Skip to main content
SpringerLink
Log in

Sign-changing solutions for a modified nonlinear Schrödinger equation in \({\mathbb {R}}^N\)

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We consider the modified nonlinear Schrödinger equation

$$\begin{aligned} -\Delta u+V(x)u-\Delta (u^2)u =|u|^{p-2}u\ \ \text{ in }\ {\mathbb {R}}^N, \end{aligned}$$

where \(N\ge 3,\ V\) is a given potential and \(p>2\). The equation appears in modeling of superfluid film theory in plasma physics. While most existing works in the literature are only for \(p\in [4,\,4N/(N-2))\), we propose a new variational approach to deal with exponents \(p\in (2,\,4N/(N-2))\) in a unified way and obtain infinitely many sign-changing solutions.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adachi, S., Shibata, M., Watanabe, T.: Blow-up phenomena and asymptotic profiles of ground states of quasilinear elliptic equations with \(H^{1}\)-supercritical nonlinearities. J. Differential Equations 256, 1492–1514 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambrosetti, A., Malchiodi, A.: Perturbation methods and semilinear elliptic problems on \({\mathbb{R}}^n\). Progress in Mathematics, vol. 240. Birkhäuser, Basel (2006)

  3. Ambrosetti, A., Struwe, M.: Existence of steady vortex rings in an ideal fluid. Arch. Rational Mech. Anal. 108, 97–109 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arcoya, D., Boccardo, L., Orsina, L.: Critical points for functionals with quasilinear singular Euler-Lagrange equations. Calc. Var. Partial Differential Equations 47, 159–180 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bartsch, T., Liu, Z., Weth, T.: Nodal solutions of a \(p\)-Laplacian equation. Proc. London Math. Soc. 91, 129–152 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bass, F.G., Nasonov, N.N.: Nonlinear electromagnetic-spin waves. Phys. Rep. 189, 165–223 (1990)

    Article  Google Scholar 

  7. Benci, V., Fortunato, D., Masiello, A.: On the geodesic connectedness of Lorentzian manifolds. Math. Z. 217, 73–93 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations, I. Existence of a ground state. Arch. Rational Mech. Anal. 82, 313–345 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  9. Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations, II. Existence of infinitely many solutions. Arch. Rational Mech. Anal. 82, 347–375 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brizhik, L., Eremko, A., Piette, B., Zakrzewski, W.J.: Electron self-trapping in a discrete two-dimensional lattice. Phys. D 159, 71–90 (2001)

    Article  MATH  Google Scholar 

  11. Brizhik, L., Piette, B., Zakrzewski, W.J.: Spontaneously localized electron states in a discrete anisotropic two-dimensional lattice. Phys. D 146, 275–288 (2000)

    Article  MATH  Google Scholar 

  12. Cassani, D., do Ó, J.M., Moameni, A.: Existence and concentration of solitary waves for a class of quasilinear Schrödinger equations. Comm. Pure Appl. Anal 9, 281–306 (2010)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Colin, M., Jeanjean, L., Squassina, M.: Stability and instability results for standing waves of quasi-linear Schrödinger equations. Nonlinearity 23, 1353–1385 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Deng, Y., Peng, S., Yan, S.: Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations. J. Differential Equations 260, 1228–1262 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. do Ó, J.M., Miyagaki, O.H., Soares, S.H.M.: Soliton solutions for quasilinear Schrödinger equations with critical growth. J. Differential Equations 248, 722–744 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fang, X., Szulkin, A.: Multiple solutions for a quasilinear Schrödinger equation. J. Differential Equations 254, 2015–2032 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Felmer, P., Silva, E.: Homoclinic and periodic orbits for Hamiltonian systems. Ann. Scuola Norm. Sup. Pisa Cl. Sci 26, 285–301 (1998)

    MathSciNet  MATH  Google Scholar 

  19. Hartmann, B., Zakrzewski, W.J.: Electrons on hexagonal lattices and applications to nanotubes. Phys. Rev. B 68, 184302 (2003)

    Article  Google Scholar 

  20. Hasse, R.W.: A general method for the solution of nonlinear soliton and kink Schrödinger equations. Z. Phys. B 37, 83–87 (1980)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Hirata, J., Ikoma, N., Tanaka, K.: Nonlinear scalar field equations in \({\mathbb{R}}^N\): mountain pass and symmetric mountain pass approaches. Topol. Methods Nonlinear Anal. 35, 253–276 (2010)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Jing, Y., Liu, Z., Wang, Z.-Q.: Multiple solutions of a parameter-dependent quasilinear elliptic equation. Calc. Var. Partial Differential Equations 55, 150 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kosevich, A.M., Ivanov, B.A., Kovalev, A.S.: Magnetic solitons. Phys. Rep. 194, 117–238 (1990)

    Article  Google Scholar 

  26. Liu, H., Zhao, L.: Existence results for quasilinear Schrödinger equations with a general nonlinearity. Comm. Pure Appl. Anal. 19, 3429–3444 (2020)

    Article  MATH  Google Scholar 

  27. Liu, J., Liu, X., Wang, Z.-Q.: Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method. Comm. Partial Differential Equations 39, 2216–2239 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Liu, J., Liu, X., Wang, Z.-Q.: Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete Contin. Dyn. Syst. Ser. S 14, 1779–1799 (2021)

    MathSciNet  MATH  Google Scholar 

  29. Liu, J., Wang, Y., Wang, Z.-Q.: Soliton solutions for quasilinear Schrödinger equations, II. J. Differential Equations 187, 473–493 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Liu, J., Wang, Y., Wang, Z.-Q.: Solutions for quasilinear Schrödinger equations via the Nehari method. Comm. Partial Differential Equations 29, 879–901 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Liu, J., Wang, Z.-Q.: Soliton solutions for quasilinear Schrödinger equations, I. Proc. Amer. Math. Soc. 131, 441–448 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  32. Liu, J., Wang, Z.-Q.: Multiple solutions for quasilinear elliptic equations with a finite potential well. J. Differential Equations 257, 2874–2899 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Liu, X., Liu, J., Wang, Z.-Q.: Quasilinear elliptic equations via perturbation method. Proc. Amer. Math. Soc. 141, 253–263 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Liu, X., Liu, J., Wang, Z.-Q.: Quasilinear elliptic equations with critical growth via perturbation method. J. Differential Equations 254, 102–124 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Liu, Z., Wang, Z.-Q.: Sign-changing solutions of nonlinear elliptic equations. Front. Math. China 3, 221–238 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Liu, Z., Wang, Z.-Q.: On Clark’s theorem and its applications to partially sublinear problems. Ann. Inst. H. Poincaré Anal. Non-linéaire 32, 1015–1037 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Liu, Z., Wang, Z.-Q., Zhang, J.: Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system. Ann. Mat. Pura Appl. 195, 775–794 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. Liu, Z., Zhang, Z., Huang, S.: Existence and nonexistence of positive solutions for a static Schrödinger-Poisson-Slater equation. J. Differential Equations 266, 5912–5941 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  40. Makhankov, V.G., Fedyanin, V.K.: Non-linear effects in quasi-one-dimensional models of condensed matter theory. Phys. Rep. 104, 1–86 (1984)

    Article  MathSciNet  Google Scholar 

  41. Minty, G.J.: Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29, 341–346 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  42. Poppenberg, M., Schmitt, K., Wang, Z.-Q.: On the existence of soliton solutions to quasilinear Schrödinger equations. Calc. Var. Partial Differential Equations 14, 329–344 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  43. Quispel, G.R.W., Capel, H.W.: Equation of motion for the Heisenberg spin chain. Phys. A 110, 41–80 (1982)

    Article  MathSciNet  Google Scholar 

  44. Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conference Series in Mathematics, No. 65, AMS, Providence (1986)

  45. Ruiz, D., Siciliano, G.: Existence of ground states for a modified nonlinear Schrödinger equation. Nonlinearity 23, 1221–1233 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  46. Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of 2-spheres. Ann. of Math. 113, 1–24 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  47. Severo, U.B., Gloss, E., da Silva, E.D.: On a class of quasilinear Schrödinger equations with superlinear or asymptotically linear terms. J. Differential Equations 263, 3550–3580 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  48. Silva, E.A.B., Vieira, G.F.: Quasilinear asymptotically periodic Schrödinger equations with critical growth. Calc. Var. Partial Differential Equations 39, 1–33 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  49. Strauss, W.A.: Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55, 149–162 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  50. Szulkin, A., Zou, W.: Homoclinic orbits for asymptotically linear Hamiltonian systems. J. Funct. Anal. 187, 25–41 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  51. Uhlenbeck, K.: Morse theory by perturbation methods with applications to harmonic maps. Trans. Amer. Math. Soc. 267, 569–583 (1981)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to express sincere thanks to their advisor Prof. Zhaoli Liu for his guidance, encouragement and helpful suggestions. Y. Jing is supported by National Natural Science Foundation of China (No. 11901017). H. Liu is supported by Natural Science Foundation of Zhejiang Province (No. LY21A010020) and National Natural Science Foundation of China (No. 12171204).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Technology and Business University, Beijing, China

    Yongtao Jing

  2. Institute of Mathematics, Jiaxing University, Zhejiang, China

    Haidong Liu

Authors
  1. Yongtao Jing
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Haidong Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Haidong Liu.

Additional information

Communicated by P. H. Rabinowitz.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jing, Y., Liu, H. Sign-changing solutions for a modified nonlinear Schrödinger equation in \({\mathbb {R}}^N\). Calc. Var. 61, 144 (2022). https://doi.org/10.1007/s00526-022-02266-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-022-02266-9

Mathematics Subject Classification

Search

Navigation