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Consider the following perturbed PDE involving two Hardy-Sobolev critical exponents: $$ \\begin{cases} &\\Delta u+\\lambda_1 \\frac{u^{2^*(s_1)-1}}{|x|^{s_1}}+\\lambda_2\\frac{u^{2^*(s_2)-1}}{|x|^{s_2}}+\\lambda_3\\frac{u^p}{|x|^{s_3}}=0\\;\\quad \\hbox{in}\\;\\Omega,\\\\ &u(x)>0\\;\\hbox{in}\\;\\Omega,\\;\\, u(x)=0\\;\\hbox{on}\\;\\partial\\Omega, \\end{cases} $$ where $0<s_2<s_1<2, 0\\leq s_3<2, 2^*(s_i):=\\frac{2(N-s_i"},"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":"1504.00730","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-03T02:42:03Z","cross_cats_sorted":[],"title_canon_sha256":"36a721f1229c5cfae8895df04b48fd6f66c8d06d43b2ec09fa1bcfe77a22bf5b","abstract_canon_sha256":"5766b35f9417764e3935e41ed0a8f84d7fd7543a74d062df5c332b9ab98f8d75"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:37:18.880393Z","signature_b64":"4M1aDczN8v6GEHcWrlhPCq2U6yyudvZFpZrbB9T5tR3of5n+vyg4LVsb0wXDJBMcwvaCzKW043pdFKnq7RqECQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f1b33610a3e0bda93bb613d136b4338faf5bfe33442c3b3fe691ba12d7ae4eff","last_reissued_at":"2026-05-18T01:37:18.879623Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:37:18.879623Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A perturbed nonlinear elliptic PDE with two Hardy-Sobolev critical exponents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Wenming Zou, Xuexiu Zhong","submitted_at":"2015-04-03T02:42:03Z","abstract_excerpt":"Let $\\Omega$ be a $C^1$ open bounded domain in $\\R^N$ ($N\\geq 3$) with $0\\in \\partial \\Omega$. Suppose that $\\partial\\Omega$ is $C^2$ at $0$ and the mean curvature of $\\partial\\Omega$ at $0$ is negative. Consider the following perturbed PDE involving two Hardy-Sobolev critical exponents: $$ \\begin{cases} &\\Delta u+\\lambda_1 \\frac{u^{2^*(s_1)-1}}{|x|^{s_1}}+\\lambda_2\\frac{u^{2^*(s_2)-1}}{|x|^{s_2}}+\\lambda_3\\frac{u^p}{|x|^{s_3}}=0\\;\\quad \\hbox{in}\\;\\Omega,\\\\ &u(x)>0\\;\\hbox{in}\\;\\Omega,\\;\\, u(x)=0\\;\\hbox{on}\\;\\partial\\Omega, \\end{cases} $$ where $0<s_2<s_1<2, 0\\leq s_3<2, 2^*(s_i):=\\frac{2(N-s_i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.00730","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1504.00730","created_at":"2026-05-18T01:37:18.879751+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.00730v2","created_at":"2026-05-18T01:37:18.879751+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.00730","created_at":"2026-05-18T01:37:18.879751+00:00"},{"alias_kind":"pith_short_12","alias_value":"6GZTMEFD4C62","created_at":"2026-05-18T12:29:07.941421+00:00"},{"alias_kind":"pith_short_16","alias_value":"6GZTMEFD4C62SO5W","created_at":"2026-05-18T12:29:07.941421+00:00"},{"alias_kind":"pith_short_8","alias_value":"6GZTMEFD","created_at":"2026-05-18T12:29:07.941421+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6GZTMEFD4C62SO5WCPITNNBTR6","json":"https://pith.science/pith/6GZTMEFD4C62SO5WCPITNNBTR6.json","graph_json":"https://pith.science/api/pith-number/6GZTMEFD4C62SO5WCPITNNBTR6/graph.json","events_json":"https://pith.science/api/pith-number/6GZTMEFD4C62SO5WCPITNNBTR6/events.json","paper":"https://pith.science/paper/6GZTMEFD"},"agent_actions":{"view_html":"https://pith.science/pith/6GZTMEFD4C62SO5WCPITNNBTR6","download_json":"https://pith.science/pith/6GZTMEFD4C62SO5WCPITNNBTR6.json","view_paper":"https://pith.science/paper/6GZTMEFD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.00730&json=true","fetch_graph":"https://pith.science/api/pith-number/6GZTMEFD4C62SO5WCPITNNBTR6/graph.json","fetch_events":"https://pith.science/api/pith-number/6GZTMEFD4C62SO5WCPITNNBTR6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6GZTMEFD4C62SO5WCPITNNBTR6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6GZTMEFD4C62SO5WCPITNNBTR6/action/storage_attestation","attest_author":"https://pith.science/pith/6GZTMEFD4C62SO5WCPITNNBTR6/action/author_attestation","sign_citation":"https://pith.science/pith/6GZTMEFD4C62SO5WCPITNNBTR6/action/citation_signature","submit_replication":"https://pith.science/pith/6GZTMEFD4C62SO5WCPITNNBTR6/action/replication_record"}},"created_at":"2026-05-18T01:37:18.879751+00:00","updated_at":"2026-05-18T01:37:18.879751+00:00"}