{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:435WH35YAUK573VGU3ZVAJS34X","short_pith_number":"pith:435WH35Y","schema_version":"1.0","canonical_sha256":"e6fb63efb80515dfeea6a6f350265be5f51fa44e480ff00bb5a44292059cc2d7","source":{"kind":"arxiv","id":"1504.01133","version":2},"attestation_state":"computed","paper":{"title":"A nonlinear elliptic PDE with multiple Hardy-Sobolev critical exponents in $\\mathbb{R}^N$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Wenming Zou, Xuexiu Zhong","submitted_at":"2015-04-05T16:02:15Z","abstract_excerpt":"In this paper, we will study the following PDE in $\\mathbb{R}^N$ involving multiple Hardy-Sobolev critical exponents: $$ \\begin{cases} \\Delta u+\\sum_{i=1}^{l}\\lambda_i \\frac{u^{2^*(s_i)-1}}{|x|^{s_i}}+u^{2^*-1}=0\\;\\hbox{in}\\;\\mathbb{R}^N, u\\in D_{0}^{1,2}(\\mathbb{R}^N), \\end{cases} $$ where $0<s_1<s_2<\\cdots<s_l<2, 2^\\ast:=\\frac{2N}{N-2}, \\; 2^\\ast(s):=\\frac{2(N-s)}{N-2}$ and there exists some $k\\in [1, l]$ such that $\\lambda_i>0$ for $1\\leq i\\leq k$; $\\lambda_i<0$ for $k+1\\leq i\\leq l$. We develop an interesting way to study this class of equations involving mixed sign parameters. We prove th"},"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":"1504.01133","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-05T16:02:15Z","cross_cats_sorted":[],"title_canon_sha256":"a339d9a40fdac62d4f1f3cf106059eea13d23e4193efd6ad392796cd61009a08","abstract_canon_sha256":"0d27022c862c66207fcfbf361d8ed2cb972932e3ef37623110b250b9b44f4405"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:27:10.260424Z","signature_b64":"/Ris/NiQ7HBOHjmS/ULg4X/4eYTkExNM+TvBczhEVQ1WEVJ48mUKoUvVhUAiASpMt+MLi94ORgMMwhYi2Ld1Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e6fb63efb80515dfeea6a6f350265be5f51fa44e480ff00bb5a44292059cc2d7","last_reissued_at":"2026-05-18T00:27:10.259741Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:27:10.259741Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A nonlinear elliptic PDE with multiple Hardy-Sobolev critical exponents in $\\mathbb{R}^N$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Wenming Zou, Xuexiu Zhong","submitted_at":"2015-04-05T16:02:15Z","abstract_excerpt":"In this paper, we will study the following PDE in $\\mathbb{R}^N$ involving multiple Hardy-Sobolev critical exponents: $$ \\begin{cases} \\Delta u+\\sum_{i=1}^{l}\\lambda_i \\frac{u^{2^*(s_i)-1}}{|x|^{s_i}}+u^{2^*-1}=0\\;\\hbox{in}\\;\\mathbb{R}^N, u\\in D_{0}^{1,2}(\\mathbb{R}^N), \\end{cases} $$ where $0<s_1<s_2<\\cdots<s_l<2, 2^\\ast:=\\frac{2N}{N-2}, \\; 2^\\ast(s):=\\frac{2(N-s)}{N-2}$ and there exists some $k\\in [1, l]$ such that $\\lambda_i>0$ for $1\\leq i\\leq k$; $\\lambda_i<0$ for $k+1\\leq i\\leq l$. We develop an interesting way to study this class of equations involving mixed sign parameters. 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