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Existence of nonradial solutions, by contrast, is known only for $N\\geq 4$, $\\alpha =2$, $f\\left( u\\right) =u^{(N+2)/(N-2)}$ and $A$ large enough. Here "},"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":"1708.01228","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-08-03T17:20:40Z","cross_cats_sorted":[],"title_canon_sha256":"09bf17e23592faf68ee57bb296951e1ffbb2e523cd4a793ab74d45fa2e6f40dd","abstract_canon_sha256":"a8d152f0a11cc09517849a0501b032537a8c17418aa1f122a88937055d87ee69"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:22.445491Z","signature_b64":"u47YZ2BTSHjgHlQ8TYtBo+15YgbTmTqgbhBno5+sG+6Bj+11yknk5jULEC7S4lskhr54tAcXyFZdu6Wq3mOUDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"91eac5e2ee920bfd6d2be6f108638547374e14727d15e8a513d6dc307638e7bf","last_reissued_at":"2026-05-18T00:14:22.445075Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:22.445075Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multiple nonradial solutions for a nonlinear elliptic problem with singular and decaying radial potential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Sergio Rolando","submitted_at":"2017-08-03T17:20:40Z","abstract_excerpt":"Many existence and nonexistence results are known for nonnegative radial solutions $u\\in D^{1,2}(\\mathbb{R}^{N})\\cap L^{2}(\\mathbb{R}^{N},\\left|x\\right| ^{-\\alpha }dx)$ to the equation \\[ -\\triangle u+\\dfrac{A}{\\left| x\\right| ^{\\alpha }}u=f\\left( u\\right) \\quad \\textrm{in }\\mathbb{R}^{N},\\quad N\\geq 3,\\quad A,\\alpha >0, \\] with nonlinearites satisfying $\\left| f\\left( u\\right) \\right| \\leq \\left(\\mathrm{const.}\\right) u^{p-1}$ for some $p>2$. Existence of nonradial solutions, by contrast, is known only for $N\\geq 4$, $\\alpha =2$, $f\\left( u\\right) =u^{(N+2)/(N-2)}$ and $A$ large enough. 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