{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:DWL5MNCFU5LGWSMWIJGL6SLKUR","short_pith_number":"pith:DWL5MNCF","schema_version":"1.0","canonical_sha256":"1d97d63445a7566b4996424cbf496aa44b3d03df23e567a38d1dd8bc33c7b76e","source":{"kind":"arxiv","id":"1608.01123","version":1},"attestation_state":"computed","paper":{"title":"Existence, nonexistence, symmetry and uniqueness of ground state for critical Schr\\\"odinger system involving Hardy term","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Senping Luo, Wenming Zou","submitted_at":"2016-08-03T09:10:21Z","abstract_excerpt":"We study the following elliptic system with critical exponent: \\begin{displaymath} \\begin{cases}-\\Delta u_j-\\frac{\\lambda_j}{|x|^2}u_j=u_j^{2^*-1}+\\sum\\limits_{k\\neq j}\\beta_{jk}\\alpha_{jk}u_j^{\\alpha_{jk}-1}u_k^{\\alpha_{kj}},\\;\\;x\\in\\R^N, u_j\\in D^{1,2}(\\R^N),\\quad u_j>0 \\;\\; \\hbox{in} \\quad \\R^N\\setminus \\{0\\},\\quad j=1,...,r.\\end{cases}\\end{displaymath} Here $N\\geq 3, r\\geq2, 2^*=\\frac{2N}{N-2}, \\lambda_j\\in (0, \\frac{(N-2)^2}{4})$ for all $ j=1,...,r $; $\\beta_{jk}=\\beta_{kj}$; \\; $\\alpha_{jk}>1, \\alpha_{kj}>1,$ satisfying $\\alpha_{jk}+\\alpha_{kj}=2^* $ for all $k\\neq j$. Note that the non"},"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":"1608.01123","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-08-03T09:10:21Z","cross_cats_sorted":[],"title_canon_sha256":"0b3f8c8009567580e69324f6bd026d45920edd2ac8ac3ce69bc052d38bc61f3e","abstract_canon_sha256":"05558b5326f073c3e09eb413c44683b24a871104f8a95822c4effc7f2ef0d362"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:09:56.536748Z","signature_b64":"6sDpxqznk9W2AlKuNaAdj7QXmQaG6t55kma8ze4hAV30Ay2nWR1OCLJNbqMTd+mZNM4oYh4eu3DCrUs+5vN0DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1d97d63445a7566b4996424cbf496aa44b3d03df23e567a38d1dd8bc33c7b76e","last_reissued_at":"2026-05-18T01:09:56.536120Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:09:56.536120Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Existence, nonexistence, symmetry and uniqueness of ground state for critical Schr\\\"odinger system involving Hardy term","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Senping Luo, Wenming Zou","submitted_at":"2016-08-03T09:10:21Z","abstract_excerpt":"We study the following elliptic system with critical exponent: \\begin{displaymath} \\begin{cases}-\\Delta u_j-\\frac{\\lambda_j}{|x|^2}u_j=u_j^{2^*-1}+\\sum\\limits_{k\\neq j}\\beta_{jk}\\alpha_{jk}u_j^{\\alpha_{jk}-1}u_k^{\\alpha_{kj}},\\;\\;x\\in\\R^N, u_j\\in D^{1,2}(\\R^N),\\quad u_j>0 \\;\\; \\hbox{in} \\quad \\R^N\\setminus \\{0\\},\\quad j=1,...,r.\\end{cases}\\end{displaymath} Here $N\\geq 3, r\\geq2, 2^*=\\frac{2N}{N-2}, \\lambda_j\\in (0, \\frac{(N-2)^2}{4})$ for all $ j=1,...,r $; $\\beta_{jk}=\\beta_{kj}$; \\; $\\alpha_{jk}>1, \\alpha_{kj}>1,$ satisfying $\\alpha_{jk}+\\alpha_{kj}=2^* $ for all $k\\neq j$. 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