Classification of Positive Solitary Solutions of the Nonlinear Choquard Equation

Abstract

In this paper, we settle the longstanding open problem concerning the classification of all positive solutions to the nonlinear stationary Choquard equation $$\Delta u-u+2u\left(\frac{1}{|x|}*|u|^2\right)=0, \quad u\in H^1(\mathbb{R}^3),$$ 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 $$u=K_1*F\left(u,K_2*u\right)$$ .