Semi-classical states for the nonlinear Choquard equations: existence, multiplicity and concentration at a potential well

DOI10.4171/rmi/1105zbMath1431.35169arXiv1708.02356OpenAlexW2971900296MaRDI QIDQ2280514

Silvia Cingolani, Kazunaga Tanaka

Publication date: 18 December 2019

Published in: Revista Matemática Iberoamericana (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1708.02356

zbMATH Keywords

positive solutionspotential wellsemiclassical statesnonlinear Choquard equationnon-local nonlinearitiesrelative cup-length

Mathematics Subject Classification ID

Variational methods applied to PDEs (35A15) Abstract critical point theory (Morse theory, Lyusternik-Shnirel'man theory, etc.) in infinite-dimensional spaces (58E05) NLS equations (nonlinear Schrödinger equations) (35Q55) PDEs in connection with quantum mechanics (35Q40) Existence problems for PDEs: global existence, local existence, non-existence (35A01) Variational methods for second-order elliptic equations (35J20) Positive solutions to PDEs (35B09)

Related Items (28)

Multiplicity and concentration of solutions for fractional Kirchhoff–Choquard equation with critical growth ⋮ Infinitely many non-radial positive solutions for Choquard equations ⋮ On the fractional NLS equation and the effects of the potential Well's topology ⋮ Existence and local uniqueness of normalized peak solutions for a Schrödinger-Newton system ⋮ Semi-classical states for elliptic system near saddle points of potentials ⋮ On some qualitative aspects for doubly nonlocal equations ⋮ Multi-peak solutions of a class of fractional \(p\)-Laplacian equations ⋮ The Benci-Cerami problem for the fractional Choquard equation with critical exponent ⋮ Semi-classical states for the Choquard equations with doubly critical exponents: Existence, multiplicity and concentration ⋮ Semiclassical states to the nonlinear Choquard equation with critical growth ⋮ Semi-classical analysis around local maxima and saddle points for degenerate nonlinear Choquard equations ⋮ Multibump positive solutions for Choquard equation with double potentials in ℝ3$$ {\mathrm{\mathbb{R}}}^3 $$ ⋮ Existence and asymptotic behavior of solutions for generalized Choquard systems ⋮ Localized nodal solutions for system of critical Choquard equations ⋮ Multiple solutions for a fractional Choquard problem with slightly subcritical exponents on bounded domains ⋮ Unnamed Item ⋮ Unnamed Item ⋮ Unnamed Item ⋮ Ground State Solutions for the Nonlinear Choquard Equation with Prescribed Mass ⋮ Localized nodal solutions for semiclassical Choquard equations ⋮ Nondegeneracy of ground states and multiple semiclassical solutions of the Hartree equation for general dimensions ⋮ Multi-bump standing waves for nonlinear Schrödinger equations with a general nonlinearity: the topological effect of potential wells ⋮ Multiple bound states of higher topological type for semi-classical Choquard equations ⋮ Standing waves for the pseudo-relativistic Hartree equation with Berestycki-Lions nonlinearity ⋮ Concentration phenomena for the Schrödinger-Poisson system in \(\mathbb{R}^2\) ⋮ Ground state solution for a class of Choquard equations involving general critical growth term ⋮ Construction of infinitely many solutions for a critical Choquard equation via local Pohožaev identities ⋮ Ground state solutions for critical Choquard equation with singular potential: existence and regularity

Cites Work

• Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential
• Multiple solutions to a magnetic nonlinear Choquard equation
• Semi-classical states for the Choquard equation
• Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity
• Multiple complex-valued solutions for nonlinear magnetic Schrödinger equations
• A guide to the Choquard equation
• Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities
• Nonlinear scalar field equations. I: Existence of a ground state
• The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems
• The effect of the domain's configuration space on the number of nodal solutions of singularly perturbed elliptic equations
• Uniqueness of ground states for pseudorelativistic Hartree equations
• Classification of positive solitary solutions of the nonlinear Choquard equation
• A note on Schrödinger-Newton systems with decaying electric potential
• Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology
• On the point-particle (Newtonian) limit of the nonlinear Hartree equation
• Semiclassical states of nonlinear Schrödinger equations
• Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations
• Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential
• Multiple solutions of the forced double pendulum equation
• Local mountain passes for semilinear elliptic problems in unbounded domains
• On gravity's role in quantum state reduction
• Intertwining semiclassical solutions to a Schrödinger-Newton system
• Nonlinear scalar field equations in \(\mathbb{R}^N\): Mountain pass and symmetric mountain pass approaches
• Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions
• Multi-bump positive solutions for a nonlinear elliptic problem in expanding tubular domains
• Positive and sign changing solutions to a nonlinear Choquard equation
• Dual variational methods in critical point theory and applications
• Multiplicity of positive solutions of nonlinear Schrödinger equations concentrating at a potential well
• Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics
• Standing waves for nonlinear Schrödinger equations with a general nonlinearity
• Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities
• Existence of Semiclassical Bound States of Nonlinear Schrödinger Equations with Potentials of the Class (V)_a
• Quantum computation, entanglement and state reduction
• Strongly interacting bumps for the Schrödinger–Newton equations
• On a nonlinear elliptic equation involving the critical sobolev exponent: The effect of the topology of the domain
• The Choquard equation and related questions
• Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation
• Spherically-symmetric solutions of the Schrödinger-Newton equations
• A remark on least energy solutions in $\mathbf {R}^N$
• Existence of groundstates for a class of nonlinear Choquard equations
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item

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