{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:DB33SJDGU66TLXPVZGEEB5XREU","short_pith_number":"pith:DB33SJDG","schema_version":"1.0","canonical_sha256":"1877b92466a7bd35ddf5c98840f6f1250295e4482916ac5b21ee146e246802d2","source":{"kind":"arxiv","id":"1606.07734","version":1},"attestation_state":"computed","paper":{"title":"Explicit solutions and multiplicity results for some equations with the $p$-Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.AP","authors_text":"Philip Korman","submitted_at":"2016-06-24T15:53:46Z","abstract_excerpt":"We derive explicit ground state solutions for several equations with the $p$-Laplacian in $R^n$, including (here $\\varphi (z)=z|z|^{p-2}$, with $p>1$) \\[ \\varphi \\left(u'(r)\\right)' +\\frac{n-1}{r} \\varphi \\left(u'(r)\\right)+u^M+u^Q=0 \\,. \\] The constant $M>0$ is assumed to be below the critical power, while $Q=\\frac{M p-p+1}{p-1}$ is above the critical power. This explicit solution is used to give a multiplicity result, similarly to C.S. Lin and W.-M. Ni [11]. We also give the $p$-Laplace version of G. Bratu's solution [3].\n  In another direction, we present a change of variables which removes"},"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":"1606.07734","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-06-24T15:53:46Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"d0b2128d5a21d523b198506a35ca6cece288d538b4931b7bd7190cb26e221f37","abstract_canon_sha256":"53da969e0fa4b0ef30fd03d00bb9cdfcb702881bab1b8f622d17911f142ee921"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:55.807292Z","signature_b64":"IWhKhqDJeItYoE2SiD9n5fIFAXN1KHrDRoFPmkI3odSIuzSTjubPLDPADq8AfjuPlpWKm/5JONzpEHcSujDIBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1877b92466a7bd35ddf5c98840f6f1250295e4482916ac5b21ee146e246802d2","last_reissued_at":"2026-05-18T01:11:55.806961Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:55.806961Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Explicit solutions and multiplicity results for some equations with the $p$-Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.AP","authors_text":"Philip Korman","submitted_at":"2016-06-24T15:53:46Z","abstract_excerpt":"We derive explicit ground state solutions for several equations with the $p$-Laplacian in $R^n$, including (here $\\varphi (z)=z|z|^{p-2}$, with $p>1$) \\[ \\varphi \\left(u'(r)\\right)' +\\frac{n-1}{r} \\varphi \\left(u'(r)\\right)+u^M+u^Q=0 \\,. \\] The constant $M>0$ is assumed to be below the critical power, while $Q=\\frac{M p-p+1}{p-1}$ is above the critical power. This explicit solution is used to give a multiplicity result, similarly to C.S. Lin and W.-M. Ni [11]. We also give the $p$-Laplace version of G. Bratu's solution [3].\n  In another direction, we present a change of variables which removes"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.07734","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":"1606.07734","created_at":"2026-05-18T01:11:55.807017+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.07734v1","created_at":"2026-05-18T01:11:55.807017+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.07734","created_at":"2026-05-18T01:11:55.807017+00:00"},{"alias_kind":"pith_short_12","alias_value":"DB33SJDGU66T","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_16","alias_value":"DB33SJDGU66TLXPV","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_8","alias_value":"DB33SJDG","created_at":"2026-05-18T12:30:09.641336+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/DB33SJDGU66TLXPVZGEEB5XREU","json":"https://pith.science/pith/DB33SJDGU66TLXPVZGEEB5XREU.json","graph_json":"https://pith.science/api/pith-number/DB33SJDGU66TLXPVZGEEB5XREU/graph.json","events_json":"https://pith.science/api/pith-number/DB33SJDGU66TLXPVZGEEB5XREU/events.json","paper":"https://pith.science/paper/DB33SJDG"},"agent_actions":{"view_html":"https://pith.science/pith/DB33SJDGU66TLXPVZGEEB5XREU","download_json":"https://pith.science/pith/DB33SJDGU66TLXPVZGEEB5XREU.json","view_paper":"https://pith.science/paper/DB33SJDG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.07734&json=true","fetch_graph":"https://pith.science/api/pith-number/DB33SJDGU66TLXPVZGEEB5XREU/graph.json","fetch_events":"https://pith.science/api/pith-number/DB33SJDGU66TLXPVZGEEB5XREU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DB33SJDGU66TLXPVZGEEB5XREU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DB33SJDGU66TLXPVZGEEB5XREU/action/storage_attestation","attest_author":"https://pith.science/pith/DB33SJDGU66TLXPVZGEEB5XREU/action/author_attestation","sign_citation":"https://pith.science/pith/DB33SJDGU66TLXPVZGEEB5XREU/action/citation_signature","submit_replication":"https://pith.science/pith/DB33SJDGU66TLXPVZGEEB5XREU/action/replication_record"}},"created_at":"2026-05-18T01:11:55.807017+00:00","updated_at":"2026-05-18T01:11:55.807017+00:00"}