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We assume that there exists a bounded domain $\\Omega \\subset \\R^N$ such that \\[m_0 \\equiv \\inf_{x \\in \\Omega} V(x) < \\inf_{x \\in \\partial \\Omega} V(x) \\] and we set $K = \\{x \\in \\Omega \\ | \\ V(x) = m_0\\}$. For $\\e >0$ small we prove the existence of at least ${\\cuplength}(K) + 1$ solutions to (\\ref{eq:0.1}) concentrating, a"},"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":"1305.3685","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-05-16T05:29:13Z","cross_cats_sorted":[],"title_canon_sha256":"28626bf12ce07554d8ca5dd4695f977d9cc2a943fb2f6f5bbb3a618466db8f34","abstract_canon_sha256":"6d0d4c28b6b0a06398d31343af50a2a039b11d4631901370e50cfdc715c3b0ea"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:25:34.203062Z","signature_b64":"VAwXePW/tYbYaW87+j8LlxDBaE2LM6anjJtj/gnQ7uamT45r1jdA2pjmppigWaUlblVpwXo/hHBshbulloetBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1cf9a1148f059e7867443cb802fbb1941aaf5f3ae2058effa711d7b16c7b8bbb","last_reissued_at":"2026-05-18T03:25:34.202613Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:25:34.202613Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multiplicity of positive solutions of nonlinear Schr\\\"odinger \\'equations concentrating at a potential well","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kazunaga Tanaka, Louis Jeanjean, Silvia Cingolani","submitted_at":"2013-05-16T05:29:13Z","abstract_excerpt":"We consider singularly perturbed nonlinear Schr\\\"odinger equations \\be \\label{eq:0.1} - \\varepsilon^2 \\Delta u + V(x)u = f(u), \\ \\ u > 0, \\ \\ v \\in H^1(\\R^N) \\ee where $V \\in C(\\R^N, \\R)$ and $f$ is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain $\\Omega \\subset \\R^N$ such that \\[m_0 \\equiv \\inf_{x \\in \\Omega} V(x) < \\inf_{x \\in \\partial \\Omega} V(x) \\] and we set $K = \\{x \\in \\Omega \\ | \\ V(x) = m_0\\}$. For $\\e >0$ small we prove the existence of at least ${\\cuplength}(K) + 1$ solutions to (\\ref{eq:0.1}) concentrating, a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.3685","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":"1305.3685","created_at":"2026-05-18T03:25:34.202682+00:00"},{"alias_kind":"arxiv_version","alias_value":"1305.3685v1","created_at":"2026-05-18T03:25:34.202682+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.3685","created_at":"2026-05-18T03:25:34.202682+00:00"},{"alias_kind":"pith_short_12","alias_value":"DT42CFEPAWPH","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_16","alias_value":"DT42CFEPAWPHQZ2E","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_8","alias_value":"DT42CFEP","created_at":"2026-05-18T12:27:43.054852+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/DT42CFEPAWPHQZ2EHS4AF65RSQ","json":"https://pith.science/pith/DT42CFEPAWPHQZ2EHS4AF65RSQ.json","graph_json":"https://pith.science/api/pith-number/DT42CFEPAWPHQZ2EHS4AF65RSQ/graph.json","events_json":"https://pith.science/api/pith-number/DT42CFEPAWPHQZ2EHS4AF65RSQ/events.json","paper":"https://pith.science/paper/DT42CFEP"},"agent_actions":{"view_html":"https://pith.science/pith/DT42CFEPAWPHQZ2EHS4AF65RSQ","download_json":"https://pith.science/pith/DT42CFEPAWPHQZ2EHS4AF65RSQ.json","view_paper":"https://pith.science/paper/DT42CFEP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1305.3685&json=true","fetch_graph":"https://pith.science/api/pith-number/DT42CFEPAWPHQZ2EHS4AF65RSQ/graph.json","fetch_events":"https://pith.science/api/pith-number/DT42CFEPAWPHQZ2EHS4AF65RSQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DT42CFEPAWPHQZ2EHS4AF65RSQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DT42CFEPAWPHQZ2EHS4AF65RSQ/action/storage_attestation","attest_author":"https://pith.science/pith/DT42CFEPAWPHQZ2EHS4AF65RSQ/action/author_attestation","sign_citation":"https://pith.science/pith/DT42CFEPAWPHQZ2EHS4AF65RSQ/action/citation_signature","submit_replication":"https://pith.science/pith/DT42CFEPAWPHQZ2EHS4AF65RSQ/action/replication_record"}},"created_at":"2026-05-18T03:25:34.202682+00:00","updated_at":"2026-05-18T03:25:34.202682+00:00"}