Uniqueness of radial solutions for the fractional Laplacian (Q2816393)

In the interesting paper under review the authors prove general uniqueness results for radial solutions to linear and nonlinear equations involving the fractional Laplacian \((-\Delta)^s\) with \(s\in(0,1)\) for any space dimensions \(N\geq1.\)NEWLINENEWLINEMore precisely, by extending a monotonicity formula due to \textit{X. Cabré} and \textit{Y. Sire} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 31, No. 1, 23--53 (2014; Zbl 1286.35248)], it is shown that the linear equation NEWLINE\[NEWLINE (-\Delta)^su+Vu=0\quad \text{in}\;\mathbb{R}^N NEWLINE\]NEWLINE has at most one radial and bounded solution vanishing at infinity, provided that the potential \(V\) is radial and nondecreasing. In particular, this implies that all radial eigenvalues of the corresponding fractional Schrödinger operator \((-\Delta)^s+V\) are simple.NEWLINENEWLINEThese linear results are further combined with topological bounds for a related problem on the upper half-space \(\mathbb{R}^{N+1}_+\) to get uniqueness and nondegeneracy of ground state solutions to the nonlinear equation NEWLINE\[NEWLINE (-\Delta)^sQ+Q-|Q|^\alpha Q=0\quad \text{in}\;\mathbb{R}^N NEWLINE\]NEWLINE for arbitrary space dimensions \(N\geq1\) and all admissible exponents \(\alpha>0.\) This generalizes the nondegeneracy and uniqueness result for dimension \(N=1\) obtained by the first two authors [Acta Math. 210, No. 2, 261--318 (2013; Zbl 1307.35315)] and, in particular, the uniqueness result of \textit{C. J. Amick} and \textit{J. F. Toland} [Acta Math. 167, No. 1--2, 107--126 (1991; Zbl 0755.35108)] regarding for solitary waves of the Benjamin-Ono equation.