{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:435WH35YAUK573VGU3ZVAJS34X","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"0d27022c862c66207fcfbf361d8ed2cb972932e3ef37623110b250b9b44f4405","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-05T16:02:15Z","title_canon_sha256":"a339d9a40fdac62d4f1f3cf106059eea13d23e4193efd6ad392796cd61009a08"},"schema_version":"1.0","source":{"id":"1504.01133","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.01133","created_at":"2026-05-18T00:27:10Z"},{"alias_kind":"arxiv_version","alias_value":"1504.01133v2","created_at":"2026-05-18T00:27:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.01133","created_at":"2026-05-18T00:27:10Z"},{"alias_kind":"pith_short_12","alias_value":"435WH35YAUK5","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"435WH35YAUK573VG","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"435WH35Y","created_at":"2026-05-18T12:29:02Z"}],"graph_snapshots":[{"event_id":"sha256:a0029c5ee430403ff29bb4fbc0435aa18e933f78c55c10188a142ea03301d213","target":"graph","created_at":"2026-05-18T00:27:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper, we will study the following PDE in $\\mathbb{R}^N$ involving multiple Hardy-Sobolev critical exponents: $$ \\begin{cases} \\Delta u+\\sum_{i=1}^{l}\\lambda_i \\frac{u^{2^*(s_i)-1}}{|x|^{s_i}}+u^{2^*-1}=0\\;\\hbox{in}\\;\\mathbb{R}^N, u\\in D_{0}^{1,2}(\\mathbb{R}^N), \\end{cases} $$ where $0<s_1<s_2<\\cdots<s_l<2, 2^\\ast:=\\frac{2N}{N-2}, \\; 2^\\ast(s):=\\frac{2(N-s)}{N-2}$ and there exists some $k\\in [1, l]$ such that $\\lambda_i>0$ for $1\\leq i\\leq k$; $\\lambda_i<0$ for $k+1\\leq i\\leq l$. We develop an interesting way to study this class of equations involving mixed sign parameters. We prove th","authors_text":"Wenming Zou, Xuexiu Zhong","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-05T16:02:15Z","title":"A nonlinear elliptic PDE with multiple Hardy-Sobolev critical exponents in $\\mathbb{R}^N$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.01133","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6ae9eff67ec23b314108c6f1c68cb01551089eeee44e4b88fdf20e45eb60e514","target":"record","created_at":"2026-05-18T00:27:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"0d27022c862c66207fcfbf361d8ed2cb972932e3ef37623110b250b9b44f4405","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-05T16:02:15Z","title_canon_sha256":"a339d9a40fdac62d4f1f3cf106059eea13d23e4193efd6ad392796cd61009a08"},"schema_version":"1.0","source":{"id":"1504.01133","kind":"arxiv","version":2}},"canonical_sha256":"e6fb63efb80515dfeea6a6f350265be5f51fa44e480ff00bb5a44292059cc2d7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e6fb63efb80515dfeea6a6f350265be5f51fa44e480ff00bb5a44292059cc2d7","first_computed_at":"2026-05-18T00:27:10.259741Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:27:10.259741Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/Ris/NiQ7HBOHjmS/ULg4X/4eYTkExNM+TvBczhEVQ1WEVJ48mUKoUvVhUAiASpMt+MLi94ORgMMwhYi2Ld1Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:27:10.260424Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.01133","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6ae9eff67ec23b314108c6f1c68cb01551089eeee44e4b88fdf20e45eb60e514","sha256:a0029c5ee430403ff29bb4fbc0435aa18e933f78c55c10188a142ea03301d213"],"state_sha256":"b06311b2072fc11d285b507d439430cf2fcafdce8b353d3932ab7c2f69f46080"}