Acta Applicandae Mathematicae

, Volume 164, Issue 1, pp 123–135 | Cite as

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

  • Yaotian Shen
  • Youjun WangEmail author


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


Quasilinear Schrödinger equations Mountain pass theorem (AR) condition 

Mathematics Subject Classification

35J20 35J60 


  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) CrossRefGoogle 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) zbMATHCrossRefGoogle 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) MathSciNetzbMATHCrossRefGoogle 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) MathSciNetzbMATHGoogle 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) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43, 270–291 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Costa, D.G., Magalhães, C.A.: Variational elliptic problems which are nonquadratic at infinity. Nonlinear Anal. 23, 1401–1412 (1994) MathSciNetzbMATHCrossRefGoogle 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) MathSciNetzbMATHCrossRefGoogle 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) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Yang, J., Wang, Y.J., Abdelgadir, A.A.: Soliton solutions for quasilinear Schrödinger equations. J. Math. Phys. 54, 071502 (2013) MathSciNetzbMATHCrossRefGoogle 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) zbMATHGoogle 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) MathSciNetzbMATHCrossRefGoogle 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) zbMATHGoogle 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) MathSciNetCrossRefGoogle 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) zbMATHCrossRefGoogle Scholar
  18. 18.
    Moschetto, D.: Existence and multiplicity results for a nonlinear stationary Schrödinger equation. Ann. Pol. Math. 99, 39–43 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Zou, W.: Variant fountain theorems and their applications. Manuscr. Math. 104, 343–358 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Liu, Z.L., Wang, Z.Q.: On the Ambrosetti–Rabinowitz superlinear condition. Adv. Nonlinear Stud. 4, 561–572 (2004) MathSciNetzbMATHCrossRefGoogle 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) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996) zbMATHCrossRefGoogle Scholar
  23. 23.
    Schechter, M.: Linking Methods in Critical Point Theory. Birkhäuser, Boston (1999) zbMATHCrossRefGoogle Scholar
  24. 24.
    Struwe, M.: Variational Methods. Springer, Berlin (2007) zbMATHGoogle Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsSouth China University of TechnologyGuangzhouP.R. China

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