Dual variational methods for a nonlinear Helmholtz system

Abstract

This paper considers a pair of coupled nonlinear Helmholtz equations

$$\begin{aligned} {\left\{ \begin{array}{ll} -\,\Delta u - \mu u = a(x) \left( |u|^\frac{p}{2} + b(x) |v|^\frac{p}{2} \right) |u|^{\frac{p}{2} - 2}u,\\ -\,\Delta v - \nu v = a(x) \left( |v|^\frac{p}{2} + b(x) |u|^\frac{p}{2} \right) |v|^{\frac{p}{2} - 2}v \end{array}\right. } \end{aligned}$$

on \(\mathbb {R}^N\) where \(\frac{2(N+1)}{N-1}< p < 2^*\). The existence of nontrivial strong solutions in \(W^{2, p}(\mathbb {R}^N)\) is established using dual variational methods. The focus lies on necessary and sufficient conditions on the parameters deciding whether or not both components of such solutions are nontrivial.

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

References

  1. 1.

    Ambrosetti, A., Colorado, E.: Bound and ground states of coupled nonlinear Schrödinger equations. Compt. Rend. Math. 342(7), 453–458 (2006)

    Article  MATH  Google Scholar 

  2. 2.

    Córdoba, A.: Singular integrals, maximal functions and fourier restriction to spheres: the disk multiplier revisited. Adv. Math. 290, 208–235 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Evéquoz, G.: A dual approach in Orlicz spaces for the nonlinear Helmholtz equation. Zeitschrift für angewandte Mathematik und Physik 66(6), 2995–3015 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Evéquoz, G.: Existence and asymptotic behavior of standing waves of the nonlinear Helmholtz equation in the plane. arXiv:1606.00788v2 (2016)

  5. 5.

    Evéquoz, G.: Multiple standing waves for the nonlinear Helmholtz equation concentrating in the high frequency limit. Annali di Matematica Pura ed Applicata (1923-) 196(6), 2023–2042 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Evéquoz, G.: On the periodic and asymptotically periodic nonlinear Helmholtz equation. Nonlinear Anal. Theory Methods Appl. 152, 88–101 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Evéquoz, G., Weth, T.: Real solutions to the nonlinear Helmholtz equation with local nonlinearity. Arch. Ration. Mech. Anal. 211(2), 359–388 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Evéquoz, G., Weth, T.: Dual variational methods and nonvanishing for the nonlinear Helmholtz equation. Adv. Math. 280, 690–728 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Evéquoz, G., Weth, T.: Branch continuation inside the essential spectrum for the nonlinear Schrödinger equation. J. Fixed Point Theory Appl. 19(1), 475–502 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Evéquoz, G., Yeşil,T. : Dual ground state solutions for the critical nonlinear Helmholtz equation. arXiv:1707.00959v1 (2017)

  11. 11.

    Gutiérrez, S.: Non trivial \({L}^q\) solutions to the Ginzburg–Landau equation. Mathematische Annalen 328(1), 1–25 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Jakszto, M.: Another proof that \({L}^p\)-bounded pointwise convergence implies weak convergence. Real Anal. Exch. 36(2), 479–482 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Maia, L., Montefusco, E., Pellacci, B.: Positive solutions for a weakly coupled nonlinear Schrödinger system. J. Differ. Equ. 229(2), 743–767 (2006)

    Article  MATH  Google Scholar 

  14. 14.

    Mandel, R.: Minimal energy solutions for repulsive nonlinear Schrödinger systems. J. Differ. Equ. 257(2), 450–468 (2014)

    Article  MATH  Google Scholar 

  15. 15.

    Mandel, R.: Minimal energy solutions and infinitely many bifurcating branches for a class of saturated nonlinear Schrödinger systems. Adv. Nonlinear Stud. 16(1), 95–113 (2015)

    MATH  Google Scholar 

  16. 16.

    Mandel, R.: Minimal energy solutions for cooperative nonlinear Schrödinger systems. Nonlinear Differ. Equ. Appl. NoDEA 22(2), 239–262 (2015)

    Article  MATH  Google Scholar 

  17. 17.

    Mandel, R., Montefusco, E., Pellacci, B.: Oscillating solutions for nonlinear Helmholtz equations. Zeitschrift für angewandte Mathematik und Physik 6, 121 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton, New Jersey (1970)

    Google Scholar 

  19. 19.

    Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Monographs in harmonic analysis 3, Princeton mathematical series, 43, 2nd edn. Princeton University Press, Princeton (1995)

    Google Scholar 

  20. 20.

    Szulkin, A., Weth, T.: Ground state solutions for some indefinite variational problems. J. Funct. Anal. 257(12), 3802–3822 (2009)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Rainer Mandel.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mandel, R., Scheider, D. Dual variational methods for a nonlinear Helmholtz system. Nonlinear Differ. Equ. Appl. 25, 13 (2018). https://doi.org/10.1007/s00030-018-0504-z

Download citation

Mathematics Subject Classification

  • Primary 35J50
  • Secondary 35J05

Keywords

  • Nonlinear Helmholtz sytem
  • Dual variational methods