{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:ILM5NUC5U6YVN3VQYRRSRVNENU","short_pith_number":"pith:ILM5NUC5","canonical_record":{"source":{"id":"1303.5149","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-03-21T03:06:45Z","cross_cats_sorted":[],"title_canon_sha256":"cef9a8a835ca02f97e3f0ea6cfa71136625055a7534e9ae31e79092d8402f130","abstract_canon_sha256":"239d13dade47dd619467d2ac1e0e185cd29aff6b5f30a4e644c83a6c6f9f372e"},"schema_version":"1.0"},"canonical_sha256":"42d9d6d05da7b156eeb0c46328d5a46d20cf5d78cffb08213a289556e00c2adb","source":{"kind":"arxiv","id":"1303.5149","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.5149","created_at":"2026-05-18T03:25:58Z"},{"alias_kind":"arxiv_version","alias_value":"1303.5149v2","created_at":"2026-05-18T03:25:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.5149","created_at":"2026-05-18T03:25:58Z"},{"alias_kind":"pith_short_12","alias_value":"ILM5NUC5U6YV","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"ILM5NUC5U6YVN3VQ","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"ILM5NUC5","created_at":"2026-05-18T12:27:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:ILM5NUC5U6YVN3VQYRRSRVNENU","target":"record","payload":{"canonical_record":{"source":{"id":"1303.5149","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-03-21T03:06:45Z","cross_cats_sorted":[],"title_canon_sha256":"cef9a8a835ca02f97e3f0ea6cfa71136625055a7534e9ae31e79092d8402f130","abstract_canon_sha256":"239d13dade47dd619467d2ac1e0e185cd29aff6b5f30a4e644c83a6c6f9f372e"},"schema_version":"1.0"},"canonical_sha256":"42d9d6d05da7b156eeb0c46328d5a46d20cf5d78cffb08213a289556e00c2adb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:25:58.317992Z","signature_b64":"xehjv/87ee6LPqfp+LcckIbjB/zEYJNGd7sGwKY3NBG3SFYGTlI/WMrHaWrtJ5JHj/UchB84ukZ9F2IBss6HAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"42d9d6d05da7b156eeb0c46328d5a46d20cf5d78cffb08213a289556e00c2adb","last_reissued_at":"2026-05-18T03:25:58.317436Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:25:58.317436Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1303.5149","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:25:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VzCys85pDwIHWpTZFG3PUXgmqhNojzm/op6k9KnHAr6JcCJ6my4RH1EtiSZ7YEP3qeCBBxxRtz6EC3HWeSBGCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T05:13:12.229077Z"},"content_sha256":"1051dcc0ff8e95a17219fa3a856b2c19b66fc9439a6bb341f627d36f9e363dac","schema_version":"1.0","event_id":"sha256:1051dcc0ff8e95a17219fa3a856b2c19b66fc9439a6bb341f627d36f9e363dac"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:ILM5NUC5U6YVN3VQYRRSRVNENU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stable solutions and finite Morse index solutions of nonlinear elliptic equations with Hardy potential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Wonjeong Jeong, Youngae Lee","submitted_at":"2013-03-21T03:06:45Z","abstract_excerpt":"We are concerned with Liouville-type results of stable solutions and finite Morse index solutions for the following nonlinear elliptic equation with Hardy potential: \\begin{displaymath} \\Delta u+\\dfrac{\\mu}{|x|^2}u+|x|^l |u|^{p-1}u=0 \\qquad \\textrm{in}\\ \\ \\Omega, \\end{displaymath} where $\\Omega=\\RN$, $\\RN\\setminus\\{0\\}$ for $N\\geq3$, $p>1$, $l>-2$ and $\\mu<(N-2)^2/4$. Our results depend crucially on a new critical exponent $p=p_c(l,\\mu)$ and the parameter $\\mu$ in Hardy term. We prove that there exist no nontrivial stable solution and finite Morse index solution for $1<p<p_c(l,\\mu)$. We also o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.5149","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:25:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dyUnM6pN89fVpNwln4vcPmhdK4qtEFS+P64bGkV91xKejT2yTTQ32z1BnrvbnblZepZlnrkoysjgyX5qPfTFCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T05:13:12.229816Z"},"content_sha256":"cf70321568459c26fd1408018470ebda9279d411944f8d3be6466853d59d17ca","schema_version":"1.0","event_id":"sha256:cf70321568459c26fd1408018470ebda9279d411944f8d3be6466853d59d17ca"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ILM5NUC5U6YVN3VQYRRSRVNENU/bundle.json","state_url":"https://pith.science/pith/ILM5NUC5U6YVN3VQYRRSRVNENU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ILM5NUC5U6YVN3VQYRRSRVNENU/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":"2026-06-11T05:13:12Z","links":{"resolver":"https://pith.science/pith/ILM5NUC5U6YVN3VQYRRSRVNENU","bundle":"https://pith.science/pith/ILM5NUC5U6YVN3VQYRRSRVNENU/bundle.json","state":"https://pith.science/pith/ILM5NUC5U6YVN3VQYRRSRVNENU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ILM5NUC5U6YVN3VQYRRSRVNENU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:ILM5NUC5U6YVN3VQYRRSRVNENU","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":"239d13dade47dd619467d2ac1e0e185cd29aff6b5f30a4e644c83a6c6f9f372e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-03-21T03:06:45Z","title_canon_sha256":"cef9a8a835ca02f97e3f0ea6cfa71136625055a7534e9ae31e79092d8402f130"},"schema_version":"1.0","source":{"id":"1303.5149","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.5149","created_at":"2026-05-18T03:25:58Z"},{"alias_kind":"arxiv_version","alias_value":"1303.5149v2","created_at":"2026-05-18T03:25:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.5149","created_at":"2026-05-18T03:25:58Z"},{"alias_kind":"pith_short_12","alias_value":"ILM5NUC5U6YV","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"ILM5NUC5U6YVN3VQ","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"ILM5NUC5","created_at":"2026-05-18T12:27:46Z"}],"graph_snapshots":[{"event_id":"sha256:cf70321568459c26fd1408018470ebda9279d411944f8d3be6466853d59d17ca","target":"graph","created_at":"2026-05-18T03:25:58Z","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 are concerned with Liouville-type results of stable solutions and finite Morse index solutions for the following nonlinear elliptic equation with Hardy potential: \\begin{displaymath} \\Delta u+\\dfrac{\\mu}{|x|^2}u+|x|^l |u|^{p-1}u=0 \\qquad \\textrm{in}\\ \\ \\Omega, \\end{displaymath} where $\\Omega=\\RN$, $\\RN\\setminus\\{0\\}$ for $N\\geq3$, $p>1$, $l>-2$ and $\\mu<(N-2)^2/4$. Our results depend crucially on a new critical exponent $p=p_c(l,\\mu)$ and the parameter $\\mu$ in Hardy term. We prove that there exist no nontrivial stable solution and finite Morse index solution for $1<p<p_c(l,\\mu)$. We also o","authors_text":"Wonjeong Jeong, Youngae Lee","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-03-21T03:06:45Z","title":"Stable solutions and finite Morse index solutions of nonlinear elliptic equations with Hardy potential"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.5149","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":"sha256:1051dcc0ff8e95a17219fa3a856b2c19b66fc9439a6bb341f627d36f9e363dac","target":"record","created_at":"2026-05-18T03:25:58Z","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":"239d13dade47dd619467d2ac1e0e185cd29aff6b5f30a4e644c83a6c6f9f372e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-03-21T03:06:45Z","title_canon_sha256":"cef9a8a835ca02f97e3f0ea6cfa71136625055a7534e9ae31e79092d8402f130"},"schema_version":"1.0","source":{"id":"1303.5149","kind":"arxiv","version":2}},"canonical_sha256":"42d9d6d05da7b156eeb0c46328d5a46d20cf5d78cffb08213a289556e00c2adb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"42d9d6d05da7b156eeb0c46328d5a46d20cf5d78cffb08213a289556e00c2adb","first_computed_at":"2026-05-18T03:25:58.317436Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:25:58.317436Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xehjv/87ee6LPqfp+LcckIbjB/zEYJNGd7sGwKY3NBG3SFYGTlI/WMrHaWrtJ5JHj/UchB84ukZ9F2IBss6HAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:25:58.317992Z","signed_message":"canonical_sha256_bytes"},"source_id":"1303.5149","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1051dcc0ff8e95a17219fa3a856b2c19b66fc9439a6bb341f627d36f9e363dac","sha256:cf70321568459c26fd1408018470ebda9279d411944f8d3be6466853d59d17ca"],"state_sha256":"ef24243bdc8d75a22302a22ab7cd7579dd10732be56bb0ef6c1c8a29d8038724"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cmne1TQ2+20LlwfZo4DIhnRX4PIJLFVfLJong/r15t5DGcfh6KKbApqT4YWdz2SPrjUYw+miOKqHhMSEz0IUBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T05:13:12.233240Z","bundle_sha256":"bbe761fb20ffb264448fc68dff908581a403ce955682c1e171cca87354c5ea8f"}}