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The existence of nontrivial strong solutions in $W^{2, p}(\\mathbb{R}^N)$ is established using dual variational methods.\n  The focus lies on necessary and sufficient conditions on the parameters deciding whethe"},"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":"1710.04526","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-10-12T14:08:21Z","cross_cats_sorted":[],"title_canon_sha256":"46118e6b0140da3bb97a7a7652de0472743df8edb1f95c2f407ce0cf18795e51","abstract_canon_sha256":"0da30c41cfba3b3dd41865315cfaf43a69ac064447855b5000a0acdaebeb7aa3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:08:31.333497Z","signature_b64":"dst2nyodUkMPnQ7//2ZWG9v8uUdLjMIhnl+ma6RijPp9LG+NFTuGcKpITR9AbWxpgATAIknP1X2cClmwvOlEAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c8d029fee91a16edd7d6f16a27abbb750d37489e45f105fa2b0b68df9cc4bb31","last_reissued_at":"2026-05-18T00:08:31.333083Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:08:31.333083Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dual Variational Methods for a nonlinear Helmholtz system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Dominic Scheider, Rainer Mandel","submitted_at":"2017-10-12T14:08:21Z","abstract_excerpt":"This paper considers a pair of coupled nonlinear Helmholtz equations \\begin{align*}\n  -\\Delta u - \\mu u = a(x) \\left( |u|^\\frac{p}{2} + b(x) |v|^\\frac{p}{2} \\right)|u|^{\\frac{p}{2} - 2}u, \\end{align*} \\begin{align*}\n  -\\Delta v - \\nu v = a(x) \\left( |v|^\\frac{p}{2} + b(x) |u|^\\frac{p}{2} \\right)|v|^{\\frac{p}{2} - 2}v \\end{align*} on $\\mathbb{R}^N$ where $\\frac{2(N+1)}{N-1} < p < 2^\\ast$. The existence of nontrivial strong solutions in $W^{2, p}(\\mathbb{R}^N)$ is established using dual variational methods.\n  The focus lies on necessary and sufficient conditions on the parameters deciding whethe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.04526","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":"1710.04526","created_at":"2026-05-18T00:08:31.333143+00:00"},{"alias_kind":"arxiv_version","alias_value":"1710.04526v2","created_at":"2026-05-18T00:08:31.333143+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.04526","created_at":"2026-05-18T00:08:31.333143+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZDICT7XJDILO","created_at":"2026-05-18T12:31:59.375834+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZDICT7XJDILO3V6W","created_at":"2026-05-18T12:31:59.375834+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZDICT7XJ","created_at":"2026-05-18T12:31:59.375834+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/ZDICT7XJDILO3V6W6FVCPK53OU","json":"https://pith.science/pith/ZDICT7XJDILO3V6W6FVCPK53OU.json","graph_json":"https://pith.science/api/pith-number/ZDICT7XJDILO3V6W6FVCPK53OU/graph.json","events_json":"https://pith.science/api/pith-number/ZDICT7XJDILO3V6W6FVCPK53OU/events.json","paper":"https://pith.science/paper/ZDICT7XJ"},"agent_actions":{"view_html":"https://pith.science/pith/ZDICT7XJDILO3V6W6FVCPK53OU","download_json":"https://pith.science/pith/ZDICT7XJDILO3V6W6FVCPK53OU.json","view_paper":"https://pith.science/paper/ZDICT7XJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1710.04526&json=true","fetch_graph":"https://pith.science/api/pith-number/ZDICT7XJDILO3V6W6FVCPK53OU/graph.json","fetch_events":"https://pith.science/api/pith-number/ZDICT7XJDILO3V6W6FVCPK53OU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZDICT7XJDILO3V6W6FVCPK53OU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZDICT7XJDILO3V6W6FVCPK53OU/action/storage_attestation","attest_author":"https://pith.science/pith/ZDICT7XJDILO3V6W6FVCPK53OU/action/author_attestation","sign_citation":"https://pith.science/pith/ZDICT7XJDILO3V6W6FVCPK53OU/action/citation_signature","submit_replication":"https://pith.science/pith/ZDICT7XJDILO3V6W6FVCPK53OU/action/replication_record"}},"created_at":"2026-05-18T00:08:31.333143+00:00","updated_at":"2026-05-18T00:08:31.333143+00:00"}