 2014, 94:156-1706. [17] Gu G., Wu X., Yu Y., et al., Multiplicity and concentration behavior of positive solutions for a fractional kirchhoff equation in R3 (in Chinese), Sci. Sin. Math., 2018:??,??. [18] He X., Zou W., Multiplicity of concentrating solutions for a class of fractional kirchhoff equation, Manuscripta Math. https://doi.org/10.1007/s00229-018-1017-0. [19] He X., Zou W., Existence and concentration behavior of positive solutions for a Kirchhoff equation in R3, J. Differential Equations, 2012, 252(2):1813-1834. [20] He X., Zou W., Ground states for nonlinear Kirchhoff equations with critical growth, Ann. Mat. Pura Appl., 2014, 193(2):473-500. [21] He Y., Li G., Standing waves for a class of Kirchhoff type problems in R3 involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 2015, 54(3):3067-3106. [22] He Y., Li G., Peng S., Concentrating bound states for Kirchhoff type problems in R3 involving critical Sobolev exponents, Adv. Nonlinear Stud., 2014, 14(2):483-510. [23] Kirchhoff G., Vorlesungenüber Mechanik, Birkhäuser, Basel, 1883. [24] Lions L. J., On some questions in boundary value problems of mathematical physics, 1978, 30:284-346. [25] Ma T., Muñoz J., Rivera J. E., Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett., 2003, 16(2):243-248. [26] Mao A., Zhang Z., Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., 2009, 70(3):1275-1287. [27] Molica Bisci G., Radulescu V., Servadei R., Variational methods for nonlocal fractional problems, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2016. [28] Moser J., A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math., 1960, 13:457-468. [29] Di Nezza E., Palatucci G., Valdinoci E., Hitchhiker's guide to the fractional sobolev spaces, Bull. des Sci. Math., 2012, 136:521-573. [30] Ono K., Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings, J. Differential Equations, 1997, 137(2):273-301. [31] Perera K., Zhang Z., Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations, 2006, 221(1):246-255. [32] Pucci P., Saldi S., Critical stationary Kirchhoff equations in RN involving nonlocal operators, Rev. Mat. Iberoam., 2016, 32(1):1-22. [33] Pucci P., Xiang M., Zhang B., Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in RN, Calc. Var. Partial Differential Equations, 2015, 54(3):27852806. [34] Pucci P., Xiang M., Zhang B., Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations, Adv. Nonlinear Anal., 2016, 5(1):27-55. [35] Rabinowitz H. P., On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 1992, 43(2):270-291. [36] Rabinowitz H. P., Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 2007, 60(1):67-112. [37] Wang J., Tian L., Xu J., et al., Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations, 2012, 253(7):2314-2351. [38] Willem M., Minimax Theorems, Birkhäuser, Boston, 1996. [39] Wu X., Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in RN, Nonlinear Anal. Real World Appl., 2011, 12(2):1278-1287. [40] Yu Y., Zhao F., Zhao L., The concentration behavior of ground state solutions for a fractional SchrödingerPoisson system, Calc. Var. Partial Differential Equations, 2017, 56(4):116-125.
|
[1] | 王文波, 周见文, 李永昆, 李全清. 临界或超临界增长分数阶Schrödinger—Poisson方程正解的存在性[J]. 数学学报, 2021, 64(2): 269-280. | [2] | 沈烈军. 带有临界增长的Kirchhoff型问题的基态解[J]. 数学学报, 2018, 61(2): 197-216. | [3] | 夏滨. 带逆平方势的非线性Schrödinger方程的有限时间性态[J]. 数学学报, 2017, 60(5): 799-814. | [4] | 郑神州章腊萍. 次临界增长$P$-调和组的处处内部正则性[J]. Acta Mathematica Sinica, English Series, 2008, 51(5): 1001-101. | [5] | 李周欣;沈尧天;. 含临界指数的类p-Laplacian方程无穷多解的存在性[J]. Acta Mathematica Sinica, English Series, 2008, 51(4): 663-670. | [6] | 李晓光;张健;. 二维空间中一类具临界幂的耦合非线性波动系统的爆破解[J]. Acta Mathematica Sinica, English Series, 2008, 51(4): 769-778. | [7] | 张平正;. 非线性Schrdinger方程基态解的一致集中[J]. Acta Mathematica Sinica, English Series, 2008, 51(1): 165-170. | [8] | 李晓光;张健;. 带调和势的非线性Schrdinger方程爆破解的L~2集中率[J]. Acta Mathematica Sinica, English Series, 2006, 49(4): 909-914. | [9] | 耿堤. 含非对称临界非线性项的p-Laplace方程的多解问题[J]. Acta Mathematica Sinica, English Series, 2004, 47(4): 751-762. | [10] | 冉启康;方爱农. R~N上临界增长的椭圆方程无穷多解的存在性[J]. Acta Mathematica Sinica, English Series, 2002, 45(4): 773-782. | [11] | 余澍祥. 二维流形上周期解的存在性[J]. Acta Mathematica Sinica, English Series, 1980, 23(5): 712-719. | [12] | 陈兰荪;王明淑. 二次系统极限环的相对位置与个数[J]. Acta Mathematica Sinica, English Series, 1979, 22(6): 751-758. |
|
PDF全文下载地址:
http://www.actamath.com/Jwk_sxxb_cn/CN/article/downloadArticleFile.do?attachType=PDF&id=23722
非齐型空间上分数型Marcinkiewicz积分算子的加权估计林海波,王宸雁中国农业大学理学院北京100083WeightedEstimatesforFractionalTypeMarcinkiewiczIntegralOperatorsonNon-homogeneousSpacesHaiBoLIN ... 中科院数学与系统科学研究院 本站小编 Free考研考试 2021-12-27Zalcman引理在随机迭代函数族动力系统中的应用黄小杰1,2,刘芝秀11南昌工程学院理学院江西330099;2复旦大学计算机科学技术学院上海200433AnApplicationofZalcmanLemmainDynamicalSystemsofRandomIteratedFunctionFami ... 中科院数学与系统科学研究院 本站小编 Free考研考试 2021-12-27变量核奇异积分和分数次微分加权范不等式杨沿奇,陶双平西北师范大学数学与统计学院兰州730070WeightedNormInequalitiesofVariableSingularIntegralsandFractionalDifferentiationYanQiYANG,ShuangPingTAOC ... 中科院数学与系统科学研究院 本站小编 Free考研考试 2021-12-27分数阶Schrdinger-Kirchhoff方程无穷多高能量解的存在性徐家发1,刘立山2,蒋继强21.重庆师范大学数学科学学院重庆401331;2.曲阜师范大学数学科学学院曲阜273165ExistenceofInfinitelyManyHighEnergySolutionsforFr ... 中科院数学与系统科学研究院 本站小编 Free考研考试 2021-12-27伪自伴量子系统的酉演化与绝热定理黄永峰1,2,曹怀信1,王文华31陕西师范大学数学与信息科学学院西安710119;2昌吉学院数学系昌吉831100;3陕西师范大学民族教育学院西安710119UnitaryEvolutionandAdiabaticTheoremofPseudoSelf-adjoint ... 中科院数学与系统科学研究院 本站小编 Free考研考试 2021-12-27一类次线性弱耦合系统无穷多个周期解的存在性王超盐城师范学院数学与统计学院盐城224002TheExistenceofInfinitePeriodicSolutionsofaClassofSub-linearSystemswithWeakCouplingChaoWANGSchoolofMathemat ... 中科院数学与系统科学研究院 本站小编 Free考研考试 2021-12-27具年龄结构和非局部扩散的三种群Lotka-Volterra竞争合作系统行波解稳定性张丽娟,霍振香,任晴晴,王福昌防灾科技学院,廊坊065201StabilityoftheTravelingWaveSolutionsforThreeSpeciesLotka-VolterraCompetitive-co ... 中科院数学与系统科学研究院 本站小编 Free考研考试 2021-12-27带有非紧条件的拟线性Schrdinger-Poisson系统非平凡解的存在性陈丽珍1,冯晓晶2,李刚31.山西财经大学应用数学学院,太原,030006;2.山西大学数学科学学院,太原,030006;3.扬州大学数学科学学院,扬州,225002TheExistenceofNontrivia ... 中科院数学与系统科学研究院 本站小编 Free考研考试 2021-12-27双参数奇摄动非线性抛物型系统的广义尖层解冯依虎1,2,侯磊2,莫嘉琪31.亳州学院电子与信息工程系,亳州236800;2.上海大学数学系,上海200436;3.安徽师范大学数学与统计学院,芜湖241003TheGeneralizedSpikeLayerSolutiontoSingularPertur ... 中科院数学与系统科学研究院 本站小编 Free考研考试 2021-12-27时标上二阶拟线性延迟阻尼动态系统的动力学行为分析李继猛1,杨甲山21.邵阳学院理学院,邵阳422004;2.梧州学院大数据与软件工程学院,梧州543002DynamicalBehaviorofSecond-orderQuasilinearDelayDampedDynamicEquationsonTi ... 中科院数学与系统科学研究院 本站小编 Free考研考试 2021-12-27
|