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 -Δ u+u=(I_α*uᵖ)uᵖ⁻¹ in RN, which have long remained open problems. For N≥3, α∈(0,N), and N+α/N<p<N+α/N-2,we prove that every positive H¹-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 H¹(RN) 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≥2 when α∈(2,N). The cases α∈(0,2) and α∈(2,N) with $p<2$ remain open.