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We show that if $V$ attains local minimum and $\\Gamma$ attains global maximum at the same point or $V$ attains global minimum and $\\Gamma$ attains local maximum at the same point, then there exists a positive solution for sufficiently small $\\hslash>0$."},"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":"1808.09148","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-08-28T07:31:28Z","cross_cats_sorted":[],"title_canon_sha256":"554c8fdae3257c08e6aa247b47c741c8b429ed0055746a497a3ca194acd38dbd","abstract_canon_sha256":"f83aabc479835061fa66e7acd895babb893299b5758a1fc93f012a9e8757cac4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:21.350217Z","signature_b64":"OrR8bnHmlA5INNNsEvxMJEgktkg9aP6wEmjtUEfLT1s9oLl9KlC9bHamMKL9qNl9VxMV8lOzAJZ1zW1HoP+EDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"29d1baac5a25b22fb0a1f1df0ba08cd89442575267fad206f5dbd2874b5e2755","last_reissued_at":"2026-05-18T00:05:21.349756Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:21.349756Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Note on semiclassical states for the Schr\\\"{o}dinger equation with nonautonomous nonlinearities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Bartosz Bieganowski, Jaros{\\l}aw Mederski","submitted_at":"2018-08-28T07:31:28Z","abstract_excerpt":"We consider the following Schr\\\"{o}dinger equation $$ - \\hslash ^2 \\Delta u + V(x)u = \\Gamma(x) f(u) \\quad \\mathrm{in} \\ \\mathbb{R}^N, $$ where $u \\in H^1 (\\mathbb{R}^N)$, $u > 0$, $\\hslash > 0$ and $f$ is superlinear and subcritical nonlinear term. We show that if $V$ attains local minimum and $\\Gamma$ attains global maximum at the same point or $V$ attains global minimum and $\\Gamma$ attains local maximum at the same point, then there exists a positive solution for sufficiently small $\\hslash>0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.09148","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":"1808.09148","created_at":"2026-05-18T00:05:21.349832+00:00"},{"alias_kind":"arxiv_version","alias_value":"1808.09148v1","created_at":"2026-05-18T00:05:21.349832+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.09148","created_at":"2026-05-18T00:05:21.349832+00:00"},{"alias_kind":"pith_short_12","alias_value":"FHI3VLC2EWZC","created_at":"2026-05-18T12:32:22.470017+00:00"},{"alias_kind":"pith_short_16","alias_value":"FHI3VLC2EWZC7MFB","created_at":"2026-05-18T12:32:22.470017+00:00"},{"alias_kind":"pith_short_8","alias_value":"FHI3VLC2","created_at":"2026-05-18T12:32:22.470017+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/FHI3VLC2EWZC7MFB6HPQXIEM3C","json":"https://pith.science/pith/FHI3VLC2EWZC7MFB6HPQXIEM3C.json","graph_json":"https://pith.science/api/pith-number/FHI3VLC2EWZC7MFB6HPQXIEM3C/graph.json","events_json":"https://pith.science/api/pith-number/FHI3VLC2EWZC7MFB6HPQXIEM3C/events.json","paper":"https://pith.science/paper/FHI3VLC2"},"agent_actions":{"view_html":"https://pith.science/pith/FHI3VLC2EWZC7MFB6HPQXIEM3C","download_json":"https://pith.science/pith/FHI3VLC2EWZC7MFB6HPQXIEM3C.json","view_paper":"https://pith.science/paper/FHI3VLC2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1808.09148&json=true","fetch_graph":"https://pith.science/api/pith-number/FHI3VLC2EWZC7MFB6HPQXIEM3C/graph.json","fetch_events":"https://pith.science/api/pith-number/FHI3VLC2EWZC7MFB6HPQXIEM3C/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FHI3VLC2EWZC7MFB6HPQXIEM3C/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FHI3VLC2EWZC7MFB6HPQXIEM3C/action/storage_attestation","attest_author":"https://pith.science/pith/FHI3VLC2EWZC7MFB6HPQXIEM3C/action/author_attestation","sign_citation":"https://pith.science/pith/FHI3VLC2EWZC7MFB6HPQXIEM3C/action/citation_signature","submit_replication":"https://pith.science/pith/FHI3VLC2EWZC7MFB6HPQXIEM3C/action/replication_record"}},"created_at":"2026-05-18T00:05:21.349832+00:00","updated_at":"2026-05-18T00:05:21.349832+00:00"}