{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:LDDBGQD362VLDGDXT5PORTJAS7","short_pith_number":"pith:LDDBGQD3","schema_version":"1.0","canonical_sha256":"58c613407bf6aab198779f5ee8cd2097f7b1147e89d83a1db2eac58370f01ae2","source":{"kind":"arxiv","id":"1405.0162","version":2},"attestation_state":"computed","paper":{"title":"A note on semilinear elliptic equation with biharmonic operator and multiple critical nonlinearities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mousomi Bhakta","submitted_at":"2014-05-01T14:06:37Z","abstract_excerpt":"We study the existence and non-existence of nontrivial weak solution of $$ {\\Delta^2u-\\mu\\frac{u}{|x|^{4}} = \\frac{|u|^{q_{\\beta}-2}u}{|x|^{\\beta}}+|u|^{q-2}u\\quad\\textrm{in ${\\mathbb R}^N$,}} $$ where $N\\geq 5$, $q_{\\beta}=\\frac{2(N-\\beta)}{N-4}$, $0<\\beta<4$, $1<q\\leq 2^{**}$ and $\\mu<\\mu_1:=\\big(\\frac{N(N-4)}{4}\\big)^2$. Using Pohozaev type of identity, we prove the non-existence result when $1<q< 2^{**}$. On the other hand when the equation has multiple critical nonlinearities i.e. $q=2^{**}$ and $-(N-2)^2\\leq\\mu<\\mu_1$, we establish the existence of nontrivial solution using the Mountain-"},"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":"1405.0162","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-05-01T14:06:37Z","cross_cats_sorted":[],"title_canon_sha256":"03b06b8652e103c42771dd67e12d7a2477c62fbe04372b6a1fbd821267f53d6b","abstract_canon_sha256":"cca277c1325d7501cdcfddd5a34aa0e1c36c805a094355a1adccf0d10c921b95"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:11.091494Z","signature_b64":"VOauHHevdbEaHs9Pr+mviNis7rV4vO9Ifc6mYkYGK0ZNbvDNGlG/RX5f1X/X4OzolmzKYIaGyzLbWEkOdkuyCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"58c613407bf6aab198779f5ee8cd2097f7b1147e89d83a1db2eac58370f01ae2","last_reissued_at":"2026-05-18T01:10:11.091097Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:11.091097Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on semilinear elliptic equation with biharmonic operator and multiple critical nonlinearities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mousomi Bhakta","submitted_at":"2014-05-01T14:06:37Z","abstract_excerpt":"We study the existence and non-existence of nontrivial weak solution of $$ {\\Delta^2u-\\mu\\frac{u}{|x|^{4}} = \\frac{|u|^{q_{\\beta}-2}u}{|x|^{\\beta}}+|u|^{q-2}u\\quad\\textrm{in ${\\mathbb R}^N$,}} $$ where $N\\geq 5$, $q_{\\beta}=\\frac{2(N-\\beta)}{N-4}$, $0<\\beta<4$, $1<q\\leq 2^{**}$ and $\\mu<\\mu_1:=\\big(\\frac{N(N-4)}{4}\\big)^2$. Using Pohozaev type of identity, we prove the non-existence result when $1<q< 2^{**}$. On the other hand when the equation has multiple critical nonlinearities i.e. $q=2^{**}$ and $-(N-2)^2\\leq\\mu<\\mu_1$, we establish the existence of nontrivial solution using the Mountain-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.0162","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":"1405.0162","created_at":"2026-05-18T01:10:11.091157+00:00"},{"alias_kind":"arxiv_version","alias_value":"1405.0162v2","created_at":"2026-05-18T01:10:11.091157+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.0162","created_at":"2026-05-18T01:10:11.091157+00:00"},{"alias_kind":"pith_short_12","alias_value":"LDDBGQD362VL","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_16","alias_value":"LDDBGQD362VLDGDX","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_8","alias_value":"LDDBGQD3","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/LDDBGQD362VLDGDXT5PORTJAS7","json":"https://pith.science/pith/LDDBGQD362VLDGDXT5PORTJAS7.json","graph_json":"https://pith.science/api/pith-number/LDDBGQD362VLDGDXT5PORTJAS7/graph.json","events_json":"https://pith.science/api/pith-number/LDDBGQD362VLDGDXT5PORTJAS7/events.json","paper":"https://pith.science/paper/LDDBGQD3"},"agent_actions":{"view_html":"https://pith.science/pith/LDDBGQD362VLDGDXT5PORTJAS7","download_json":"https://pith.science/pith/LDDBGQD362VLDGDXT5PORTJAS7.json","view_paper":"https://pith.science/paper/LDDBGQD3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1405.0162&json=true","fetch_graph":"https://pith.science/api/pith-number/LDDBGQD362VLDGDXT5PORTJAS7/graph.json","fetch_events":"https://pith.science/api/pith-number/LDDBGQD362VLDGDXT5PORTJAS7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LDDBGQD362VLDGDXT5PORTJAS7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LDDBGQD362VLDGDXT5PORTJAS7/action/storage_attestation","attest_author":"https://pith.science/pith/LDDBGQD362VLDGDXT5PORTJAS7/action/author_attestation","sign_citation":"https://pith.science/pith/LDDBGQD362VLDGDXT5PORTJAS7/action/citation_signature","submit_replication":"https://pith.science/pith/LDDBGQD362VLDGDXT5PORTJAS7/action/replication_record"}},"created_at":"2026-05-18T01:10:11.091157+00:00","updated_at":"2026-05-18T01:10:11.091157+00:00"}