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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1709.07685","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-09-22T10:45:16Z","cross_cats_sorted":[],"title_canon_sha256":"c1125713bc57da76cc8f6c59a22b052c194c688b0c22f511eecc1dd56b22c6f8","abstract_canon_sha256":"2fd47f714bfc5420b7e1d8ac05a74f254f5d551e718ec059cbf6fec2a6076e75"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:34:32.567248Z","signature_b64":"GE5lm+/MLdiTsoFOQz05Se5AIltCzhdQ+gt4i7ooiy24hFchQSytR0Q9h4C3cGSDccZEAMKJfvgYzzZ0y2D6CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"16359e142103b01562ca14ff08bcf4911b4ff7cfd115ab83e2fc02e4ad28009a","last_reissued_at":"2026-05-18T00:34:32.566901Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:34:32.566901Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $x_1 \\neq x_2$. 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