A Class of Quasilinear Schrödinger Equations with Improved (AR) Condition

Abstract

By improving the classical (AR) condition, we establish the existence of positive solutions for a class of quasilinear Schrödinger equations.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Borisov, A.B., Borovskiy, A.V., et al.: Observation of relativistic and charge-displacement self-channeling of intense subpicosecond ultraviolet (248 nm) radiation in plasmas. Phys. Rev. Lett. 68, 2309 (1992)

    Article  Google Scholar 

  2. 2.

    Bouard, A., Hayashi, N., Saut, J.: Global existence of small solutions to a relativistic nonlinear Schrödinger equation. Commun. Math. Phys. 189, 73–105 (1997)

    MATH  Article  Google Scholar 

  3. 3.

    Colin, M.: On the local well-posedness of quasilinear Schrödinger equations in arbitrary space dimension. Commun. Partial Differ. Equ. 27, 325–354 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Colin, M.: Stability of stationary waves for a quasilinear Schrödinger equation in dimension 2. Adv. Differ. Equ. 8(1), 1–28 (2003)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Lions, P.L.: The concentration compactness principle in the calculus of variations. The locally compact case. Part I and II. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 109–145, 223–283 (1984)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43, 270–291 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Costa, D.G., Magalhães, C.A.: Variational elliptic problems which are nonquadratic at infinity. Nonlinear Anal. 23, 1401–1412 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Shen, Y.T., Guo, X.K.: Discussion of nontrivial critical points of the functional \({\int _{\varOmega }F(x,u, Du)dx}\). Acta Math. Sci. 10, 249–258 (1990)

    Google Scholar 

  10. 10.

    Shen, Y.T., Wang, Y.J.: Soliton solutions for generalized quasilinear Schrödinger equations. Nonlinear Anal. 80, 194–201 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Shen, Y.T., Wang, Y.J.: Standing waves for a class of quasilinear Schrödinger equations. Complex Var. Elliptic Equ. 61(6), 817–842 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Yang, J., Wang, Y.J., Abdelgadir, A.A.: Soliton solutions for quasilinear Schrödinger equations. J. Math. Phys. 54, 071502 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Jeanjean, L.: On the existence of bounded Palais-Smale sequences and applications to a Landesman-Lazer-type problem set on \(\mathbb{R}^{N}\). Proc. R. Soc. Edinb. 129A, 787–809 (1999)

    MATH  MathSciNet  Article  Google Scholar 

  14. 14.

    Jeanjean, L., Tanaka, K.: Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. Calc. Var. Partial Differ. Equ. 21, 287–318 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Jeanjean, L., Tanaka, K.: A remark on least energy solutions in \(\mathbb{R}^{N}\). Proc. Am. Math. Soc. 131, 2399–2408 (2003)

    MATH  MathSciNet  Article  Google Scholar 

  16. 16.

    Ding, W.Y., Ni, W.M.: On the existence of positive entire solutions of a semilinear elliptic equation. Arch. Ration. Mech. Anal. 91, 183–208 (1986)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Ding, Y., Szulkin, A.: Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms. J. Differ. Equ. 222(1), 137–163 (2006)

    MATH  Article  Google Scholar 

  18. 18.

    Moschetto, D.: Existence and multiplicity results for a nonlinear stationary Schrödinger equation. Ann. Pol. Math. 99, 39–43 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Zou, W.: Variant fountain theorems and their applications. Manuscr. Math. 104, 343–358 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Liu, Z.L., Wang, Z.Q.: On the Ambrosetti–Rabinowitz superlinear condition. Adv. Nonlinear Stud. 4, 561–572 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Tang, X.H.: Infinitely many solutions for semilinear Schrödinger equations with sign-changing potential and nonlinearity. J. Math. Anal. Appl. 401, 407–415 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)

    Google Scholar 

  23. 23.

    Schechter, M.: Linking Methods in Critical Point Theory. Birkhäuser, Boston (1999)

    Google Scholar 

  24. 24.

    Struwe, M.: Variational Methods. Springer, Berlin (2007)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Youjun Wang.

Additional information

The second author is supported by the Fundamental Research Funds for the Central Universities (No. 2018MS59) and Natural Science Foundation of Guangdong (No. 2018A0303130196).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Shen, Y., Wang, Y. A Class of Quasilinear Schrödinger Equations with Improved (AR) Condition. Acta Appl Math 164, 123–135 (2019). https://doi.org/10.1007/s10440-018-00228-y

Download citation

Keywords

  • Quasilinear Schrödinger equations
  • Mountain pass theorem
  • (AR) condition

Mathematics Subject Classification

  • 35J20
  • 35J60