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In addition, we prove that in dimension $N=2$, any semi-stable radial weak solution of $-\\Delta u=f(u)$, posed in $B_1$ with Dirichlet data $u|_{\\partial B_1}=0$, is regular."},"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":"1405.1241","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-05-06T12:17:12Z","cross_cats_sorted":[],"title_canon_sha256":"01b7f4b30351fbbf4ea6a23ca2ceee64a3232ba5b0157550f7521a5b0df2124e","abstract_canon_sha256":"51885d537f12a068517de9f8566859ddc2ababf0a9c1428cb272fe8dd23189f4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:52:33.201016Z","signature_b64":"kXBihIQa9EcRpocY5Ig31mrubyx/2FeTk+JSD+O6/7eIesk8a0jUY5OtLQDrJCSShjcAgIQjfMiiHOey5KRkCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"72cc9203ff775a28a9637d94665347e654152a83455eb4659cf39c7c1be7cd8a","last_reissued_at":"2026-05-18T02:52:33.200366Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:52:33.200366Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Non-energy semi-stable radial solutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Salvador Villegas","submitted_at":"2014-05-06T12:17:12Z","abstract_excerpt":"This paper is devoted to the study of semi-stable radial solutions $u\\notin H^1(B_1)$ of $-\\Delta u=f(u) \\mbox{in} \\overline{B_1}\\setminus \\{0\\}=\\{x\\in \\mathbb{R}^N : 0<\\vert x\\vert\\leq 1\\}$, where $f\\in C^1(\\mathbb{R})$ and $N\\geq 2$. 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