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The Existence of Arbitrary Multiple Nodal Solutions for a Class of Quasilinear Schrödinger Equations

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Abstract

This paper is concerned to studying the quasilinear Schrödinger equation:

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

where V(x) is a given potential, \(\gamma >0\) and either \(p\in (2,2^*),2^*=\frac{2N}{N-2}\) for \(N\geqslant 4\) or \(p\in (2,4)\) for \(N=3\). We establish the existence of arbitrary multiple nodal solutions for the above equations.

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References

  1. Abdolrazaghi, F., Razani, A.: A unique weak solution for a kind of coupled system of fractional Schrödinger equations. Opus. Math. 40, 313–322 (2020)

    Article  Google Scholar 

  2. Alves, C.O., Wang, Y., Shen, Y.: Soliton solutions for a class of quasilinear Schrödinger equations with a parameter. J. Differ. Equ. 259, 318–343 (2015)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  5. Choudhuri, D., Saoudi, K.: Existence of multiple solutions to Schrödinger–Poisson system in a nonlocal set up in \(\mathbb{R} ^3\). Z. Angew. Math. Phys. 73, 1–17 (2022)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Davydova, T.A., Fishchuk, A.I.: Upper hybrid nonlinear wave structures. Ukr. J. Phys. 40, 487 (1995)

    Google Scholar 

  8. 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 

  9. Jing, Y., Liu, H.: Sign-changing solutions for a modified nonlinear Schrödinger equation in \(\mathbb{R} ^N\). Calc. Var. Partial Differ. Equ. 61, 144 (2022)

    Article  Google Scholar 

  10. Jing, Y., Liu, Z., Wang, Z.-Q.: Multiple solutions of a parameter-dependent quasilinear elliptic equation. Calc. Var. Partial Differ. Equ. 55, 1–26 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  12. Kurihara, S.: Exact soliton solution for superfluid film dynamics. J. Phys. Soc. Jpn. 50, 3801–3805 (1981)

    Article  MathSciNet  Google Scholar 

  13. Liu, J., Wang, Z.-Q.: Soliton solutions for quasilinear Schrödinger equations I. Proc. Am. Math. Soc. 187, 441–448 (2003)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  16. Liu, J.-Q., Wang, Y.-Q., Wang, Z.-Q.: Quasilinear equations via elliptic regularization method. Adv. Nonlinear Stud. 13, 517–531 (2003)

    Article  MathSciNet  Google Scholar 

  17. Liu, J.-Q., Wang, Y.-Q., Wang, Z.-Q.: Solutions for quasilinear Schrödinger equations via the Nehari method. Commun. Partial Differ. Equ. 29, 879–901 (2004)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Liu, J., Liu, X., Wang, Z.-Q.: Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method. Commun. Partial Differ. Equ. 39, 2216–2239 (2014)

    Article  MathSciNet  Google Scholar 

  20. 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  Google Scholar 

  21. 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 

  22. Moussaoui, A., Saoudi, K.: Existence and location of nodal solutions for quasilinear convection-absorption Neumann problems. arXiv preprint arXiv:2304.00647

  23. Nakamura, A.: Damping and modification of exciton solitary waves. J. Phys. Soc. Jpn. 42, 1824–1835 (1977)

    Article  Google Scholar 

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

    Article  Google Scholar 

  25. 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 

  26. Razani, A., Cowan, C.: q-Laplace equation involving the gradient on general bounded and exterior domains. NoDEA Nonlinear Differ. Equ. Appl. 31 (2024)

  27. Razani, A., Gustavo Costa, S.A., Giovany Figueiredo, M.: A study on a class of weighted elliptic problems with indefinite nonlinearities. Appl. Anal. 24 (2023)

  28. Razani, A.: Non-existence of solution of Haraux–Weissler equation on a strictly starshped domain. Miskolc Math. Notes 24, 395–402 (2023)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Wang, Y.: Solitary solutions for a class of quasilinear Schrödinger equations in \(\mathbb{R} ^3\). Z. Angew. Math. Phys. 67, 1–17 (2016)

    Article  Google Scholar 

  31. Wang, Y., Shen, Y.: Existence and asymptotic behavior of positive solutions for a class of quasilinear Schrödinger equations. Adv. Nonlinear Stud. 18, 131–150 (2018)

    Article  MathSciNet  Google Scholar 

  32. Wang, Y., Zou, W.: Bound states to critical quasilinear Schrödinger equations. NoDEA Nonlinear Differ. Equ. Appl. 19, 19–47 (2012)

    Article  Google Scholar 

  33. Wang, X., Brown, D.W., Lindenberg, K., West, B.J.: Alternative formulation of Davydov’s theory of energy transport in biomolecular systems. Phys. Rev. A 37, 3557–3566 (1998)

    Article  Google Scholar 

  34. Willem, M.: Minimax Theorems. Birkhäuser, Basel (1996)

    Book  Google Scholar 

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

  36. Zhang, X., Huang, C.: Asymptotic behavior of multiple solutions for quasilinear Schrödinger equations. Electron. J. Qual. Theory Differ. Equ. 64, 1–28 (2022)

    Google Scholar 

  37. Zhang, H., Liu, Z., Tang, C., Zhang, J.: Existence and multiplicity of sign-changing solutions for quasilinear Schrödinger equations with sub-cubic nonlinearity. J. Differ. Equ. 365, 199–234 (2023)

    Article  Google Scholar 

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Acknowledgements

The authors would like to thank excellent work in references.

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Authors and Affiliations

  1. College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, People’s Republic of China

    Kun Wang, Chen Huang & Gao Jia

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  1. Kun Wang
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  2. Chen Huang
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  3. Gao Jia
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Kun Wang, Chen Huang wrote the main manuscript text, and Gao Jia revised the paper. All authors reviewed the manuscript.

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Correspondence to Chen Huang.

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Wang, K., Huang, C. & Jia, G. The Existence of Arbitrary Multiple Nodal Solutions for a Class of Quasilinear Schrödinger Equations. Qual. Theory Dyn. Syst. 23, 148 (2024). https://doi.org/10.1007/s12346-024-01010-2

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