Uniqueness of radial solutions for the fractional Laplacian

Abstract

We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian $(-\Delta {)}^s$ with $s\in (0,1)$ for any space dimensions $N\ge 1$. By extending a monotonicity formula found by Cabre and Sire \cite{CaSi-10}, we show that the linear equation $(-\Delta {)}^{s}u+Vu=0$ in ${\mathbb{R}}^N$ has at most one radial and bounded solution vanishing at infinity, provided that the potential $V$ is a radial and non-decreasing. In particular, this result implies that all radial eigenvalues of the corresponding fractional Schrödinger operator $H=(-\Delta {)}^s+V$ are simple. Furthermore, by combining these findings on linear equations with topological bounds for a related problem on the upper half-space ${\mathbb{R}}_{+}^{N+1}$, we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation $(-\Delta {)}^{s}Q+Q-|Q{|}^{\alpha }Q=0$ in ${\mathbb{R}}^N$ for arbitrary space dimensions $N\ge 1$ and all admissible exponents $\alpha >0$. This generalizes the nondegeneracy and uniqueness result for dimension N=1 recently obtained by the first two authors in \cite{FrLe-10} and, in particular, the uniqueness result for solitary waves of the Benjamin--Ono equation found by Amick and Toland \cite{AmTo-91}.

38 pages; revised version; various typos corrected; proof of Lemma 8.1 corrected; discussion of case κ_* =1 in the proof of Theorem 2 corrected with new Lemma A.2; accepted for publication in Comm. Pure. Appl. Math

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