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\\end{cases} $where $\\Omega$ is a bounded domain in $R^N$, $0\\in\\partial\\Omega$, all the principal curvatures of $\\partial\\Omega$ at $0$ are negative and $\\mu\\geq 0, \\ \\ a>0, \\ \\ N\\geq 7, \\ \\ 0<t<2, \\ \\ 2^{\\star}=\\frac{2N}{N-2}$ and $2^{\\star}(t)=\\frac{2(N-t)}{N-2}$."},"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":"1410.7880","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-29T05:43:44Z","cross_cats_sorted":[],"title_canon_sha256":"53bb3368f53b670044d6d2b7f4f0dc6d2ab701b6035f7328fd3bb7395081ff9d","abstract_canon_sha256":"90bc842ad2a5e2afff61e6a30766fe47bb3ff32f773636c861086ed66fbfbc15"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:39:07.980891Z","signature_b64":"fumXEXJplSGqnSdG1QSmWeMOhiEOqFedXoIzqfyW9us5nxfh8F4n3D7GA9N+iCB64emhW1wh0+aDrl1yrjNHBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9f87a70d11174119777f7d84a18cc37d79f4782618bd164594bed2b505548c3","last_reissued_at":"2026-05-18T02:39:07.980213Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:39:07.980213Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Infinitely many sign changing solutions of an elliptic problem involving critical Sobolev and Hardy-Sobolev exponent","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mousomi Bhakta","submitted_at":"2014-10-29T05:43:44Z","abstract_excerpt":"We study the existence and multiplicity of sign changing solutions of the following equation $ \\begin{cases} -\\Delta u = \\mu |u|^{2^{\\star}-2}u+\\frac{|u|^{2^{*}(t)-2}u}{|x|^t}+a(x)u \\quad\\text{in}\\quad \\Omega, u=0 \\quad\\text{on}\\quad\\partial\\Omega, \\end{cases} $where $\\Omega$ is a bounded domain in $R^N$, $0\\in\\partial\\Omega$, all the principal curvatures of $\\partial\\Omega$ at $0$ are negative and $\\mu\\geq 0, \\ \\ a>0, \\ \\ N\\geq 7, \\ \\ 0<t<2, \\ \\ 2^{\\star}=\\frac{2N}{N-2}$ and 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