Positive solutions to the nonlinear Choquard equation: symmetry, uniqueness, and nondegeneracy

We study the symmetry, uniqueness, and nondegeneracy of positive solutions to the Choquard equation $-Δu+u=(I_α*u^p)u^{p-1}$ in $\mathbb{R}^N$, which have long remained open problems. For $N\geq3$, $α\in(0,N)$, and $\frac{N+α}{N}<p<\frac{N+α}{N-2}$,we prove that every positive $H^1$-solution is radially symmetric and strictly decreasing in the radial variable. The symmetry proof relies on the moving-plane method, where the sublinear singularity is controlled by analyzing the far-field behavior of the nonlocal term. Uniqueness and nondegeneracy are established through a functional framework hinging on a nonnegative joint form and a Schur complement argument.Radial nondegeneracy and Morse index one are obtained by ruling out the spectral interval $(0,p-1]$ via a strict cumulative Riesz identity, which leads to uniqueness through a Nehari two-well argument. Nondegeneracy in the full space $H^1(\mathbb{R}^N)$ follows from the spherical-harmonic decomposition of the linearized form. These results hold for the full range of $p$ when $α=2$, and for $p\geq2$ when $α\in(2,N)$. The cases $α\in(0,2)$ and $α\in(2,N)$ with $p<2$ remain open.