On a fourth‐order nonlinear Helmholtz equation

DOI10.1112/jlms.12196zbMath1422.35068arXiv1803.08249OpenAlexW2962704872WikidataQ128896790 ScholiaQ128896790MaRDI QIDQ4967967

Jean-Baptiste Castéras, Rainer Mandel, Denis Bonheure

Publication date: 11 July 2019

Published in: Journal of the London Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1803.08249

zbMATH Keywords

existence of soutionsfourth order nonlinear Helmholtz equation

Mathematics Subject Classification ID

Existence problems for PDEs: global existence, local existence, non-existence (35A01) Variational methods for higher-order elliptic equations (35J35) Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian (35J91)

Related Items (6)

Time-harmonic solutions for Maxwell's equations in anisotropic media and Bochner-Riesz estimates with negative index for non-elliptic surfaces ⋮ Existence and multiplicity of solutions for a class of Helmholtz systems ⋮ Normalized solutions for a fourth-order Schrödinger equation with a positive second-order dispersion coefficient ⋮ Uncountably many solutions for nonlinear Helmholtz and curl-curl equations ⋮ Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime ⋮ Radial and non-radial multiple solutions to a general mixed dispersion NLS equation

Cites Work

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• Refined energy inequality with application to well-posedness for the fourth order nonlinear Schrödinger type equation on torus
• Branch continuation inside the essential spectrum for the nonlinear Schrödinger equation
• Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equations of fourth order in dimensions \(d\geqslant 9\)
• Regularity of very weak solutions for elliptic equation of divergence form
• Oscillating solutions for nonlinear Helmholtz equations
• The focusing energy-critical fourth-order Schrödinger equation with radial data
• Global well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small data in modulation and Sobolev spaces
• Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case
• Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation
• The cubic fourth-order Schrödinger equation
• The nonlinear Schrödinger equation. Self-focusing and wave collapse
• Non trivial \(L^q\) solutions to the Ginzburg-Landau equation
• Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion
• Elliptic partial differential equations of second order
• Time decay for the nonlinear beam equation
• Dual variational methods for a nonlinear Helmholtz system
• A note on the local regularity of distributional solutions and subsolutions of semilinear elliptic systems
• On the restriction conjecture
• Infinitely many large amplitude homoclinic orbits for a class of autonomous Hamiltonian systems
• Dual variational methods and nonvanishing for the nonlinear Helmholtz equation
• Real solutions to the nonlinear Helmholtz equation with local nonlinearity
• Existence and asymptotic behavior of standing waves of the nonlinear Helmholtz equation in the plane
• On Representation Formulas and Radiation Conditions
• Scattering theory for the fourth-order Schrödinger equation in low dimensions
• REGULARITY OF VERY WEAK SOLUTIONS FOR NONHOMOGENEOUS ELLIPTIC EQUATION
• THE MASS-CRITICAL FOURTH-ORDER SCHRÖDINGER EQUATION IN HIGH DIMENSIONS
• Self-Focusing with Fourth-Order Dispersion
• The Fourth-Order Dispersive Nonlinear Schrödinger Equation: Orbital Stability of a Standing Wave
• Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I
• Travelling Waves in a Suspension Bridge
• Orbitally Stable Standing Waves of a Mixed Dispersion Nonlinear Schrödinger Equation
• Dispersion estimates for fourth order Schrödinger equations
• Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime
• Classical Fourier Analysis
• Large-Amplitude Periodic Oscillations in Suspension Bridges: Some New Connections with Nonlinear Analysis
• Blowup for Biharmonic NLS
• Waveguide solutions for a nonlinear Schrödinger equation with mixed dispersion
• Strong Instability of Ground States to a Fourth Order Schrödinger Equation
• A restriction theorem for the Fourier transform

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