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

Song Fan; Gui-Dong Li; Jianjun Zhang

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arXiv · 2611.02980

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

Song Fan·Gui-Dong Li·Jianjun Zhang

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Abstract

We study positive solutions $Q\in H^1(\mathbb R^N)$ of \[-ΔQ+Q=(I_α*Q^p)Q^{p-1},\] where $I_α$ is the Riesz potential of order $α$, $N\ge1$, $0<α 1+α/N$, and $(N-2)p<N+α$. We prove that every such solution is radially symmetric and strictly decreasing about some point. For $p\ge2$, we establish uniqueness up to translations and show that the real linearized operator has Morse index one and kernel spanned by $\partial_1Q,\ldots,\partial_NQ$. The symmetry proof uses the method of moving planes. Uniqueness follows from nondegeneracy and compactness through a continuation argument.

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BibTeX RIS

Song Fan, Gui-Dong Li, Jianjun Zhang. 2026-11-02. Symmetry, uniqueness and nondegeneracy of positive solutions of the Choquard equation. https://arxiv.org/abs/2611.02980

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