Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics

Abstract

We consider a semilinear elliptic problem [- Δu + u = (I_α\ast \abs{u}^p) \abs{u}^{p - 2} u \quad\text{in (\mathbb{R}^N),}] where (I_α) is a Riesz potential and (p>1). This family of equations includes the Choquard or nonlinear Schrödinger-Newton equation. For an optimal range of parameters we prove the existence of a positive groundstate solution of the equation. We also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point. Finally, we derive the decay asymptotics at infinity of the groundstates.

23 pages, updated bibliography

References in corpus (1)

• Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains

Cited by in corpus (117)

• A guide to the Choquard equation
• Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent
• Nodal solutions for the Choquard equation
• Semi-classical states for the Choquard equation
• Limiting Bourgain-Brezis estimates for systems: theme and variations
• Stationary waves with prescribed $L^2$-norm for the planar Schrödinger-Poisson system
• Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains
• Groundstates and radial solutions to nonlinear Schrödinger-Poisson-Slater equations at the critical frequency
• 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
• On fractional Schrodinger systems of Choquard type
• Choquard equations under confining external potentials
• Groundstates for Choquard type equations with Hardy-Littlewood-Sobolev lower critical exponent
• Ground state solutions for non-autonomous fractional Choquard equations
• The logarithmic Choquard equation: sharp asymptotics and nondegeneracy of the groundstate
• Multiplicity and concentration results for a fractional Choquard equation via penalization method
• Classification of nonnegative solutions to static Schrödinger-Hartree and Schrödinger-Maxwell equations with combined nonlinearities
• Classification of nonnegative solutions to static Schrödinger-Hartree-Maxwell type equations
• 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
• Semiclassical stationary states for nonlinear Schrödinger equations under a strong external magnetic field
• Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in ${\mathbb{R}}^2$
• Existence of solutions for a fractional Choquard--type equation in $\mathbb{R}$ with critical exponential growth
• Groundstates of the Choquard equations with a sign-changing self-interaction potential
• Multiplicity and concentration results for a fractional Schrödinger-Poisson type equation with magnetic field
• Existence and asymptotic behavior of the least energy solutions for fractional Choquard equations with potential well
• Multiple nodal solutions of nonlinear Choquard equations
• Interpolation inequalities between Sobolev and Morrey-Campanato spaces: A common gateway to concentration-compactness and Gagliardo-Nirenberg interpolation inequalities
• Properties of groundstates of nonlinear Schrödinger equations under a weak constant magnetic field
• Multiple and concentration of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent
• Traveling waves for some nonlocal 1D Gross-Pitaevskii equations with nonzero conditions at infinity
• Normalized solutions for a Choquard equation with exponential growth in ${\mathbb{R}}^2$
• On fractional Schrödinger equations with Hartree type nonlinearities
• The Fröhlich Polaron at Strong Coupling -- Part I: The Quantum Correction to the Classical Energy
• On the classification of the spectrally stable standing waves of the Hartree problem
• Limit profiles for singularly perturbed Choquard equations with local repulsion
• The existence of a nontrivial weak solution to a double critical problem involving fractional Laplacian in ${\mathbb{R}}^n$ with a Hardy term
• Semiclassical Sobolev constants for the electro-magnetic Robin Laplacian
• Existence of solutions for critical Choquard equations via the concentration compactness method
• On uniqueness and non-degeneracy of anisotropic polarons
• Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities
• Ergodic Mean-Field Games with aggregation of Choquard-type
• Semiclassical states for Choquard type equations with critical growth: critical frequency case
• The Lieb-Yau Conjecture for Ground States of Pseudo-Relativistic Boson Stars
• Nonlocal Hardy type inequalities with optimal constants and remainder terms
• Ground states and semiclassical states of nonlinear Choquard equations involving Hardy-Littlewood-Sobolev critical growth
• Uniqueness and Nondegeneracy of Ground States for Choquard Equations in three dimensions
• Stability and instability of standing waves for a generalized Choquard equation with potential
• Nonlinear scalar field equation with competing nonlocal terms
• Coron problem for nonlocal equations invloving Choquard nonlinearity
• Multiple positive solutions for degenerate Kirchhoff equations with singular and Choquard nonlinearity
• Concentration Behavior of Nonlinear Hartree-type Equation with almost Mass Critical Exponent
• Quasilinear Choquard equations involving N-Laplacian and critical exponential nonlinearity
• On the asymptotic decay of the Schrödinger--Newton ground state
• Existence and uniqueness of ground states for $p$ - Choquard model in 3D
• Mass minimizers and concentration for nonlinear Choquard equations in ${\mathbb{R}}^N$
• Nondegeneracy of ground states and multiple semiclassical solutions of the Hartree equation for general dimensions
• Normalized solutions to a Choquard equation involving mixed local and nonlocal operators
• Asymptotic decay of solutions for sublinear fractional Choquard equations
• Groundstates for nonlinear fractional Choquard equations with general nonlinearities
• Uniqueness of positive solutions with Concentration for the Schrödinger-Newton problem
• Scattering of radial data in the focusing NLS and generalized Hartree equations
• Saddle solutions for the fractional Choquard equation
• Non-existence and multiplicity of positive solutions for Choquard equations with critical combined nonlinearities
• Choquard equations with critical exponential nonlinearities in the zero mass case
• Existence of solutions for a class of nonlinear Choquard equations with critical growth
• Positive solutions with prescribed mass for a planar Choquard equation with critical growth
• Semirelativistic Choquard equations with singular potentials and general nonlinearities arising from Hartree-Fock theory
• Limit profiles and uniqueness of ground states to the nonlinear Choquard equations
• Normalized ground states for the fractional nonlinear Schrödinger equations
• Stable blow-up dynamics in the $L^2$-critical and $L^2$-supercritical generalized Hartree equation
• High and low perturbations of the critical Choquard equation on the Heisenberg group
• Standing waves for nonlinear Hartree type equations: existence and qualitative properties
• Schrödinger-Poisson systems with zero mass in the Sobolev limiting case
• The Fröhlich Polaron at Strong Coupling -- Part II: Energy-Momentum Relation and Effective Mass
• Orbital stability of solitary waves for the generalized Choquard model
• Schrödinger equation in dimension two with competing logarithmic self-interaction
• Singularly perturbed critical Choquard equations
• Nonlinear Schrödinger-Poisson systems in dimension two: the zero mass case
• Groundstates for Choquard equations with the upper critical exponent
• On nodal solutions of a nonlocal Choquard equation in a bounded domain
• Critical growth elliptic problems involving Hardy-Littlewood-Sobolev critical exponent in non-contractible domains
• Uniqueness and nondegeneracy of solutions for a critical nonlocal equation
• Isolated singularities of positive solutions for Choquard equations in sublinear case
• On the critical Choquard equation with potential well
• Symmetric properties for Choquard equations involving fully nonlinear nonlocal operator
• The existence of least energy nodal solutions for some class of Kirchhoff equations and Choquard equations
• Soliton dynamics for the generalized Choquard equation
• Positive bound states to nonlinear Choquard equations in the presence of nonsymmetric potentials
• Semiclassical analysis for pseudo-relativistic Hartree equations
• Nonlocal Pertubations of Fractional Choquard Equation
• Some a priori estimates for a critical Schrodinger-Newton equation
• Semi-classical states for the nonlinear Choquard equations: existence, multiplicity and concentration at a potential well
• Positive solutions for double singularly perturbed Schroedinger Maxwell systems
• Generalized Schrödinger-Newton system in dimension $N\ge 3$: critical case
• On a class of mixed Choquard-Schrödinger-Poisson system
• Existence of a nontrivial solution for a strongly indefinite periodic Schrodinger-Poisson system
• Maximizers for Gagliardo-Nirenberg inequalities and related non-local problems
• Quantitative Properties on the Steady States to A Schrödinger-Poisson-Slater System
• Existence of Positive Solutions for a class of Quasilinear Schrödinger Equations of Choquard type
• Orbital stability of standing waves for Schrödinger type equations with slowly decaying linear potential
• Classification of solutions to conformally invariant systems with mixed order and exponentially increasing or nonlocal nonlinearity
• Nonlocal problem with critical exponential nonlinearity of convolution type: A non-resonant case
• On doubly nonlocal $p$-fractional coupled elliptic system
• Ground states of the planar nonlinear Schrödinger--Newton system with a point interaction
• Nonlocal problems with Hardy-Littlewood-Sobolev critical exponent and Hardy potential
• Multiple ${\mathbb{S}}^1$-orbits for the Schrödinger-Newton system
• Three solutions for a fractional elliptic problem with asymmetric critical Choquard nonlinearity
• Ground state solutions for a nonlinear Choquard equation
• Standing waves with prescribed mass for the Schrödinger equations with van der Waals type potentials
• Nondegeneracy of solutions for a critical Hartree equation
• Concentration phenomena for a fractional Choquard equation with magnetic field
• Existence, regularity, asymptotic decay and radiality of solutions to some extension problems
• Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity
• Existence and nonexistence of solutions to Choquard equations