A guide to the Choquard equation

Abstract

We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations $$-\Delta u+V(x)u=(|x{|}^{-(N-\alpha )}\ast |u{|}^p)|u{|}^{p-2}u\text{in RN},$$ and some of its variants and extensions.

39 pages

References in corpus (10)

• Gravitation and quantummechanical localization of macroobjects
• The Schrödinger-Newton equation and its foundations
• Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent
• Nodal solutions for the Choquard equation
• The Schrödinger-Newton equation as non-relativistic limit of self-gravitating Klein-Gordon and Dirac fields
• Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains
• Least action nodal solutions for the quadratic Choquard equation
• The one-dimensional Schrödinger-Newton equations
• The Schrödinger-Newton equations beyond Newton
• The Schrodinger-Newton System with Self-Field Coupling

Cited by in corpus (50)

• Multiple normalized solutions for a Sobolev critical Schrödinger-Poisson-Slater equation
• Fractional Choquard Equation with Critical Nonlinearities
• Choquard equations with critical nonlinearities
• Odd symmetry of least energy nodal solutions for the Choquard equation
• Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent
• Groundstates for Choquard type equations with Hardy-Littlewood-Sobolev lower critical exponent
• Multiplicity and concentration results for a fractional Choquard equation via penalization method
• Existence of groundstates for a class of nonlinear Choquard equations in the plane
• High energy positive solutions for a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponents
• Standing waves with a critical frequency for nonlinear Choquard equations
• Positive solutions for quasilinear elliptic inequalities and systems with nonlocal terms
• Groundstates of the Choquard equations with a sign-changing self-interaction potential
• A nonrelativistic limit for AdS perturbations
• Higher order evolution inequalities with nonlinear convolution terms
• On the radial linear stability of nonrelativistic $\ell$-boson stars
• Normalized solutions for a Choquard equation with exponential growth in ${\mathbb{R}}^2$
• On fractional Schrödinger equations with Hartree type nonlinearities
• Limit profiles for singularly perturbed Choquard equations with local repulsion
• Optimal parabolic upper bound for the energy-momentum relation of a strongly coupled polaron
• Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities
• Ergodic Mean-Field Games with aggregation of Choquard-type
• Nonlocal planar Schrödinger-Poisson systems in the fractional Sobolev limiting case
• Nonlinear Perturbations of a periodic magnetic Choquard equation with Hardy-Littlewood-Sobolev critical exponent
• Multiple positive solutions for degenerate Kirchhoff equations with singular and Choquard nonlinearity
• Nonlinear scalar field equation with competing nonlocal terms
• Quasilinear Choquard equations involving N-Laplacian and critical exponential nonlinearity
• Existence and stability of standing waves for coupled nonlinear Hartree type equations
• On the asymptotic decay of the Schrödinger--Newton ground state
• Existence and uniqueness of ground states for $p$ - Choquard model in 3D
• Asymptotic decay of solutions for sublinear fractional Choquard equations
• Schrödinger-Newton-Hooke system in higher dimensions. Part I: Stationary states
• Normalized solutions to a Choquard equation involving mixed local and nonlocal operators
• Saddle solutions for the fractional Choquard equation
• Choquard equations with critical exponential nonlinearities in the zero mass case
• Positive solutions with prescribed mass for a planar Choquard equation with critical growth
• On a class of singular Hamiltonian Choquard-type elliptic systems with critical exponential growth
• Standing waves for nonlinear Hartree type equations: existence and qualitative properties
• On elliptic problems with Choquard term and singular nonlinearity
• A Choquard type equation with a singular absorption nonlinearity in two dimension
• Schrödinger-Poisson systems with zero mass in the Sobolev limiting case
• Schrödinger equation in dimension two with competing logarithmic self-interaction
• Fujita type results for quasilinear parabolic inequalities with nonlocal terms
• Infinitely many solutions for $p$-fractional Choquard type equations involving general nonlocal nonlinearities with critical growth via the concentration compactness method
• Nonlinear Schrödinger-Poisson systems in dimension two: the zero mass case
• Compact embedding from variable-order Sobolev space to $L^{q(x)}(\Omega )$ and its application to Choquard equation with variable order and variable critical exponent
• Polyharmonic inequalities with nonlocal terms
• Nonlinear Inequalities with Double Riesz Potentials
• Ground states of the planar nonlinear Schrödinger--Newton system with a point interaction
• Singular solutions for coercive quasilinear elliptic inequalities with nonlocal terms
• Nonlocal problem with critical exponential nonlinearity of convolution type: A non-resonant case