您好, 访客   登录/注册

带有临界增长的Kirchhoff方程极小能量变号解的存在性

来源:用户上传      作者:

  摘 要:为了深入研究Kirchhoff方程的性质,讨论了带有Hartree项和临界增长非线性项的Kirchhoff方程极小能量变号解的存在性。利用能量泛函在变号Nehari流形上的下确界Cλ收敛于0,得到空间E紧嵌入L6(R3)这一技术性结果。结果表明,利用限制变分方法和定量形变引理获得极小化序列对应的极小值点是该问题的非平凡解。研究方法在理论证明方面得到了良好的结果,对研究其他Kirchhoff方程解的存在性有一定的指导意义。
  关键词:非线性泛函分析;Kirchhoff方程;Hartree非线性项;临界增长;变分方法;变号解
  中图分类号:O175 文献标识码:A
  文章编号:1008-1542(2020)04-0327-07
  doi:10.7535/hbkd.2020yx04005
  3 结 语
  本文研究了带有Hartree项和临界增长非线性项的Kirchhoff方程极小能量变号解的存在性。利用变分方法和精细的分析技巧获得了紧嵌入的结果。然而,文中的理论证明是在特定的势函数、非线性项和一个参数下进行的,并未考虑其他势函数和多参数的情形。未来研究中,将尝试解决这类问题解的存在性以及解的渐进行为。
  参考文献/References:
  [1] BENCI V, FORTUNATO D. An eigenvalue problem for the Schrdinger-Maxwell equations[J]. Topological Methods in Nonlinear Analysis,1998,11(2):283-293.
  [2] SHI Qihong, PENG Congming.Wellposedness for semirelativistic Schrdinger equation with power-type non-linearity[J]. Nonlinear Analysis: Theory, Methods & Applications, 2019, 178:133-144.
  [3] SHI Qihong, WANG Shu. Klein-Gordon-Zakharov system in energy space: Blow-up profile and subsonic limit[J]. Mathematical Methods in the Applied Sciences, 2019, 42(9):3211-3221.
  [4] 巴振, 贺小明.一类Kirchhoff-Poisson方程变号解的存在性[J].中央民族大学学报(自然科学版), 2015, 24(2):82-87.
  BA Zhen, HE Xiaoming. Existence of sign-changing solutions for a class of Kirchhoff-Poisson equation in bounded domains[J]. Journal of MUC (Natural Science Edition),2015, 24(2):82-87.
  [5] 郝亚文, 李宇华.渐近周期的Kirchhoff-Schrdinger-Poisson系统非平凡解的存在性[J].河南科技大學学报(自然科学版), 2017, 38(3):95-99.
  HAO Yawen, LI Yuhua. Nontrivial solution existence for asymptotically periodic Kirchhoff-Schrdinger-Poisson system[J]. Journal of Henan University of Science and Technology(Natural Science), 2017, 38(3):95-99.
  [6] 郝剑伟, 黄永艳.带有Hartree和对数非线性项的Schrdinger方程非平凡解的存在性[J].河北科技大学学报, 2019, 40(6):482-487.
  HAO Jianwei, HUANG Yongyan. Existence of nontrivial solution of Schrdinger equations with Hartree and logarithmic nonlinearities[J]. Journal of Hebei University of Science and Technology, 2019, 40(6):482-487.
  [7] LI Gongbao, PENG Shuangjie, YAN Shusen. Infinitely many positive solutions for the nonlinear Schrdinger-Poisson system[J]. Communications in Contemporary Mathematics, 2010, 12(6):1069-1092.
  [8] 马锋, 赵雷嘎.一类非线性Schrdinger-Poisson方程解的存在性[J].北京化工大学学报(自然科学版), 2015,42(2):119-124.
  MA Feng, ZHAO Leiga. Existence of solutions for a class of Schrdinger-Poisson equations[J]. Journal of Beijing University of Chemical Technology(Natural Science), 2015, 42(2):119-124.   [9] 余胜龙, 唐春雷.带有负的非局部项的Schrdinger-Poisson方程的正基态解[J].西南大学学报(自然科学版), 2013, 35(6):68-71.
  YU Shenglong, TANG Chunlei. On positive ground state solution to the Schrdinger-Poisson system with the negative non-local term[J]. Journal of Southwest University (Natural Science Edition), 2019, 40(6):68-71.
  [10]赵桂兰.一类Kirchhoff-Schrdinger-Poisson系统正解的存在性[J].纺织高校基础科学学报, 2015, 28(2):198-203.
  ZHAO Guilan. Existence of a positive solution to a class of Kirchhoff-Schrdinger-Poisson system[J]. Basic Sciences Journal of Textile Universities, 2015,28(2):198-203.
  [11]SUN Jijiang, MA Shiwang. Ground state solutions for some Schrdinger-Poisson systems with periodic potentials[J].Journal of Differen-tial Equations, 2016,260(3):2119-2149.
  [12]IANNI I. Sign-changing radial solutions for the Schrdinger-Poisson-Slater problem[J]. Topological Methods in Nonlinear Analysis, 2013, 41(2):365-385.
  [13]LIANG Zhanping, XU Jing, ZHU Xiaoli. Revisit to sign-changing solutions for the nonlinear Schrdinger-Poissonsystem in R3[J]. Journal of Mathematical Analysis and Applications, 2016, 435(1):783-799.
  [14]ALVES C O, SOUTO M A S. Existence of least energy nodal solution for a Schrdinger-Poisson system in bounded domains[J]. Zeitschrift Für Angewandte Mathematik Und Physik, 2014, 65(6):1153-1166.
  [15]ALVES C O, SOUTO M A O, SOARES S H M. A sign-changing solution for the Schrdinger-Poisson equation in R3[J].The Rocky Mountain Journal of Mathematics, 2017, 47(1):1-25.
  [16]WANG Zhengping, ZHOU Huansong. Sign-changing solutions for the nonlinear Schrdinger-Poisson system in R3[J]. Calculus of Variations and Partial Differential Equations, 2015,52(3/4):927-943.
  [17]SHUAI Wei, WANG Qingfang. Existence and asymptotic behavior of sign-changing solutions for the nonlinear Schrdinger-Poisson system in R3 [J].Zeitschrift Für Angewandte Mathematik Und Physik, 2015, 66(6):3267-3282.
  [18]WANG Dabing, ZHANG Huabo, GUAN Wen. Existence of least-energy sign-changing solutions for Schrdinger-Poisson system with critical growth[J].Journal of Mathematical Analysis and Applications,2019, 479:2284-2301.
  [19]WANG Dabing. Least energy sign-changing solutions of Kirchhoff-type equation with critical growth[J].Journal of Mathematical Physics, 2020, 61(1):1-7.
  [20]BARTSCH T, WANG Zhiqiang. Existence and multiplicity results for some superlinear elliptic problems on RN [J]. Communications in Partial Differential Equations, 1995, 20(9/10):1725-1741.
  [21]LIEB E, LOSS M. Analysis, Graduate Studies in Mathematics[M]. [S.l.]:[s.n.], 2001.
  [22]周民強.实变函数论[M]. 北京:北京大学出版社, 2008.
  [23]LI Fuyi, GAO Chunjuan, ZHU Xiaoli. Existence and concentration of sign-changing solutions to Kirchhoff-typesystem with Hartree-type nonlinearity[J]. Journal of Mathematical Analysis and Applications, 2017, 448(1):60-80.
转载注明来源:https://www.xzbu.com/1/view-15308523.htm