{"paper":{"title":"On the Neumann Problem of Hardy-Sobolev critical equations with the multiple singularities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chun-Hsiung Hsia, Gyeongha Hwang, Masato Hashizume","submitted_at":"2017-09-22T10:45:16Z","abstract_excerpt":"Let $N \\geq 3$ and $\\Omega \\subset \\mathbb{R}^N$ be $C^2$ bounded domain. We study the existence of positive solution $u \\in H^1(\\Omega)$ of \\begin{align*} \\left\\{ \\begin{array}{l} -\\Delta u + \\lambda u = \\frac{|u|^{2^*(s)-2}u}{|x-x_1|^s} + \\frac{|u|^{2^*(s)-2}u}{|x-x_2|^s}\\text{ in }\\Omega\\\\ \\frac{\\partial u}{\\partial \\nu} = 0 \\text{ on }\\partial\\Omega, \\end{array}\\right. \\end{align*} where $0 < s <2$, $2^*(s) = \\frac{2(N-s)}{N-2}$ and $x_1, x_2 \\in \\overline{\\Omega}$ with $x_1 \\neq x_2$. First, we show the existence of positive solutions to the equation provided the positive $\\lambda$ is sma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.07685","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"}