Multiple Sign-Changing Solutions for a Class of Schrödinger Equations with Saturable Nonlinearity


In this paper, we construct sign-changing radial solutions for a class of Schrödinger equations with saturable nonlinearity which arises from several models in mathematical physics. More precisely, for any given nonnegative integer k, by using a minimization argument, we first obtain a sign-changing minimizer with k nodes of a constrained minimization problem, and show, by a deformation lemma and Miranda’s theorem, that the minimizer is the desired solution.

This is a preview of subscription content, access via your institution.


  1. [1]

    Akhmediev N, Ankiewicz A. Partially coherent solitons on a finite background. Phys Rev Lett, 1999, 82(13): 2661–2664

    Article  Google Scholar 

  2. [2]

    Agrawal G P, Kivshar Y S. Optical solitons: from fibers to photonic crystals. Academic Press, 2003

  3. [3]

    Akhmediev N, Królinowski W, Snyder A. Partially coherent solitons of variable shape. Phys Rev Lett, 1998, 81(21): 4632–4635

    Article  Google Scholar 

  4. [4]

    Berestycki H, Lions P L. Nonlinear scalar field equations, I, existence of a ground state. Arch Ration Mech Anal, 1983, 82(4): 313–345

    MathSciNet  Article  Google Scholar 

  5. [5]

    Berestycki H, Lions P L. Nonlinear scalar field equations, II, existence of infinitely many solutions. Arch Ration Mech Anal, 1983, 82(4): 347–375

    MathSciNet  Article  Google Scholar 

  6. [6]

    Bartsch T, Willem M. Infinitely many radial solutions of a semilinear elliptic problem on ℝN. Arch Ration Mech Anal, 1993, 124(3): 261–276

    MathSciNet  Article  Google Scholar 

  7. [7]

    Castro A, Cossio J, Neuberger J M. A sigh-changing solution for a superlinear Dirichlet problem. Rocky Mountain J Math, 1997, 27(4): 1041–1053

    MathSciNet  Article  Google Scholar 

  8. [8]

    Conti M, Merizzi L, Terracini S. Radial solutions of superlinear equations on RN, Part I, A global variational approach. Arch Ration Mech Anal, 2000, 153(4): 291–316

    MathSciNet  Article  Google Scholar 

  9. [9]

    Cao D, Li S, Liu Z. Nodal solutions for a supercritical semilinear problem with variable exponent. Cal Var PDEs, 2018, 57(2): 38

    MathSciNet  Article  Google Scholar 

  10. [10]

    Cao D, Zhu X. On the existence and nodal character of solutions of semilinear elliptic equation. Acta Mathematica Scientia, 1988, 8B(3): 285–300

    MathSciNet  Google Scholar 

  11. [11]

    Cerami G, Solimini S, Struwe M. Some existence results for superlinear elliptic boundary problems involving critical exponents. J Funct Anal, 1986, 69(3): 289–306

    MathSciNet  Article  Google Scholar 

  12. [12]

    Deng Y. The existence and nodal character of the solutions in ℝn for semilinear elliptic equation involving critical Sobolev exponent. Acta Mathematica Scientia, 1989, 9B(4): 385–402

    MathSciNet  Article  Google Scholar 

  13. [13]

    Deng Y, Peng S, Shuai W. Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in ℝ3. J Funct Anal, 2015, 269(11): 3500–3527

    MathSciNet  Article  Google Scholar 

  14. [14]

    Deng Y, Peng S, Wang J. Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent. J Math Phys, 2013, 54(1): 011504

    MathSciNet  Article  Google Scholar 

  15. [15]

    Deng Y, Peng S, Wang J. Nodal soliton solutions for generalized quasilinear Schrödinger equations. J Math Phys, 2014, 55(5): 051501

    MathSciNet  Article  Google Scholar 

  16. [16]

    Kulpa W. The Poincaré-Miranda theorem. Amer Math Mon, 1997, 104(6): 545–550

    MATH  Google Scholar 

  17. [17]

    Lin T C, Belić M R, Petrović M S, Chen G. Ground states of nonlinear Schrödinger systems with saturable nonlinearity in ℝ2 for two counterpropagating beams. J Math Phys, 2014, 55(1): 011505

    MathSciNet  Article  Google Scholar 

  18. [18]

    Lin T C, Belić M R, Petrović M S, Aleksić N B, Chen G. Ground-state counterpropagating solitons in photorefractive media with saturable nonlinearity. J Opt Soc Am B, 2013, 30(4): 1036–1040

    Article  Google Scholar 

  19. [19]

    Lin T C, Belić M R, Petrović M S, Hajaiej H, Chen G. The virial theorem and ground state energy estimates of nonlinear Schrödinger equations in ℝ2 with square root and saturable nonlinearities in nonlinear optics. Cal Var PDEs, 2017, 56(5): 147

    Article  Google Scholar 

  20. [20]

    Liu T C, Wang X, Wang Z Q. Orbital stability and energy estimate of ground states of saturable nonlinear Schrödinger equations with intensity functions in ℝ2. Journal Diff Equations, 2017, 263(8): 4750–4786

    Article  Google Scholar 

  21. [21]

    Litchinitser N M, Królikowski W, Akhmediev N N, Agrawal G P. Asymmetric partially coherent solitons in saturable nonlinear media. Phys Rev E, 1999, 60(2): 2377–2380

    Article  Google Scholar 

  22. [22]

    Liu Z, Wang Z Q. On the Ambrosetti-Rabinowitz superlinear condition. Adv Nonlinear Stud, 2004, 4(4): 563–574

    MathSciNet  Article  Google Scholar 

  23. [23]

    Maia L A, Miyagaki O H, Soares S. A sign-changing solution for an asymptotically linear Schröodinger equation. Proc Edin Math Soc, 2015, 58(3): 697–716

    Article  Google Scholar 

  24. [24]

    Miranda C. Un’osservazione su un teorema di Brouwer. Boll Un Mat Ital, 1940, 3(2): 5–7

    MathSciNet  MATH  Google Scholar 

  25. [25]

    Nehari Z. Characteristic values associated with a class of nonlinear second order differential equations. Acta Math, 1961, 105(3/4): 141–175

    MathSciNet  Article  Google Scholar 

  26. [26]

    Ostrovskaya E A, Kivshar Y S. Multi-hump optical solitons in a saturable medium. J Opt B: Quantum Semiclassical Opt, 1999, 1(1): 77–83

    Article  Google Scholar 

  27. [27]

    Pohozaev S. Eigenfunctions of the equations Δu + λf(u) = 0. Dokl Akad Nauk SSSR, 1965, 165(1): 36–39

    MathSciNet  Google Scholar 

  28. [28]

    Ryder G H. Boundary value problem for a class of nonlinear differential equations. Pacific J Math, 1967, 22(3): 477–503

    MathSciNet  Article  Google Scholar 

  29. [29]

    Struwe M. Superlinear elliptic boundary value problems with rotational symmetry. Arch Math, 1982, 39(3): 233–240

    MathSciNet  Article  Google Scholar 

  30. [30]

    Stuart C A. Guidance properties of nonlinear planar waveguides. Arch Ration Mech Anal, 1993, 125(2): 145–200

    MathSciNet  Article  Google Scholar 

  31. [31]

    Serrin J, Tang M. Uniqueness of ground states for quasilinear elliptic equations. Indiana Univ Math J, 2000, 49(3): 897–923

    MathSciNet  Article  Google Scholar 

  32. [32]

    Stuart C A, Zhou H S. Applying the mountain pass theorem to an asymptotically linear elliptic equation on RN. Commun PDEs, 1999, 24(9/10): 1731–1758

    Article  Google Scholar 

  33. [33]

    Stegeman G I, Christodoulides D N, Segev M. Optical spatial solitons: historical Perspectives. IEEE J Sel Top Quantum Electron, 2000, 6(6): 1419–1427

    Article  Google Scholar 

  34. [34]

    Szulkin A, Weth T. The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications. Boston: International Press, 2010: 597–632

    Google Scholar 

  35. [35]

    Wang X, Liu T C, Wang Z Q. Existence and concentration of ground states for saturable nonlinear Schröodinger equations with intensity functions in ℝ2. Nonlinear Anal, 2018, 173: 19–36

    MathSciNet  Article  Google Scholar 

  36. [36]

    Willem M. Minimax Theorems. Basel: Birkhöaser, 1996

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Zhongyuan Liu.

Additional information

The author is supported by National Natural Science Foundation of China (11971147), China Postdoctoral Science Foundation (2019M662475) and Henan Postdoctoral Research Grant (201902026).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Liu, Z. Multiple Sign-Changing Solutions for a Class of Schrödinger Equations with Saturable Nonlinearity. Acta Math Sci 41, 493–504 (2021).

Download citation

Key words

  • Sign-changing solutions
  • saturable nonlinearity
  • Nehari manifold
  • variational methods

2010 MR Subject Classification

  • 35J61
  • 35J66
  • 35Q55
  • 35Q60