Advertisement

Acta Mathematica Scientia

, Volume 39, Issue 2, pp 580–596 | Cite as

A Nontrivial Solution of a Quasilinear Elliptic Equation Via Dual Approach

  • Xianyong Yang
  • Wei Zhang
  • Fukun ZhaoEmail author
Article
  • 1 Downloads

Abstract

In this article, we are concerned with the existence of solutions of a quasilinear elliptic equation in ℝN which includes the so-called modified nonlinear Schrödinger equation as a special case. Combining the dual approach and the nonsmooth critical point theory, we obtain the existence of a nontrivial solution.

Key words

nontrivial solution quasilinear elliptic equation nonsmooth critical point theory dual approach 

2010 MR Subject Classification

35J20 35J62 49J52 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The authors would like to thank Dr. KeWu for helpful suggestions on the present article, as well as bringing their attention to [28] and [23].

References

  1. [1]
    Brüll L, Lange H. Solitary waves for quasilinear Schrödinger equations. Exposition Math, 1986, 4(3): 279–288MathSciNetzbMATHGoogle Scholar
  2. [2]
    Canino A, Degiovanni M. Nonsmooth critical point theory and quasilinear elliptic equations//Topological Methods in Differential Equations and Inclusions. Netherlands: Springer, 1995: 1–50Google Scholar
  3. [3]
    Chen J H, Tang X H, Cheng B T. Existence and nonexistence of positive solutions for a class of generalized quasilinear Schrödinger equations involving Kirchhoff-type perturbation with critical Sobolev exponent. J Math Phys, 2018, 59(2): 021505, 24MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Chen S T, Tang X H. Ground state solutions for generalized quasilinear Schrödinger equations with variable potentials and Berestycki-Lions nonlinearities. J Math Phys, 2018, 59(8): 081508, 18MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Colin M, Jeanjean L. Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal, 2004, 56(2): 213–226MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Deng Y B, Guo Y X, Liu J Q. Existence of solutions for quasilinear elliptic equations with Hardy potential. J Math Phys, 2016, 57(3): 031503, 15MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Deng Y B, Peng S J, Yan S S. Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations. J Differential Equations, 2016, 260(2): 1228–1262MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    do Ó J M B, Miyagaki O H, Soares S H M. Soliton solutions for quasilinear Schrödinger equations with critical growth. J Differential Equations, 2010, 248(4): 722–744MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Deng Z Y, Huang Y S. On positive G-symmetric solutions of a weighted quasilinear elliptic equation with critical Hardy-Sobolev exponent. Acta Math Sci, 2014, 34B(5): 1619–1633MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Gasiński L, Papageorgiou N S. Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems. Boca Raton, FL: Chapman & Hall/CRC, 2005zbMATHGoogle Scholar
  11. [11]
    Jabri Y. The Mountain Pass Theorem. Cambridge: Cambridge University Press, 2003CrossRefzbMATHGoogle Scholar
  12. [12]
    Li Q Q, Wu X. Existence of nontrivial solutions for generalized quasilinear Schrödinger equations with critical or supercritical growths. Acta Math Sci, 2017, 37B(6): 1870–1880CrossRefzbMATHGoogle Scholar
  13. [13]
    Li Z X, Shen Y T. Nonsmooth critical point theorems and its applications to quasilinear Schrödinger equations. Acta Math Sci, 2016, 36B(1): 73–86MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Kurihara S. Exact soliton solution for superfluid film dynamics. J Phys Soc Japan, 1981, 50(11): 3801–3805MathSciNetCrossRefGoogle Scholar
  15. [15]
    Liu J Q, Liu X Q, Wang Z Q. Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method. Comm Partial Differential Equations, 2014, 39(12): 2216–2239MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Liu J Q, Wang Y Q, Wang Z Q. Soliton solutions for quasilinear Schrödinger equations II. J Differential Equations, 2003, 187(2): 473–493MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Liu J Q, Wang Y Q, Wang Z Q. Solutions for quasilinear Schrödinger equations via the Nehari method. Comm Partial Differential Equations, 2004, 29(5/6): 879–901MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Liu J Q, Wang Y Q, Wang Z Q. Soliton solutions for quasilinear Schrödinger equations I. Proc Amer Math Soc, 2003, 131(2): 441–448MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Liu J Q, Wang Z Q, Guo Y X. Multibump solutions for quasilinear elliptic equations. J Funct Anal, 2012, 262(9): 4040–4102MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Liu J Q, Wang Z Q, Wu X. Multibump solutions for quasilinear elliptic equations with critical growth. J Math Phys, 2013, 54(12): 121501, 31MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Liu X Q, Liu J Q, Wang Z Q. Quasilinear elliptic equations via perturbation method. Proc Amer Math Soc, 2013. 141(1): 253–263MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Lu W D. Variational Methods in Differential Equations. Scientific Publishing House in China, 2002Google Scholar
  23. [23]
    Shen Y T, Wang Y J. Soliton solutions for generalized quasilinear Schrödinger equations. Nonlinear Anal., 2013, 80: 194–201MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Shi H X, Chen H B. Existence and multiplicity of solutions for a class of generalized quasilinear Schrödinger equations. J Math Anal Appl, 2017, 452(1): 578–594MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Wang Y J, Zou W M. Bound states to critical quasilinear Schrödinger equations. Nonlinear Differ Equ Appl, 2012, 19(1): 19–47CrossRefzbMATHGoogle Scholar
  26. [26]
    Willem M. Minimax theorems//Progress in Nonlinear Differential Equations and their Applications Vol 24, Boston, MA: Birkhäuser Boston, Inc, 1996Google Scholar
  27. [27]
    Wu K. Positive solutions of quasilinear Schrödinger equations with critical growth. Appl Math Lett, 2015, 45: 52–57MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    Wu K, Wu X. Multiplicity of solutions for a quasilinear elliptic equation. Acta Math Sci, 2016, 36B(2): 549–559MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Wu X, Wu K. Existence of positive solutions, negative solutions and high energy solutions for quasi-linear elliptic equations on ℝN. Nonlinear Anal: Real World Appl, 2014, 16: 48–64MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    Wu X, Wu K. Geometrically distinct solutions for quasilinear elliptic equations. Nonlinearity, 2014, 27(5): 987–1001MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    Yang R R, Zhang W, Liu X Q. Sign-changing solutions for p-biharmonic equations with Hardy potential in ℝN. Acta Math Sci, 2017, 37B(3): 593–606MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    Zhang J, Tang X H, Zhang W. Infinitely many solutions of quasilinear Schrödinger equation with signchanging potential. J Math Anal Appl, 2014, 420(2): 1762–1775MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  1. 1.School of Preparatory EducationYunnan Minzu UniversityKunmingChina
  2. 2.School of Mathematics and StatisticsCentral south UniversityChangshaChina
  3. 3.Department of MathematicsYunnan UniversityKunmingChina
  4. 4.Department of MathematicsYunnan Normal UniversityKunmingChina

Personalised recommendations