{"paper":{"title":"Solutions of multi-component fractional symmetric systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mostafa Fazly","submitted_at":"2015-06-04T00:41:18Z","abstract_excerpt":"We study the following elliptic system concerning the fractional Laplacian operator $$(- \\Delta)^ {s_i} u_i = H_i ( u_1,\\cdots,u_m) \\ \\ \\text{in}\\ \\ \\mathbb{R}^n,$$ when $0<s_i<1$, $u_i: \\mathbb R^n\\to R$ and $H_i$ belongs to $C^{1,\\gamma}(\\mathbb{R}^m)$ for $\\gamma > \\max(0,1-2\\min \\left \\{s_i \\right \\})$ for $1\\le i \\le m$. The above system is called symmetric when the matrix $\\mathcal H=(\\partial_j H_i(u_1,\\cdots,u_m))_{i,j=1}^m$ is symmetric. The notion of symmetric systems seems crucial to study this system with a general nonlinearity $H=(H_i)_{i=1}^m$. We establish De Giorgi type results"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.01440","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"}