{"paper":{"title":"Uniqueness and Nondegeneracy of Ground States for $(-\\Delta)^s Q + Q - Q^{\\alpha+1} = 0$ in $\\mathbb{R}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Enno Lenzmann, Rupert L. 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]. As a technical key result in this paper, we show that the associated l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.4042","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}