{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:3ML536NUBNEUL2VOYAGAC6N24Q","short_pith_number":"pith:3ML536NU","canonical_record":{"source":{"id":"1211.6686","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-28T18:05:37Z","cross_cats_sorted":[],"title_canon_sha256":"16c56b20386c5e2f978e3dc814ab921010ec83649a23ecd36682d9c1ba8555ed","abstract_canon_sha256":"6d5c0f6f67e1bf57e8fcf4632af4f59837702d2c852d3700a54cebc89d661b28"},"schema_version":"1.0"},"canonical_sha256":"db17ddf9b40b4945eaaec00c0179bae4377d35dcec35ff5b492f520f18c41b13","source":{"kind":"arxiv","id":"1211.6686","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.6686","created_at":"2026-05-18T02:18:58Z"},{"alias_kind":"arxiv_version","alias_value":"1211.6686v1","created_at":"2026-05-18T02:18:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.6686","created_at":"2026-05-18T02:18:58Z"},{"alias_kind":"pith_short_12","alias_value":"3ML536NUBNEU","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"3ML536NUBNEUL2VO","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"3ML536NU","created_at":"2026-05-18T12:26:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:3ML536NUBNEUL2VOYAGAC6N24Q","target":"record","payload":{"canonical_record":{"source":{"id":"1211.6686","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-28T18:05:37Z","cross_cats_sorted":[],"title_canon_sha256":"16c56b20386c5e2f978e3dc814ab921010ec83649a23ecd36682d9c1ba8555ed","abstract_canon_sha256":"6d5c0f6f67e1bf57e8fcf4632af4f59837702d2c852d3700a54cebc89d661b28"},"schema_version":"1.0"},"canonical_sha256":"db17ddf9b40b4945eaaec00c0179bae4377d35dcec35ff5b492f520f18c41b13","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:18:58.539029Z","signature_b64":"J3H1e+TNif0DsJK3xpbeUvhtOzo8XUJDEFZU8zxsYzG2p3ZaHpSj4y2tKgcyIoEaxJ+5HXITuu8QOM02TEJZBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"db17ddf9b40b4945eaaec00c0179bae4377d35dcec35ff5b492f520f18c41b13","last_reissued_at":"2026-05-18T02:18:58.538334Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:18:58.538334Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1211.6686","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:18:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iQJzxhA6xxzme6H3OFVx5fU67SE4hbdf8LSHi4rhkczDerrRXFXW3GD0o0UvtlUp2VbwVDrI+Z67e06Axac1DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T20:21:03.866825Z"},"content_sha256":"06b680526a6eb49ff9999f81ed01e80d2428cdcb00e052d77d268f20f16a1f8a","schema_version":"1.0","event_id":"sha256:06b680526a6eb49ff9999f81ed01e80d2428cdcb00e052d77d268f20f16a1f8a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:3ML536NUBNEUL2VOYAGAC6N24Q","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An energy constrained method for the existence of layered type solutions of NLS equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Francesca Alessio, Piero Montecchiari","submitted_at":"2012-11-28T18:05:37Z","abstract_excerpt":"We study the existence of positive solutions on $\\R^{N+1}$ to semilinear elliptic equation $-\\Delta u+u=f(u)$ where $N\\geq 1$ and $f$ is modeled on the power case $f(u)=|u|^{p-1}u$. Denoting with $c$ the mountain pass level of $\\f(u)=\\tfrac 12\\|u\\|^{2}_{H^{1}(\\R^{N})}-\\int_{\\R^{N}}F(u)\\, dx$, $u\\in H^{1}(\\R^{N})$ ($F(s)=\\int_{0}^{s}f(t)\\, dt$), we show, via a new energy constrained variational argument, that for any $b\\in [0,c)$ there exists a positive bounded solution $v_{b}\\in C^{2}(\\R^{N+1})$ such that $E_{v_{b}}(y)=\\tfrac 12\\|\\partial_{y}v_{b}(\\cdot,y)\\|^{2}_{L^{2}(\\R^{N})}-V(v_{b}(\\cdot,y"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.6686","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:18:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"85JguqfG3INiCAbuu/VPvnV51E60E8JsExu0WurOCZwaiPaklW5lIx6TikQwfXlR5TdBL6BFfLu58iL/42vTAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T20:21:03.867157Z"},"content_sha256":"8bfd7160f48c5cf01fd769ff95f359d5be19686567eb0440fb0631736ab05eb4","schema_version":"1.0","event_id":"sha256:8bfd7160f48c5cf01fd769ff95f359d5be19686567eb0440fb0631736ab05eb4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3ML536NUBNEUL2VOYAGAC6N24Q/bundle.json","state_url":"https://pith.science/pith/3ML536NUBNEUL2VOYAGAC6N24Q/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3ML536NUBNEUL2VOYAGAC6N24Q/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-20T20:21:03Z","links":{"resolver":"https://pith.science/pith/3ML536NUBNEUL2VOYAGAC6N24Q","bundle":"https://pith.science/pith/3ML536NUBNEUL2VOYAGAC6N24Q/bundle.json","state":"https://pith.science/pith/3ML536NUBNEUL2VOYAGAC6N24Q/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3ML536NUBNEUL2VOYAGAC6N24Q/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:3ML536NUBNEUL2VOYAGAC6N24Q","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":"6d5c0f6f67e1bf57e8fcf4632af4f59837702d2c852d3700a54cebc89d661b28","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-28T18:05:37Z","title_canon_sha256":"16c56b20386c5e2f978e3dc814ab921010ec83649a23ecd36682d9c1ba8555ed"},"schema_version":"1.0","source":{"id":"1211.6686","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.6686","created_at":"2026-05-18T02:18:58Z"},{"alias_kind":"arxiv_version","alias_value":"1211.6686v1","created_at":"2026-05-18T02:18:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.6686","created_at":"2026-05-18T02:18:58Z"},{"alias_kind":"pith_short_12","alias_value":"3ML536NUBNEU","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"3ML536NUBNEUL2VO","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"3ML536NU","created_at":"2026-05-18T12:26:50Z"}],"graph_snapshots":[{"event_id":"sha256:8bfd7160f48c5cf01fd769ff95f359d5be19686567eb0440fb0631736ab05eb4","target":"graph","created_at":"2026-05-18T02:18:58Z","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 existence of positive solutions on $\\R^{N+1}$ to semilinear elliptic equation $-\\Delta u+u=f(u)$ where $N\\geq 1$ and $f$ is modeled on the power case $f(u)=|u|^{p-1}u$. Denoting with $c$ the mountain pass level of $\\f(u)=\\tfrac 12\\|u\\|^{2}_{H^{1}(\\R^{N})}-\\int_{\\R^{N}}F(u)\\, dx$, $u\\in H^{1}(\\R^{N})$ ($F(s)=\\int_{0}^{s}f(t)\\, dt$), we show, via a new energy constrained variational argument, that for any $b\\in [0,c)$ there exists a positive bounded solution $v_{b}\\in C^{2}(\\R^{N+1})$ such that $E_{v_{b}}(y)=\\tfrac 12\\|\\partial_{y}v_{b}(\\cdot,y)\\|^{2}_{L^{2}(\\R^{N})}-V(v_{b}(\\cdot,y","authors_text":"Francesca Alessio, Piero Montecchiari","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-28T18:05:37Z","title":"An energy constrained method for the existence of layered type solutions of NLS equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.6686","kind":"arxiv","version":1},"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":"sha256:06b680526a6eb49ff9999f81ed01e80d2428cdcb00e052d77d268f20f16a1f8a","target":"record","created_at":"2026-05-18T02:18:58Z","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":"6d5c0f6f67e1bf57e8fcf4632af4f59837702d2c852d3700a54cebc89d661b28","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-28T18:05:37Z","title_canon_sha256":"16c56b20386c5e2f978e3dc814ab921010ec83649a23ecd36682d9c1ba8555ed"},"schema_version":"1.0","source":{"id":"1211.6686","kind":"arxiv","version":1}},"canonical_sha256":"db17ddf9b40b4945eaaec00c0179bae4377d35dcec35ff5b492f520f18c41b13","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"db17ddf9b40b4945eaaec00c0179bae4377d35dcec35ff5b492f520f18c41b13","first_computed_at":"2026-05-18T02:18:58.538334Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:18:58.538334Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"J3H1e+TNif0DsJK3xpbeUvhtOzo8XUJDEFZU8zxsYzG2p3ZaHpSj4y2tKgcyIoEaxJ+5HXITuu8QOM02TEJZBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:18:58.539029Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.6686","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:06b680526a6eb49ff9999f81ed01e80d2428cdcb00e052d77d268f20f16a1f8a","sha256:8bfd7160f48c5cf01fd769ff95f359d5be19686567eb0440fb0631736ab05eb4"],"state_sha256":"eb9ec7c471276251cba19701076e075bc4b04e562940c7a7bf2322e1a2ee6193"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0QHmjIZES4+oifuLyWweCkglljluQCiAQk3ihUQnpIA+FkALLz3Mr4HvutOjaIMdSdn0FooAgkYazq78JLPYCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T20:21:03.869053Z","bundle_sha256":"bd631d28280c83cfbbffec4bf80329856ac526e83a5e82dbb244e51dc6c384bf"}}