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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 "},"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":"1407.6232","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-07-23T14:25:41Z","cross_cats_sorted":[],"title_canon_sha256":"294901e97c0bde9ed5736a14679eafc9ec4b5fbced1e5780ae22aa4c3405be35","abstract_canon_sha256":"1be1750f5d22626fb887b746c97611dc47087283c4c8d1a83ecc4bca66f28725"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:46:57.079764Z","signature_b64":"CPn9PLE4eDpwB3OkCLmLCCbYAAxcp/k+X7hzlt3lihOLWENbdmkSHJjO/6k4XrG7UIN3LBmNmtrNJwIL0K95AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4cee90ef23a59d458a1470ccc77fc653626dc6e907fe1bf23b569335cdc2470e","last_reissued_at":"2026-05-18T02:46:57.079158Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:46:57.079158Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"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","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"David G. 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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 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.6232","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1407.6232","created_at":"2026-05-18T02:46:57.079239+00:00"},{"alias_kind":"arxiv_version","alias_value":"1407.6232v1","created_at":"2026-05-18T02:46:57.079239+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.6232","created_at":"2026-05-18T02:46:57.079239+00:00"},{"alias_kind":"pith_short_12","alias_value":"JTXJB3ZDUWOU","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_16","alias_value":"JTXJB3ZDUWOULCQU","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_8","alias_value":"JTXJB3ZD","created_at":"2026-05-18T12:28:35.611951+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/JTXJB3ZDUWOULCQUODGMO76GKN","json":"https://pith.science/pith/JTXJB3ZDUWOULCQUODGMO76GKN.json","graph_json":"https://pith.science/api/pith-number/JTXJB3ZDUWOULCQUODGMO76GKN/graph.json","events_json":"https://pith.science/api/pith-number/JTXJB3ZDUWOULCQUODGMO76GKN/events.json","paper":"https://pith.science/paper/JTXJB3ZD"},"agent_actions":{"view_html":"https://pith.science/pith/JTXJB3ZDUWOULCQUODGMO76GKN","download_json":"https://pith.science/pith/JTXJB3ZDUWOULCQUODGMO76GKN.json","view_paper":"https://pith.science/paper/JTXJB3ZD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1407.6232&json=true","fetch_graph":"https://pith.science/api/pith-number/JTXJB3ZDUWOULCQUODGMO76GKN/graph.json","fetch_events":"https://pith.science/api/pith-number/JTXJB3ZDUWOULCQUODGMO76GKN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JTXJB3ZDUWOULCQUODGMO76GKN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JTXJB3ZDUWOULCQUODGMO76GKN/action/storage_attestation","attest_author":"https://pith.science/pith/JTXJB3ZDUWOULCQUODGMO76GKN/action/author_attestation","sign_citation":"https://pith.science/pith/JTXJB3ZDUWOULCQUODGMO76GKN/action/citation_signature","submit_replication":"https://pith.science/pith/JTXJB3ZDUWOULCQUODGMO76GKN/action/replication_record"}},"created_at":"2026-05-18T02:46:57.079239+00:00","updated_at":"2026-05-18T02:46:57.079239+00:00"}