{"paper":{"title":"Symmetry of Positive Solutions for the Fractional Schr$ \\ddot{\\textrm{o}}$dinger Equations with Choquard-type Nonlinearities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Xiaoya Huang, Zhenqiu Zhang","submitted_at":"2019-06-06T02:49:45Z","abstract_excerpt":"This paper deals with the following fractional Schr$ \\ddot{\\textrm{o}}$dinger equations with Choquard-type nonlinearities \\begin{equation*} \\left\\{\\begin{array}{r@{\\ \\ }c@{\\ \\ }ll}\n  (-\\Delta)^{\\frac{\\alpha}{2}}u + u - C_{n,-\\beta} \\,(|x|^{\\beta-n}\\ast u^{p})\\, u^{p-1}& = &0 & \\mbox{in}\\ \\ \\mathbb{R}^{n}\\,, \\\\[0.05cm]\n  u & > & 0 & \\mbox{on}\\ \\ \\mathbb{R}^{n}, \\end{array}\\right. \\end{equation*} where $ 0< \\alpha,\\beta < 2, 1\\leq p <\\infty \\,\\,and\\,\\, n\\geq 2. $ First we construct a decay result at infinity and a narrow region principle for related equations. Then we establish the radial symmet"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.02388","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"}