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Write the solution in the form $u(x)= \\Sigma _{i=1}^n \\xi _i \\varphi _i+U(x)$, with $ U \\perp \\varphi _k$, $k=1, \\ldots, n$. Starting with $k=0$, when the problem is linear, we continue the solution in $k$ by keeping $\\xi =(\\xi _1, \\ldots,\\xi _n)$ fixed, but allowing for $\\mu =(\\mu _1, "},"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":"1609.05817","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-09-19T16:24:59Z","cross_cats_sorted":[],"title_canon_sha256":"42cd5d6e2b2e43508778da8d2e06acf118c8f902b418962da04ad7d87df54766","abstract_canon_sha256":"d63b615828410693da6bcffbd78e34296263583fcadfd0644c5880d69008d7f7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:25.201605Z","signature_b64":"xUmDJ6iEa5dfYcZTKJNHOxCnCwW0fX/FGhAyedrpBKjWPFOqLHj4k7LG3vLlkWVDp9NYbbmQCbsrW+wI11skDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5bf8f85cc718888bf5b7627b79a1d9b78edd26d3f27d36411611284ed14c67ac","last_reissued_at":"2026-05-18T01:04:25.200870Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:25.200870Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Curves of equiharmonic solutions, and problems at resonance","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Philip Korman","submitted_at":"2016-09-19T16:24:59Z","abstract_excerpt":"We consider the semilinear Dirichlet problem \\[ \\Delta u+kg(u)=\\mu _1 \\varphi _1+\\cdots +\\mu _n \\varphi _n+e(x) \\;\\; \\mbox{for $x \\in \\Omega$}, \\;\\; u=0 \\;\\; \\mbox{on $\\partial \\Omega$}, \\] where $\\varphi _k$ is the $k$-th eigenfunction of the Laplacian on $\\Omega$ and $e(x) \\perp \\varphi _k$, $k=1, \\ldots, n$. Write the solution in the form $u(x)= \\Sigma _{i=1}^n \\xi _i \\varphi _i+U(x)$, with $ U \\perp \\varphi _k$, $k=1, \\ldots, n$. Starting with $k=0$, when the problem is linear, we continue the solution in $k$ by keeping $\\xi =(\\xi _1, \\ldots,\\xi _n)$ fixed, but allowing for $\\mu =(\\mu _1, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.05817","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":"1609.05817","created_at":"2026-05-18T01:04:25.200995+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.05817v1","created_at":"2026-05-18T01:04:25.200995+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.05817","created_at":"2026-05-18T01:04:25.200995+00:00"},{"alias_kind":"pith_short_12","alias_value":"LP4PQXGHDCEI","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_16","alias_value":"LP4PQXGHDCEIX5NX","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_8","alias_value":"LP4PQXGH","created_at":"2026-05-18T12:30:29.479603+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/LP4PQXGHDCEIX5NXMJ5XTIOZW6","json":"https://pith.science/pith/LP4PQXGHDCEIX5NXMJ5XTIOZW6.json","graph_json":"https://pith.science/api/pith-number/LP4PQXGHDCEIX5NXMJ5XTIOZW6/graph.json","events_json":"https://pith.science/api/pith-number/LP4PQXGHDCEIX5NXMJ5XTIOZW6/events.json","paper":"https://pith.science/paper/LP4PQXGH"},"agent_actions":{"view_html":"https://pith.science/pith/LP4PQXGHDCEIX5NXMJ5XTIOZW6","download_json":"https://pith.science/pith/LP4PQXGHDCEIX5NXMJ5XTIOZW6.json","view_paper":"https://pith.science/paper/LP4PQXGH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.05817&json=true","fetch_graph":"https://pith.science/api/pith-number/LP4PQXGHDCEIX5NXMJ5XTIOZW6/graph.json","fetch_events":"https://pith.science/api/pith-number/LP4PQXGHDCEIX5NXMJ5XTIOZW6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LP4PQXGHDCEIX5NXMJ5XTIOZW6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LP4PQXGHDCEIX5NXMJ5XTIOZW6/action/storage_attestation","attest_author":"https://pith.science/pith/LP4PQXGHDCEIX5NXMJ5XTIOZW6/action/author_attestation","sign_citation":"https://pith.science/pith/LP4PQXGHDCEIX5NXMJ5XTIOZW6/action/citation_signature","submit_replication":"https://pith.science/pith/LP4PQXGHDCEIX5NXMJ5XTIOZW6/action/replication_record"}},"created_at":"2026-05-18T01:04:25.200995+00:00","updated_at":"2026-05-18T01:04:25.200995+00:00"}