{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:JTXJB3ZDUWOULCQUODGMO76GKN","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":"1be1750f5d22626fb887b746c97611dc47087283c4c8d1a83ecc4bca66f28725","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-07-23T14:25:41Z","title_canon_sha256":"294901e97c0bde9ed5736a14679eafc9ec4b5fbced1e5780ae22aa4c3405be35"},"schema_version":"1.0","source":{"id":"1407.6232","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.6232","created_at":"2026-05-18T02:46:57Z"},{"alias_kind":"arxiv_version","alias_value":"1407.6232v1","created_at":"2026-05-18T02:46:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.6232","created_at":"2026-05-18T02:46:57Z"},{"alias_kind":"pith_short_12","alias_value":"JTXJB3ZDUWOU","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_16","alias_value":"JTXJB3ZDUWOULCQU","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_8","alias_value":"JTXJB3ZD","created_at":"2026-05-18T12:28:35Z"}],"graph_snapshots":[{"event_id":"sha256:2842ce93ab9d9cea4756a424a53de126db0db32a504756f6b13852095a5bbf81","target":"graph","created_at":"2026-05-18T02:46:57Z","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":"Let $\\Omega$ be a smooth bounded domain in $\\mathbb{R}^{N}$, with $N\\geq 5$, $a>0$, $\\alpha\\geq 0$ and $2^*=\\frac{2N}{N-2}$. We show that the the exponent $q=\\frac{2(N-1)}{N-2}$ plays a critical role regarding the existence of least energy (or ground state) solutions of the Neumann problem $$ \\left\\{\\begin{array}{ll} -\\Delta u+au=u^{2^*-1}-\\alpha u^{q-1}&\\mbox{in}\\ \\Omega,\\\\ u>0&\\mbox{in}\\ \\Omega,\\\\ \\frac{\\partial u}{\\partial\\nu}=0&\\mbox{on}\\ \\partial\\Omega. \\end{array}\\right. $$ Namely, we prove that when $q=\\frac{2(N-1)}{N-2}$ there exists an $\\alpha_{0}>0$ such that the problem has a least ","authors_text":"David G. Costa, Pedro M. Gir\\~ao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-07-23T14:25:41Z","title":"Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.6232","kind":"arxiv","version":1},"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:eca04443828cc9e1ce352f5761fee8b5b0051758aaf8a4a919a93a45b1fa142b","target":"record","created_at":"2026-05-18T02:46:57Z","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":"1be1750f5d22626fb887b746c97611dc47087283c4c8d1a83ecc4bca66f28725","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-07-23T14:25:41Z","title_canon_sha256":"294901e97c0bde9ed5736a14679eafc9ec4b5fbced1e5780ae22aa4c3405be35"},"schema_version":"1.0","source":{"id":"1407.6232","kind":"arxiv","version":1}},"canonical_sha256":"4cee90ef23a59d458a1470ccc77fc653626dc6e907fe1bf23b569335cdc2470e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4cee90ef23a59d458a1470ccc77fc653626dc6e907fe1bf23b569335cdc2470e","first_computed_at":"2026-05-18T02:46:57.079158Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:46:57.079158Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CPn9PLE4eDpwB3OkCLmLCCbYAAxcp/k+X7hzlt3lihOLWENbdmkSHJjO/6k4XrG7UIN3LBmNmtrNJwIL0K95AQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:46:57.079764Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.6232","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:eca04443828cc9e1ce352f5761fee8b5b0051758aaf8a4a919a93a45b1fa142b","sha256:2842ce93ab9d9cea4756a424a53de126db0db32a504756f6b13852095a5bbf81"],"state_sha256":"7c17dc0f41cec9983fe2efb8f9f7b236fb4e755c0394da767361615323b7b832"}