{"paper":{"title":"Symmetry results in the half space for a semi-linear fractional Laplace equation through a one-dimensional analysis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"A. Quaas, B. Barrios, J. Garc\\'ia-Meli\\'an, L. Del Pezzo","submitted_at":"2017-04-09T12:51:01Z","abstract_excerpt":"In this paper we analyze the semi-linear fractional Laplace equation $$(-\\Delta)^s u = f(u) \\quad\\text{ in } \\mathbb{R}^N_+,\\quad u=0 \\quad\\text{ in } \\mathbb{R}^N\\setminus \\mathbb{R}^N_+,$$ where $\\mathbb{R}^N_+=\\{x=(x',x_N)\\in \\mathbb{R}^N:\\ x_N>0\\}$ stands for the half-space and $f$ is a locally Lipschitz nonlinearity. We completely characterize one-dimensional bounded solutions of this problem, and we prove among other things that if $u$ is a bounded solution with $\\rho:=\\sup_{\\mathbb{R}^N}u$ verifying $f(\\rho)=0$, then $u$ is necessarily one-dimensional."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.02597","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"}