{"paper":{"title":"A Liouville Theorem for a Class of Fractional Systems in $\\mathbb{R}^n_+$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jianming He, Lizhi Zhang, Mei Yu","submitted_at":"2016-11-28T14:21:33Z","abstract_excerpt":"Let $0<\\alpha,\\beta<2$ be any real number. In this paper, we investigate the following semilinear system involving the fractional Laplacian \\begin{equation*} \\left\\{\\begin{array}{lll} (-\\lap)^{\\alpha/2} u(x)=f(v(x)), & (-\\lap)^{\\beta/2} v(x)=g(u(x)), & \\qquad x\\in\\mathbb{R}^n_+, u,v\\geq0, & \\qquad x\\in\\mathbb{R}^n\\setminus\\mathbb{R}^n_+. \\end{array}\\right. \\end{equation*} Applying a direct method of moving planes for the fractional Laplacian, without any decay assumption on the solutions at infinity, we prove Liouville theorems of nonnegative solutions under some natural conditions on $f$ and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.09133","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"}