{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:EAUDXSJTUNKB26W4J3OG5C36VW","short_pith_number":"pith:EAUDXSJT","schema_version":"1.0","canonical_sha256":"20283bc933a3541d7adc4edc6e8b7eadb2aaf9bf8a3e48ccd5af71ee6b63650c","source":{"kind":"arxiv","id":"1404.3623","version":2},"attestation_state":"computed","paper":{"title":"Multiple solutions for an indefinite elliptic problem with critical growth in the gradient","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Humberto Ramos Quoirin, Louis Jeanjean","submitted_at":"2014-04-14T15:42:01Z","abstract_excerpt":"We consider the problem $(P)$, $$ -\\Delta u =c(x)u+\\mu|\\nabla u|^2 +f(x), \\quad u \\in H^1_0(\\Omega) \\cap L^{\\infty}(\\Omega),$$ where $\\Omega$ is a bounded domain of $\\mathbb{R}^N$, $N \\geq 3$, $\\mu>0, \\, c \\in \\mathcal{C}(\\overline{\\Omega}),$ and $ f \\in L^q(\\Omega)$ for some $ q>\\frac{N}{2}$ with $ f\\gneqq 0. $ Here $c$ is allowed to change sign. We show that when $c^+ \\not \\equiv 0$ and $c^+ +\\mu f$ is suitably small, this problem has at least two positive solutions. This result contrasts with the case $c \\leq 0$, where uniqueness holds. To show this multiplicity result we first transform $("},"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":"1404.3623","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-04-14T15:42:01Z","cross_cats_sorted":[],"title_canon_sha256":"cc7b196b6d8297ea8f717c07ab6c243f2e6c62afc666a06e3f60aebcb578c645","abstract_canon_sha256":"e79a48da3ee3d92bd85659c63a350ecb41430a68881c0b2de1f88f5396e6d2a1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:47:31.203609Z","signature_b64":"xGmAri2df1617Y//L2IGgO9eDOyon9l5yrxCcaZV8kmP8YOPlioRDgfblr5LhhkVw34kD8m4Hlid4LPjfJb7Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"20283bc933a3541d7adc4edc6e8b7eadb2aaf9bf8a3e48ccd5af71ee6b63650c","last_reissued_at":"2026-05-18T02:47:31.203179Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:47:31.203179Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multiple solutions for an indefinite elliptic problem with critical growth in the gradient","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Humberto Ramos Quoirin, Louis Jeanjean","submitted_at":"2014-04-14T15:42:01Z","abstract_excerpt":"We consider the problem $(P)$, $$ -\\Delta u =c(x)u+\\mu|\\nabla u|^2 +f(x), \\quad u \\in H^1_0(\\Omega) \\cap L^{\\infty}(\\Omega),$$ where $\\Omega$ is a bounded domain of $\\mathbb{R}^N$, $N \\geq 3$, $\\mu>0, \\, c \\in \\mathcal{C}(\\overline{\\Omega}),$ and $ f \\in L^q(\\Omega)$ for some $ q>\\frac{N}{2}$ with $ f\\gneqq 0. $ Here $c$ is allowed to change sign. We show that when $c^+ \\not \\equiv 0$ and $c^+ +\\mu f$ is suitably small, this problem has at least two positive solutions. This result contrasts with the case $c \\leq 0$, where uniqueness holds. To show this multiplicity result we first transform $("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.3623","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":"1404.3623","created_at":"2026-05-18T02:47:31.203239+00:00"},{"alias_kind":"arxiv_version","alias_value":"1404.3623v2","created_at":"2026-05-18T02:47:31.203239+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1404.3623","created_at":"2026-05-18T02:47:31.203239+00:00"},{"alias_kind":"pith_short_12","alias_value":"EAUDXSJTUNKB","created_at":"2026-05-18T12:28:25.294606+00:00"},{"alias_kind":"pith_short_16","alias_value":"EAUDXSJTUNKB26W4","created_at":"2026-05-18T12:28:25.294606+00:00"},{"alias_kind":"pith_short_8","alias_value":"EAUDXSJT","created_at":"2026-05-18T12:28:25.294606+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/EAUDXSJTUNKB26W4J3OG5C36VW","json":"https://pith.science/pith/EAUDXSJTUNKB26W4J3OG5C36VW.json","graph_json":"https://pith.science/api/pith-number/EAUDXSJTUNKB26W4J3OG5C36VW/graph.json","events_json":"https://pith.science/api/pith-number/EAUDXSJTUNKB26W4J3OG5C36VW/events.json","paper":"https://pith.science/paper/EAUDXSJT"},"agent_actions":{"view_html":"https://pith.science/pith/EAUDXSJTUNKB26W4J3OG5C36VW","download_json":"https://pith.science/pith/EAUDXSJTUNKB26W4J3OG5C36VW.json","view_paper":"https://pith.science/paper/EAUDXSJT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1404.3623&json=true","fetch_graph":"https://pith.science/api/pith-number/EAUDXSJTUNKB26W4J3OG5C36VW/graph.json","fetch_events":"https://pith.science/api/pith-number/EAUDXSJTUNKB26W4J3OG5C36VW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EAUDXSJTUNKB26W4J3OG5C36VW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EAUDXSJTUNKB26W4J3OG5C36VW/action/storage_attestation","attest_author":"https://pith.science/pith/EAUDXSJTUNKB26W4J3OG5C36VW/action/author_attestation","sign_citation":"https://pith.science/pith/EAUDXSJTUNKB26W4J3OG5C36VW/action/citation_signature","submit_replication":"https://pith.science/pith/EAUDXSJTUNKB26W4J3OG5C36VW/action/replication_record"}},"created_at":"2026-05-18T02:47:31.203239+00:00","updated_at":"2026-05-18T02:47:31.203239+00:00"}