Existence and asymptotic behavior of standing waves of the nonlinear Helmholtz equation in the plane

DOI10.1515/anly-2016-0023zbMath1376.35004arXiv1606.00788OpenAlexW2962699658MaRDI QIDQ2628094

Gilles Evéquoz

Publication date: 12 June 2017

Published in: Analysis (München) (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1606.00788

zbMATH Keywords

resolvent estimatesvariational methodstanding wavesnonlinear Helmholtz equation

Mathematics Subject Classification ID

Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation (35J05) Variational methods for second-order elliptic equations (35J20) Semilinear elliptic equations (35J61)

Related Items

Dual variational methods for a nonlinear Helmholtz system ⋮ Multiple standing waves for the nonlinear Helmholtz equation concentrating in the high frequency limit ⋮ The limiting absorption principle for periodic differential operators and applications to nonlinear Helmholtz equations ⋮ Existence and multiplicity of solutions for a class of Helmholtz systems ⋮ Uniform resolvent estimates for Laplace-Beltrami operator on the flat Euclidean cone ⋮ Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium ⋮ An annulus multiplier and applications to the limiting absorption principle for Helmholtz equations with a step potential ⋮ On the Helmholtz equation and Dancer’s-type entire solutions for nonlinear elliptic equations ⋮ Uncountably many solutions for nonlinear Helmholtz and curl-curl equations ⋮ On a fourth‐order nonlinear Helmholtz equation ⋮ Variational methods for breather solutions of nonlinear wave equations ⋮ The 𝑊^{𝑠,𝑝}-boundedness of stationary wave operators for the Schrödinger operator with inverse-square potential

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Nonlinear scalar field equations. II: Existence of infinitely many solutions
• Nonlinear scalar field equations. I: Existence of a ground state
• The concentration-compactness principle in the calculus of variations. The locally compact case. I
• A dual approach in Orlicz spaces for the nonlinear Helmholtz equation
• The concentration-compactness principle in the calculus of variations. The locally compact case. II
• Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators
• Existence of solutions for semilinear elliptic equations with indefinite linear part
• Entire solutions of semilinear elliptic equations
• Non trivial \(L^q\) solutions to the Ginzburg-Landau equation
• Elliptic partial differential equations of second order
• Inverse problem for a nonlinear Helmholtz equation
• Minimax theorems
• Dual variational methods in critical point theory and applications
• Dual variational methods and nonvanishing for the nonlinear Helmholtz equation
• Real solutions to the nonlinear Helmholtz equation with local nonlinearity
• Uniqueness of radiative solutions to the Schrödinger wave equation
• Spectral and scattering theory for Schrödinger operators
• Growth properties of solutions of the reduced wave equation with a variable coefficient
• Variational Methods
• Inverse acoustic and electromagnetic scattering theory
• A restriction theorem for the Fourier transform