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Multiple Solutions for a Class of Quasilinear Schrödinger Systems in \({\mathbb {R}}^{N}\)

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Abstract

In this work the symmetric mountain pass lemma is employed to establish the existence of infinitely many solutions for a class of quasilinear Schrödinger system in \({\mathbb {R}}^{N}\) involving a parameter \(\alpha \) and subcritical nonlinearities.

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References

  1. Bartsch, T., Wang, Z.Q.: Existence and multiplicity results for some superlinear elliptic problems on \(\mathbb{R}^{N}\). Commun. Partial Differ. Equ. 20, 1725–1741 (1995)

    Article  MATH  Google Scholar 

  2. Bass, F., Nasanov, N.: Nonlinear electromagnetic spin waves. Phys. Rep. 189, 165–223 (1990)

    Article  Google Scholar 

  3. Bezerra do ó, J.M., Severo, U.: Solitary waves for a class of quasilinear Schrödinger equations in dimension two. Calc. Var. Partial Diff. Equ. 38, 275–315 (2010)

    Article  MATH  Google Scholar 

  4. Brezis, H., Lieb, E.H.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88, 486–490 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, C.S.: Multiple solutions for a class of quasilinear Schrödinger equations in \(\mathbb{R}^{N}\). J. Math. Phys. 56, 071507 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  8. Duan, S.Z., Wu, X.: An existence result for a class of \(p-\)Laplacian elliptic systems involving homogeneous nonlinearities in \(\mathbb{R}^{N}\). Nonlinear Anal. 74, 4723–4737 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Evans, L.C.: Partial differential equations, graduate studies in mathematics, vol. 19. Amer. Math, Soc (1998)

  10. Fang, X.D., Szulkin, A.: Multiple solutions for a quasilinear Schrödinger equation. J. Diff. Equ. 254, 2015–2032 (2013)

    Article  MATH  Google Scholar 

  11. Guo, Y., Tang, Z.: Ground state solutions for quasilinear Schrödinger systems. J. Math. Anal. Appl. 389, 322–339 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kurihura, S.: Large-amplitude quasi-solitons in superfluids films. J. Phys. Soc. Jpn. 50, 3262–3267 (1981)

    Article  Google Scholar 

  13. Liu, J.Q., Wang, Y.Q., Wang, Z.Q.: Soliton solutions to quasilinear Schrödinger equations II. J. Diff. Equ. 187, 473–493 (2003)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  15. Liu, J.Q., Wang, Y.Q., Wang, Z.Q.: Solutions for quasilinear Schrödinger equations via Nehari method. Commun. Partial Diff. Equ. 29, 879–904 (2004)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  17. Rabinowitz, P.H.: Minimax methods in critical point theory with application to differential equations. In: CBMS Reg. Conf. Ser. Math., Vol. 65, Amer. Math. Soc., Providence, RI, (1986)

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Severo, U.: Existence of weak solutions for quasilinear elliptic equations involving the \(p\)-Laplacian. Elec. J. Diff. Equ. 56, 1–16 (2008)

    MathSciNet  MATH  Google Scholar 

  20. Severo, U., da Silva, E.: On the existence of standing wave solutions for a class of quasilinear Schrödinger systems. J. Math. Anal. Appl. 412, 763–775 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Struwe, M.: Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, third edn. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  22. Wu, X.: Multiple solutions for quasilinear Schrödinger equations with a parameter. J. Diff. Equ. 256, 2619–2632 (2014)

    Article  MATH  Google Scholar 

  23. Zhang, Y., Dong, H.H., Zhang, X.E., Yang, H.W.: Rational solutions and lump solutions to the generalized (3+1)- dimensional Shallow Water-like equation. Comput. Math. Appl. 73, 246–252 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Caisheng Chen.

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Communicated by Yong Zhou.

This work is supported by NSFC (No.11571092).

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Chen, C., Yang, H. Multiple Solutions for a Class of Quasilinear Schrödinger Systems in \({\mathbb {R}}^{N}\). Bull. Malays. Math. Sci. Soc. 42, 611–636 (2019). https://doi.org/10.1007/s40840-017-0502-z

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  • DOI: https://doi.org/10.1007/s40840-017-0502-z

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