{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:XLLHMU5DUNZD5TBI6GREI7Q6FP","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"ae80318a375f17c4888ed76c32bd89cbada395d9c10db865b08cce739626de81","cross_cats_sorted":["math-ph","math.MP","nlin.PS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-02-15T20:43:49Z","title_canon_sha256":"6172e6306f4e1eb3bdaf32734e777ba9457d3819133f79afa199555b3d901923"},"schema_version":"1.0","source":{"id":"1202.3426","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.3426","created_at":"2026-05-17T23:46:31Z"},{"alias_kind":"arxiv_version","alias_value":"1202.3426v2","created_at":"2026-05-17T23:46:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.3426","created_at":"2026-05-17T23:46:31Z"},{"alias_kind":"pith_short_12","alias_value":"XLLHMU5DUNZD","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"XLLHMU5DUNZD5TBI","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"XLLHMU5D","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:ad4f621faadc71a7b97d834a78373c598826039efd4c41a4e0ddfa34f1a9c870","target":"graph","created_at":"2026-05-17T23:46:31Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We study the leading order behaviour of positive solutions of the equation -\\Delta u +\\varepsilon u-|u|^{p-2}u+|u|^{q-2}u=0,\\qquad x\\in\\R^N, where $N\\ge 3$, $q>p>2$ and when $\\varepsilon>0$ is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of $p$, $q$ and $N$. The behavior of solutions depends sensitively on whether $p$ is less, equal or bigger than the critical Sobolev exponent $p^\\ast=\\frac{2N}{N-2}$. For $p<p^\\ast$ the solution asymptotically coincides with the solution of the equation in which the last term is absent. For $p>p^\\ast$ ","authors_text":"Cyrill B. Muratov, Vitaly Moroz","cross_cats":["math-ph","math.MP","nlin.PS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-02-15T20:43:49Z","title":"Asymptotic properties of ground states of scalar field equations with a vanishing parameter"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.3426","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:9c07eb22037f7c1c535c8bdd255e7bdc9bb0e5ce66ac171941d2323bc982897f","target":"record","created_at":"2026-05-17T23:46:31Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ae80318a375f17c4888ed76c32bd89cbada395d9c10db865b08cce739626de81","cross_cats_sorted":["math-ph","math.MP","nlin.PS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-02-15T20:43:49Z","title_canon_sha256":"6172e6306f4e1eb3bdaf32734e777ba9457d3819133f79afa199555b3d901923"},"schema_version":"1.0","source":{"id":"1202.3426","kind":"arxiv","version":2}},"canonical_sha256":"bad67653a3a3723ecc28f1a2447e1e2bd40b50aebf31b21e78b09aebd0f59e5e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bad67653a3a3723ecc28f1a2447e1e2bd40b50aebf31b21e78b09aebd0f59e5e","first_computed_at":"2026-05-17T23:46:31.486334Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:46:31.486334Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"oE1jlpfSTPlU0HT3qTq0+b+y1uUopCClUn1sNIk7ratH5ybIMlNNRfJklgFP3sBem9hPGxMVM8k1F1pASc9vBw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:46:31.486903Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.3426","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9c07eb22037f7c1c535c8bdd255e7bdc9bb0e5ce66ac171941d2323bc982897f","sha256:ad4f621faadc71a7b97d834a78373c598826039efd4c41a4e0ddfa34f1a9c870"],"state_sha256":"af48361a51146777d03aa81c6442b162cc9894ffe874df1a7a9e2cab790b99b8"}