ssRESEARCH ARTICLE

Asymptotical behavior of ground state solutions for critical quasilinear Schrödinger equation

  • Yongpeng CHEN , 1 ,
  • Yuxia GUO 2 ,
  • Zhongwei TANG 1
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  • 1. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
  • 2. Department of Mathematics, Tsinghua University, Beijing 100084, China

Received date: 03 Aug 2019

Accepted date: 15 Feb 2020

Published date: 15 Feb 2020

Copyright

2020 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature

Abstract

This paper is concerned with the existence and asymptotical behavior of positive ground state solutions for a class of critical quasilinear Schrodinger equation. By using a change of variables and variational argument, we prove the existence of positive ground state solution and discuss their asymptotical behavior.

Cite this article

Yongpeng CHEN , Yuxia GUO , Zhongwei TANG . Asymptotical behavior of ground state solutions for critical quasilinear Schrödinger equation[J]. Frontiers of Mathematics in China, 2020 , 15(1) : 21 -46 . DOI: 10.1007/s11464-020-0825-1

1
Aires J, Souto M. Existence of solutions for a quasilinear Schrödinger equation with vanishing potentials. J Math Anal Appl, 2014, 416: 924–946

DOI

2
Alves C, Barros L. Existence and multiplicity of solutions for a class of elliptic problem with critical growth. Monatsh Math, 2018, 187: 195–215

DOI

3
Bartsch T, Wang Z. Multiple positive solutions for a nonlinear Schrödinger equation. Z Angew Math Phys, 2000, 51: 366–384

DOI

4
Colin M, Jeanjean L. Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal, 2004, 56: 213–226

DOI

5
Deng Y, Peng S, Yan S. Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth. J Differential Equations, 2017, 37: 4213–4230

6
Do Ó J, Miyagaki O, Soares S. Soliton solutions for quasilinear Schrödinger equations with critical growth. J Differential Equations, 2010, 248: 722–744

DOI

7
Do Ó J, Severo U. Quasilinear Schrödinger equation involving concave and convex non-linearities. Commun Pure Appl Anal, 2009, 8: 621–644

DOI

8
Guo Y, Tang Z. Ground state solutions for the quasilinear Schrödinger equations. Nonlinear Anal, 2012, 75: 3235–3248

DOI

9
Guo Y, Tang Z. Multi-bump bound state solutions for the quasilinear Schrödinger equations with critical frequency. Pacific J Math, 2014, 270: 49–77

DOI

10
He X, Qian A, Zou W. Existence and concentration of positive solutions for quasilinear Schrödinger equations with critical growth. Nonlinearity, 2013, 26: 3137–3168

DOI

11
He Y, Li G. Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Sobolev exponents. Discrete Contin Dyn Syst, 2016, 36: 731–762

DOI

12
Li Z, Zhang Y. Solutions for a class of quasilinear Schrödinger equations with critical Sobolev exponents. J Math Phys, 2017, 58: 1–15

DOI

13
Liang S, Zhang J. Existence of multi-bump solutions for a class of quasilinear Schrödinger equations in ℝN involving critical growth. Milan J Math, 2015, 83: 55–90

14
Liu J, Wang Y, Wang Z. Soliton solutions for quasilinear Schrödinger equation, II. J Differential Equations, 2003, 187: 473–493

DOI

15
Liu J, Wang Y, Wang Z. Solutions for quasilinear Schrödinger equation via Nehari method. Comm Partial Differential Equations, 2004, 29: 879–901

DOI

16
Liu S, Zhou J. Standing waves for quasilinear Schrödinger equations with indefinite potentials. J Differential Equations, 2018, 265: 3970–3987

DOI

17
Liu X, Liu J, Wang Z. Quasilinear elliptic equations via perturbation method. Proc Amer Math Soc, 2013, 141: 253–263

DOI

18
Shen Y, Wang Y. Soliton solutions for generalized quasilinear Schrödinger equations. Nonlinear Analysis TMA, 2013, 80: 194–201

DOI

19
Silva E, Vieira G. Quasilinear asymptotically periodic Schrödinger equations with critical growth. Calc Var Partial Differential Equations, 2010, 39: 1–33

DOI

20
Wang W, Yang X, Zhao F. Existence and concentration of ground state to a quasilinear problem with competing potentials. Nonlinear Anal, 2014, 102: 120–132

DOI

21
Wang Y, Zhang Y, Shen Y. Multiple solutions for quasilinear Schrödinger equations involving critical exponent. Appl Math Comput, 2010, 216: 849–856

DOI

22
Wang Y, Zou W. Bound states to critical quasilinear Schrödinger equations. NoDEA Nonlinear Differential Equations Appl, 2012, 19: 19–47

DOI

23
Willem M. Minimax Theorem. Boston: Birkhäuser, 1996

DOI

24
Wu K. Positive solutions of quasilinear Schrödinger equations critical growth. Appl Math Lett, 2015, 45: 52–57

DOI

25
Xu L, Chen H. Ground state solutions for quasilinear Schrödinger equation via Pohozaev manifold in Orlicz space. J Differential Equations, 2018, 265: 4417–4441

DOI

26
Zeng X, Zhang Y, Zhou H. Positive solutions for a quasilinear Schrödinger equation involving Hardy potential and critical exponent. Commun Contemp Math, 2014, 16: 1–32

DOI

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