{"paper":{"title":"Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Xavier Cabre, Yannick Sire","submitted_at":"2011-11-03T11:13:42Z","abstract_excerpt":"This paper, which is the follow-up to part I, concerns the equation $(-\\Delta)^{s} v+G'(v)=0$ in $\\mathbb{R}^{n}$, with $s \\in (0,1)$, where $(-\\Delta)^{s}$ stands for the fractional Laplacian ---the infinitesimal generator of a L\\'evy process. When $n=1$, we prove that there exists a layer solution of the equation (i.e., an increasing solution with limits $\\pm 1$ at $\\pm \\infty$) if and only if the potential $G$ has only two absolute minima in $[-1,1]$, located at $\\pm 1$ and satisfying $G'(-1)=G'(1)=0$. Under the additional hypothesis $G\"(-1)>0$ and $G\"(1)>0$, we also establish its uniquenes"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.0796","kind":"arxiv","version":1},"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"}