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We establish an integral representation formula for any nonconstant classical solution satisfying certain growth at infinity. From this we prove that these solutions are radially symmetric about s"},"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":"2202.01409","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2022-02-03T04:56:34Z","cross_cats_sorted":[],"title_canon_sha256":"046e9831c6fc939eddbfe9fabdf04be0a2812e2194b48792c2ee9835830e50d5","abstract_canon_sha256":"c07e9ec26e5d5503ee37405a0d044ead5a6a7403d07379b7527123c4e293c90f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:53:54.001525Z","signature_b64":"Yi2yQ/fSc3oX20jpiuhHYANuXeRLt2yJIJ1ZqgbHqGdqbk6uPxvLbFz1KzbKEqBnIxAypttCCP/4OGsPyBmmCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"77c344eef93a34d55f829f979a4bdc9fceed3a449ec3ecb47271d51c358714e5","last_reissued_at":"2026-07-05T03:53:54.001069Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:53:54.001069Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Classification of solutions to equations involving Higher-order fractional Laplacian","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jiaqi Hu, Yuan Li, Zhenping Feng, Zhuoran Du","submitted_at":"2022-02-03T04:56:34Z","abstract_excerpt":"In this paper, we are concerned with the following equation involving higher-order fractional Lapalacian \\begin{equation*} \\left\\{\\begin{aligned} &(-\\Delta)^{p+{\\frac{\\alpha}{2}}}u(x)=u_+^\\gamma~~ \\mbox{ in }\\mathbb{R}^n,\\\\ &\\int_{\\mathbb{R}^n}u_+^\\gamma dx<+\\infty, \\end{aligned}\\right. \\end{equation*} where $p\\geq 1$ is an integer, $0<\\alp<2$, $n> 2p+\\alpha$ and $\\gamma \\in (1,\\frac{n}{n-2p-\\alp})$. We establish an integral representation formula for any nonconstant classical solution satisfying certain growth at infinity. 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