Oscillating solutions for nonlinear Helmholtz equations

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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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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Keywords

  • Nonlinear Helmholtz equations
  • Standing waves
  • Oscillating solutions

Mathematics Subject Classification

  • 35J05
  • 35J20
  • 35Q55