{"paper":{"title":"Monotonicity of solutions for some nonlocal elliptic problems in half-spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"A. Quaas, B. Barrios, J. Garcia-Melian, L. Del Pezzo","submitted_at":"2016-06-03T12:33:46Z","abstract_excerpt":"In this paper we consider classical solutions $u$ of the semilinear fractional problem $(-\\Delta)^s u = f(u)$ in $\\mathbb{R}^N_+$ with $u=0$ in $\\mathbb{R}^N \\setminus \\mathbb{R}^N_+$, where $(-\\Delta)^s$, $0<s<1$, stands for the fractional laplacian, $N\\ge 2$, $\\mathbb{R}^N_+=\\{x=(x',x_N)\\in \\mathbb{R}^N:\\ x_N>0\\}$ is the half-space and $f\\in C^1$ is a given function. With no additional restriction on the function $f$, we show that bounded, nonnegative, nontrivial classical solutions are indeed positive in $\\mathbb{R}^N_+$ and verify $$ \\frac{\\partial u}{\\partial x_N}>0 \\quad \\hbox{in } \\math"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.01061","kind":"arxiv","version":3},"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"}