We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian (−Δ)s with s ∊ (0,1) for any space dimensions N ≥ 1. By extending a monotonicity formula found by Cabré and Sire , we show that the linear equation (−Δ)su+Vu=0 in ℝN has at most one radial and bounded solution vanishing at infinity, provided that the potential V is radial and nondecreasing. In particular, this result implies that all radial eigenvalues of the corresponding fractional Schrödinger operator H = (−Δ)s + V are simple. Furthermore, by combining these findings on linear equations with topological bounds for a related problem on the upper half‐space ℝ+N+1, we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation −Δ)sQ+Q−|Q|αQ=0 in ℝN for arbitrary space dimensions N ≥ 1 and all admissible exponents α > 0. This generalizes the nondegeneracy and uniqueness result for dimension N = 1 recently obtained by the first two authors and, in particular, the uniqueness result for solitary waves of the Benjamin‐Ono equation found by Amick and Toland .© 2016 Wiley Periodicals, Inc.

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We prove uniqueness of ground states Q ∈ H 1/2 ‫ޒ(‬ 3 ) for the pseudorelativistic Hartree equation, − + m 2 Q − |x| −1 * |Q| 2 Q = −µQ, in the regime of Q with sufficiently small L 2 -mass. This result shows that a uniqueness conjecture by Lieb and Yau [1987] holds true at least for N = |Q| 2 1 except for at most countably many N . Our proof combines variational arguments with a nonrelativistic limit, leading to a certain Hartreetype equation (also known as the Choquard-Pekard or Schrödinger-Newton equation). Uniqueness of ground states for this limiting Hartree equation is well-known. Here, as a key ingredient, we prove the socalled nondegeneracy of its linearization. This nondegeneracy result is also of independent interest, for it proves a key spectral assumption in a series of papers on effective solitary wave motion and classical limits for nonrelativistic Hartree equations. IntroductionThe pseudorelativistic Hartree energy functional, given (in appropriate units) byarises in the mean-field limit of a quantum system describing many self-gravitating, relativistic bosons with rest mass m > 0. Such a physical system is often referred to as a boson star, and various models for these -at least theoretical -objects have attracted a great deal of attention in theoretical and numerical astrophysics over the past years. In order to gain some rigorous insight into the theory of boson stars, it is of particular interest to study ground states (that is, minimizers) for the variational problemwhere the parameter N > 0 plays the role of the stellar mass. Provided that problem (1-2) has indeed a ground state Q ∈ H 1/2 ‫ޒ(‬ 3 ), one readily finds that it satisfies the pseudorelativistic Hartree equation, − + m 2 Q − |x| −1 * |Q| 2 Q = −µQ, (1-3) with µ = µ(Q) ∈ ‫ޒ‬ being some Lagrange multiplier.MSC2000: 35Q55.

We study the nonlinear equationwhich is known to describe the dynamics of pseudo-relativistic boson stars in the meanfield limit. For positive mass parameters, m > 0, we prove existence of travelling solitary waves, ψ(t, x) = e itµ ϕ v (x − vt), with speed |v| < 1, where c = 1 corresponds to the speed of light in our units. Due to the lack of Lorentz covariance, such travelling solitary waves cannot be obtained by applying a Lorentz boost to a solitary wave at rest (with v = 0). To overcome this difficulty, we introduce and study an appropriate variational problem that yields the functions ϕ v ∈ H 1/2 (R 3 ) as minimizers, which we call boosted ground states. Our existence proof makes extensive use of concentration-compactness-type arguments.In addition to their existence, we prove orbital stability of travelling solitary waves ψ(t, x) = e itµ ϕ v (x − vt) and pointwise exponential decay of ϕ v (x) in x.