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We prove that $\\frac{v(x)}{\\log |x|}\\to -2$ as $|x|\\to\\infty$ and $\\alpha>2\\beta$. As a consequence under a mild condition on $v$ we prove that the solution is radially symmetric about the origin."},"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":"1209.0544","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-09-04T07:15:36Z","cross_cats_sorted":[],"title_canon_sha256":"11fbd1ba96fe8628bb19c944ff9754b53ed4050983c7ab45509b53470304817f","abstract_canon_sha256":"abf4b3e739f596331b30916c49e8c87b0b3812a59da7dfed2bc691a57383d409"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:46:17.900879Z","signature_b64":"3Xt4MOyI9Ep0d0ZNwkke0dJPYnOHkR8dYZqyd9ktAS3nVFMUCmSqkvOQGUI+C5xMA+wAn8agChg0/AqaRBlvAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"65bf4253426da249fae459e57d82552d56eb0190ea588068cb0f3ab171cfc78c","last_reissued_at":"2026-05-18T03:46:17.900073Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:46:17.900073Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Decay rate and radial symmetry of the exponential elliptic equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kin Ming Hui, SungHoon Kim","submitted_at":"2012-09-04T07:15:36Z","abstract_excerpt":"Let $n\\geq 3$, $\\alpha$, $\\beta\\in\\mathbb{R}$, and let $v$ be a solution $\\Delta v+\\alpha e^v+\\beta x\\cdot\\nabla e^v=0$ in $\\mathbb{R}^n$, which satisfies the conditions $\\lim_{R\\to\\infty}\\frac{1}{\\log R}\\int_{1}^{R}\\rho^{1-n} (\\int_{B_{\\rho}}e^v\\,dx)d\\rho\\in (0,\\infty)$ and $|x|^2e^{v(x)}\\le A_1$ in $\\R^n$. We prove that $\\frac{v(x)}{\\log |x|}\\to -2$ as $|x|\\to\\infty$ and $\\alpha>2\\beta$. As a consequence under a mild condition on $v$ we prove that the solution is radially symmetric about the origin."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.0544","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":"1209.0544","created_at":"2026-05-18T03:46:17.900177+00:00"},{"alias_kind":"arxiv_version","alias_value":"1209.0544v1","created_at":"2026-05-18T03:46:17.900177+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.0544","created_at":"2026-05-18T03:46:17.900177+00:00"},{"alias_kind":"pith_short_12","alias_value":"MW7UEU2CNWRE","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_16","alias_value":"MW7UEU2CNWRET6XE","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_8","alias_value":"MW7UEU2C","created_at":"2026-05-18T12:27:14.488303+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/MW7UEU2CNWRET6XELHSX3ASVFV","json":"https://pith.science/pith/MW7UEU2CNWRET6XELHSX3ASVFV.json","graph_json":"https://pith.science/api/pith-number/MW7UEU2CNWRET6XELHSX3ASVFV/graph.json","events_json":"https://pith.science/api/pith-number/MW7UEU2CNWRET6XELHSX3ASVFV/events.json","paper":"https://pith.science/paper/MW7UEU2C"},"agent_actions":{"view_html":"https://pith.science/pith/MW7UEU2CNWRET6XELHSX3ASVFV","download_json":"https://pith.science/pith/MW7UEU2CNWRET6XELHSX3ASVFV.json","view_paper":"https://pith.science/paper/MW7UEU2C","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1209.0544&json=true","fetch_graph":"https://pith.science/api/pith-number/MW7UEU2CNWRET6XELHSX3ASVFV/graph.json","fetch_events":"https://pith.science/api/pith-number/MW7UEU2CNWRET6XELHSX3ASVFV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MW7UEU2CNWRET6XELHSX3ASVFV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MW7UEU2CNWRET6XELHSX3ASVFV/action/storage_attestation","attest_author":"https://pith.science/pith/MW7UEU2CNWRET6XELHSX3ASVFV/action/author_attestation","sign_citation":"https://pith.science/pith/MW7UEU2CNWRET6XELHSX3ASVFV/action/citation_signature","submit_replication":"https://pith.science/pith/MW7UEU2CNWRET6XELHSX3ASVFV/action/replication_record"}},"created_at":"2026-05-18T03:46:17.900177+00:00","updated_at":"2026-05-18T03:46:17.900177+00:00"}