Classification of Positive Solitary Solutions of the Nonlinear Choquard Equation

• Published in: Archive for rational mechanics and analysis Vol. 195; no. 2; pp. 455 - 467
• Main Authors: Ma, Li, Zhao, Lin
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-02-2010; Springer; Springer Nature B.V
• Subjects: Classical Mechanics; Complex Systems; Exact sciences and technology; Fluid- and Aerodynamics; Fundamental areas of phenomenology (including applications); Mathematical and Computational Physics; Mathematical methods in physics; Numerical approximation and analysis; Ordinary and partial differential equations, boundary value problems; Physics; Physics and Astronomy; Theoretical; Elliptic System; Sobolev Inequality; Uniqueness Result; Integral Equation; Nonlinear Elliptic Equation; Hartree Fock theory; Infinitesimal transformation; Bessel functions; Generalized; Uniqueness theorem; Stationary condition; Fix point; Fixed point theorem; Integral equations; Classification; Elliptic equations; Positive solution; Modelling; Non linear effect; Riesz basis
• ISSN: 0003-9527; 1432-0673
• DOI: 10.1007/s00205-008-0208-3

In this paper, we settle the longstanding open problem concerning the classification of all positive solutions to the nonlinear stationary Choquard equation
which can be considered as a certain approximation of the Hartree–Fock theory for a one component plasma as explained in Lieb and Lieb–Simon’s papers starting from 1970s. We first prove that all the positive solutions of this equation must be radially symmetric and monotone decreasing about some fixed point. Interestingly, to use the new method of moving planes introduced by Chen–Li–Ou, we deduce the problem into an elliptic system. As a key step, we transform this differential system into a system of integral equations with the help of Riesz and Bessel potentials, and then use the method of a moving plane in an integral form. Next, using radial symmetry, we deduce the uniqueness result from Lieb’s work. Our argument can be adapted well to study the radial symmetry of positive solutions of the equation in the generalized form
.