{"paper":{"title":"Monotonicity and nonexistence results for some fractional elliptic problems in the half space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mouhamed Moustapha Fall, Tobias Weth","submitted_at":"2013-09-27T13:17:28Z","abstract_excerpt":"We study a class of fractional elliptic problems of the form $\\Ds u= f(u)$ in the half space $\\R^N_+:=\\{x \\in \\R^N\\::\\: x_1>0\\}$ with the complementary Dirichlet condition $u \\equiv 0$ in $\\R^N \\setminus \\R^N_+$. Under mild assumptions on the nonlinearity $f$, we show that bounded positive solutions are increasing in $x_1$. For the special case $f(u)=u^q$, we deduce nonexistence of positive bounded solutions in the case where $q \\ge 1$ and $q<\\frac{N-1+2s}{N-1-2s}$ if $N \\ge 1+2s$. We do not require integrability assumptions on the solutions we study."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.7230","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"}