{"paper":{"title":"Existence results of positive solutions for nonlinear cooperative elliptic systems involving fractional Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alexander Quaas, Aliang Xia","submitted_at":"2015-11-10T11:26:52Z","abstract_excerpt":"In this article, we prove existence results of positive solutions for the following nonlinear elliptic problem with gradient terms: \\begin{eqnarray*} \\left\\{\\begin{array}{l@{\\quad }l} (-\\Delta)^\\alpha u=f(x,u,v,\\nabla u, \\nabla v) &{\\rm in}\\,\\,\\Omega,\\\\ (-\\Delta)^\\alpha v=g(x,u,v,\\nabla u, \\nabla v) &{\\rm in}\\,\\,\\Omega,\\\\ u=v=0\\,\\,&{\\rm in}\\,\\,\\R^N\\setminus\\Omega,\n  \\end{array}\n  \\right.\n  \\end{eqnarray*} where $(-\\Delta)^\\alpha$ denotes the fractional Laplacian and $ \\Omega $ is a smooth bounded domain in $ \\R^N $. It shown that under some assumptions on $ f $ and $ g $, the problem has at le"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.03066","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"}