环绕定理在退化的椭圆型方程上的应用

数学物理学报 ›› 2023, Vol. 43 ›› Issue (6): 1759-1773.

环绕定理在退化的椭圆型方程上的应用

周英告(),李周欣^*()

中南大学数学与统计学院长沙 410083

收稿日期:2023-01-05 修回日期:2023-04-12 出版日期:2023-12-26 发布日期:2023-11-16
通讯作者: *李周欣，E-mail: lzx@math.pku.edu.cn
作者简介:周英告，E-mail: ygzhou@csu.edu.cn
基金资助:
湖南省研究生教育教学改革研究项目(2020JGYB031);中南大学教育教学改革研究项目(2020JGB020);中南大学教育教学改革研究项目(2022ALK006);中南大学教育教学改革研究项目(2022jy069);湖南省自然科学基金(2023JJ30645)

An Application of Linking Theorem to Degenerative Elliptic Equations

Zhou Yinggao(),Li Zhouxin^*()

School of Mathematical and Statistics, Central South University, Changsha 410083

Received:2023-01-05 Revised:2023-04-12 Online:2023-12-26 Published:2023-11-16
Supported by:
Degree and Graduate Education Reform Research Project of Hunan Province(2020JGYB031);Education and Teaching Reform Research Project of Central South University(2020JGB020);Education and Teaching Reform Research Project of Central South University(2022ALK006);Education and Teaching Reform Research Project of Central South University(2022jy069);Hunan Provincial Natural Science Foundation(2023JJ30645)

摘要/Abstract

摘要：

该文考虑一类带临界增长项的退化的椭圆型偏微分方程解的存在性, 利用变量代换的方法, 把方程转化为半线性方程, 再利用集中紧引理以及基于锥分解的环绕定理, 证明了方程解的存在性.

关键词: 椭圆型方程, 环绕定理, 临界指数

Abstract:

In this paper, we study a class of degenerative quasilinear elliptic equations with critical exponent. Using a change of variable, we reformulate the equation into a semilinear one, and prove the existence of solutions by employing Lions concentration compactness principle and linking theorem based on cone.

Key words: Elliptic equations, Linking thoerem, Critical exponent

中图分类号:

O177.91

周英告, 李周欣. 环绕定理在退化的椭圆型方程上的应用[J]. 数学物理学报, 2023, 43(6): 1759-1773.

Zhou Yinggao, Li Zhouxin. An Application of Linking Theorem to Degenerative Elliptic Equations[J]. Acta mathematica scientia,Series A, 2023, 43(6): 1759-1773.

参考文献

[1]	Adachia S, Watanable T. G-invariant positive solutions for a quasilinear Schr?dinger equation. Adv Differential Equations, 2011, 16: 289-324
[2]	Adachia S, Watanable T. Uniqueness of the ground state solutions of quasilinear Schr?dinger equations. Nonlinear Anal, 2012, 75: 819-833
[3]	Arcoya D, Boccardo L, Orsina L. Existence of critical points for some noncoercive functionals. Ann Inst H Poincaré Anal Non Linéaire, 2001, 18(4): 437-457
[4]	Brezis H, Nirenberg L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent. Commum Pure Appl Math, 1983, 36: 437-478
[5]	Brizhik L, Eremko A, Piette B, Zakrzewski W J. Static solutions of a D-dimensional modified nonlinear Schr?dinger equation. Nonlinearity, 2003, 16: 1481-1497
[6]	Colin M, Jeanjean L. Solutions for a quasilinear Schr?dinger equation: A dual approch. Nonlinear Anal, 2004, 56: 213-226
[7]	Degiovanni M, Lancelotti S. Linking solutions for p-Laplace equations with nonlinearity at critical growth. Journal of Functional Analysis, 2009, 256: 3643-3659
[8]	Degiovanni M, Magrone P. Linking solutions for quasilinear equations at critical growth involving the "1-Laplace" operator. Calc Var Partial Differential Equations, 2009, 36: 591-609
[9]	Deng Y, Peng S, Yan S. Positive soliton solutions for generalized quasilinear Schr?dinger equations with critical growth. J Differential Equations, 2015, 258: 115-147
[10]	Deng Y, Peng S, Yan S. Critical exponents and solitary wave solutions for generalized quasilinear Schr?dinger equations. J Differential Equations, 2016, 260: 1228-1262
[11]	Ghoussoub N, Yuan C. Multiple solutions for quasi-linear PDES involving the critical Sobolev and Hardy exponents. Trans Amer Math Soc, 2000, 352(12): 5703-5743
[12]	Hartmann B, Zakzeweski W. Electrons on hexagonal lattices and applications to nanotubes. Phys Rev B, 2003, 68: 184302
[13]	Kurihura S. Large-Amplitude qusi-solitons in superfuid filme. J Math Soc Japan, 1981, 50: 3262-3267
[14]	Laedke E W, Spatschek K H, Stenflo L. Evolution theorem for a class of perturbed envelope soliton solutions. J Math Phys, 1983, 24: 2764-2769
[15]	Li Z. Positive solutions for a class of singular quasilinear Schr?dinger equations with critical Sobolev exponent. J Differential Equations, 2019, 266: 7264-7290
[16]	Li Z, Yuan X, Zhang Q. Existence of critical points for noncoercive functionals with critical Sobolev exponent. Appl Anal, 2022, 101: 5358-5375
[17]	Li Z, Zhang Y. Solutions for a class of quasilinear Schr?dinger equations with critical Sobolev exponents. J Math Phys, 2017, 58: 021501
[18]	Liu J Q, Wang Z Q. Soliton solutions for quasilinear Schr?dinger equations I. Proc Amer Math Soc, 2002, 131: 441-448
[19]	Liu J Q, Wang Y Q, Wang Z Q. Soliton solutions for quasilinear Schr?dinger equations II. J Differential Equations, 2003, 187: 473-493
[20]	Lions P L. The concentration-compactness principle in the calculus of variations. The limit case, Part 2. Rev Mat Iberoam, 1985, 1(2): 45-121
[21]	Liu J Q, Wang Y Q, Wang Z Q. Solutions for the quasilinear Schr?dinger equations via the Nehari Method. Comm Partial Differential Equations, 2004, 29: 879-901
[22]	Shen Y, Li Z, Wang Y. Sign-Changing critical points for noncoercive functionals. Topol Methods Nonlinear Anal, 2014, 43(2): 373-384
[23]	Shen Y, Wang Y. Soliton solutions for generalized quasilinear Schr?diger equations. Nonlinear Anal, 2013, 80: 194-201
[24]	Silva E A B, Vieira G F. Quasilinear asymptotically periodic Sch?dinger equations with critical growth. Calc Var Partial Differential Equations, 2010, 39: 1-33
[25]	Wang Y, Zhang Y, Shen Y. Multiple solutions for quasilinear Schr?dinger equations involving critical exponent. Appl Math Comput, 2010, 216: 849-856
[26]	Wang Y, Zou W. Bound states to critical quasilinear Schr?dinger equations. NoDEA Nonlinear Differential Equations Appl, 2012, 19: 194-201
[27]	Yang J, Wang Y, Abdelgadir A A. Soliton solutions for quasilinear Schr?dinger equations. J Math Phys, 2013, 54: 071502