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Oscillating solutions for nonlinear Helmholtz equations

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Abstract

Existence results for radially symmetric oscillating solutions for a class of nonlinear autonomous Helmholtz equations are given and their exact asymptotic behaviour at infinity is established. Some generalizations to nonautonomous radial equations as well as existence results for nonradial solutions are found. Our theorems prove the existence of standing waves solutions of nonlinear Klein–Gordon or Schrödinger equations with large frequencies.

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References

  1. Ambrosetti, A., Brezis, H., Cerami, G.: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122(2), 519–543 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambrosetti, A., Malchiodi, A.: Nonlinear Analysis and Semilinear Elliptic Problems. Cambridge Studies in Advanced Mathematics, vol. 104. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  3. Benci, V., Fortunato, D.: Variational Methods in Nonlinear Field Equations. Solitary Waves Hylomorphic Solitons and Vortices. Springer Monographs in Mathematics. Springer, Cham (2014)

    MATH  Google Scholar 

  4. Boccardo, L., Escobedo, M., Peral, I.: A Dirichlet problem involving critical exponents. Nonlinear Anal. 24(11), 1639–1648 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berestycki, H., Gallouët, T., Kavian, O.: Équations de champs scalaires euclidiens non linéaires dans le plan. C. R. Acad. Sci. Paris Sér. I Math. 297(5), 307–310 (1983)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  7. Brezis, H., Oswald, L.: Remarks on sublinear elliptic equations. Nonlinear Anal. 10(1), 55–64 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, Y.F., Beckwitt, K., Wise, F.W., Malomed, B.A.: Criteria for the experimental observation of multidimensional optical solitons in saturable media. Phys. Rev. E 70, 046610 (2004)

    Article  Google Scholar 

  9. Christodoulides, D.N., Coskun, T.H., Mitchell, M., Segev, M.: Theory of incoherent self-focusing in biased photorefractive media. Phys. Rev. Lett. 78, 646–649 (1997)

    Article  Google Scholar 

  10. Evequoz, G.: A dual approach in Orlicz spaces for the nonlinear Helmholtz equation. Z. Angew. Math. Phys. 66(6), 2995–3015 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Farina, A.: Some symmetry results and Liouville-type theorems for solutions to semilinear equations. Nonlinear Anal. 121, 223–229 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gui, C., Luo, X., Zhou, F.: Asymptotic behavior of oscillating radial solutions to certain nonlinear equations. Part II. Methods Appl. Anal. 16(4), 459–468 (2009)

    MathSciNet  MATH  Google Scholar 

  16. Gui, C., Zhou, F.: Asymptotic behavior of oscillating radial solutions to certain nonlinear equations. Methods Appl. Anal. 15(3), 285–295 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Gutiérrez, S.: Non trivial \(L^q\) solutions to the Ginzburg–Landau equation. Math. Ann. 328(1–2), 125 (2004)

    MathSciNet  MATH  Google Scholar 

  18. Reichel, W., Walter, W.: Radial solutions of equations and inequalities involving the p-Laplacian. J. Inequal. Appl. 1(1), 47–71 (1997)

    MathSciNet  MATH  Google Scholar 

  19. Simon, B.: Schrödinger semigroups. Bull. Am. Math. Soc. 7(3), 447–526 (1982)

    Article  MATH  Google Scholar 

  20. Soave, N., Verzini, G.: Bounded solutions for a forced bounded oscillator without friction. J. Differ. Equ. 256, 2526–2558 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Strauss, W.A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55(2), 149162 (1977)

    Article  MathSciNet  Google Scholar 

  22. Struwe, M.: Variational Methods, Results in Mathematics and Related Areas, vol. 34, 4th edn. Springer, Berlin (2008)

    Google Scholar 

  23. Swanson, C.A.: Comparison and Oscillation Theory of Linear Differential Equations. Mathematics in Science and Engineering, vol. 48. Academic Press, New York (1968)

    Book  MATH  Google Scholar 

  24. Swanson, C.A.: Semilinear second-order elliptic oscillation. Can. Math. Bull. 22(2), 139–157 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  25. Stuart, C.A., Zhou, H.S.: Applying the mountain pass theorem to an asymptotically linear elliptic equation on \({\mathbb{R}}^N\). Commun. Partial Differ. Equ. 24(9–10), 1731–1758 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Verzini, G.: Bounded solutions to superlinear ODE’s: a variational approach. Nonlinearity 16(6), 2013–2028 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Willem, M.: Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24. Birkhäuser Boston Inc, Boston, MA (1996)

    Google Scholar 

  28. Yang, J.: Stability of vortex solitons in a photorefractive optical lattice. New J. Phys. 6(1), 47 (2004)

    Article  Google Scholar 

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Authors and Affiliations

  1. Institut für Analysis, Karlsruher Institut für Technologie, Englerstraße 2, 76131, Karlsruhe, Germany

    Rainer Mandel

  2. Dipartimento di Matematica, ”Sapienza” Università di Roma, p.le Aldo Moro 5, 00185, Rome, Italy

    Eugenio Montefusco

  3. Dipartimento di Scienze e Tecnologie, Università di Napoli ”Parthenope”, Centro Direzionale, Isola C4, 80143, Naples, Italy

    Benedetta Pellacci

Authors
  1. Rainer Mandel
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  2. Eugenio Montefusco
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  3. Benedetta Pellacci
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Corresponding author

Correspondence to Eugenio Montefusco.

Additional information

This research was partially supported by the German Research Foundation (DFG) through the grant MA 6290/2-1 and CRC 1173.

This research was partially supported by MIUR-PRIN project \(2015KB9WPT_006\) - PE1, “Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni” (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM); University Project ”Sostegno alla ricerca individuale per il trienno 2016–2018”.

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Mandel, R., Montefusco, E. & Pellacci, B. Oscillating solutions for nonlinear Helmholtz equations. Z. Angew. Math. Phys. 68, 121 (2017). https://doi.org/10.1007/s00033-017-0859-8

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  • DOI: https://doi.org/10.1007/s00033-017-0859-8

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