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Frank","submitted_at":"2010-09-21T10:15:31Z","abstract_excerpt":"We prove uniqueness of ground state solutions $Q = Q(|x|) \\geq 0$ for the nonlinear equation $(-\\Delta)^s Q + Q - Q^{\\alpha+1}= 0$ in $\\mathbb{R}$, where $0 < s < 1$ and $0 < \\alpha < \\frac{4s}{1-2s}$ for $s < 1/2$ and $0 < \\alpha < \\infty$ for $s \\geq 1/2$. Here $(-\\Delta)^s$ denotes the fractional Laplacian in one dimension. In particular, we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for $s=1/2$ and $\\alpha=1$ in [Acta Math., \\textbf{167} (1991), 107--126]. 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