Symmetry, uniqueness and nondegeneracy of positive solutions of the Choquard equation

We study positive solutions \(Q∈ H^1( R^N)\) of \[-Δ Q+Q=(I_α*Q^p)Qᵖ⁻¹,\] where \(I_α\) is the Riesz potential of order \(α\), \(N≥1\), \(0<α<N\), \(p>1+α/N\), and \((N-2)p<N+α\). We prove that every such solution is radially symmetric and strictly decreasing about some point. For \(p≥2\), we establish uniqueness up to translations and show that the real linearized operator has Morse index one and kernel spanned by \(∂_1Q,…,∂_NQ\). The symmetry proof uses the method of moving planes. Uniqueness follows from nondegeneracy and compactness through a continuation argument.