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Denote by $C_H(V)$ the Hardy constant relative to $V$ in $\\Omega$. We study positive solutions of equations (LE) $-L_{\\gamma V} u = 0$ and (NE) $-L_{\\gamma V} u+ f(u) = 0$ in $\\Omega$ when $\\gamma < C_H(V)$ and $f \\in C({\\mathbb R})$ is an odd, monotone increasing function. We establish the existence of a normalized boundary trace for pos"},"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":"1803.04214","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-03-12T12:07:39Z","cross_cats_sorted":[],"title_canon_sha256":"17cd3722329c626a66e9c4f02d3acde618262f4c07d4ec2c9552064709eb8462","abstract_canon_sha256":"9ee27803cfe85c24149d6851226a9909f9c5923e14704400f8c664f083254e45"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:21:32.296590Z","signature_b64":"prjgYvUL8nsG7PEUcAEjhhvIrVD6alCbr3Ob000H/IxbOH87nrmsd0vM/IT/y+sr+cOl9tTHt249Sjfwn/11CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8d66b0aacb497f3d6f6a533d7caad76d8a9b4058b11a92a39b5eeb4fb8f4d3f2","last_reissued_at":"2026-05-18T00:21:32.295973Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:21:32.295973Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Schr\\\"odinger equations with singular potentials: linear and nonlinear boundary value problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Moshe Marcus, Phuoc-Tai Nguyen","submitted_at":"2018-03-12T12:07:39Z","abstract_excerpt":"Let $\\Omega \\subset {\\mathbb R}^N$ ($N \\geq 3$) be a $C^2$ bounded domain and $F \\subset \\partial \\Omega$ be a $C^2$ submanifold of dimension $0 \\leq k \\leq N-2$. Put $\\delta_F(x)=dist(x,F)$, $V=\\delta_F^{-2}$ in $\\Omega$ and $L_{\\gamma V}=\\Delta + \\gamma V$. Denote by $C_H(V)$ the Hardy constant relative to $V$ in $\\Omega$. We study positive solutions of equations (LE) $-L_{\\gamma V} u = 0$ and (NE) $-L_{\\gamma V} u+ f(u) = 0$ in $\\Omega$ when $\\gamma < C_H(V)$ and $f \\in C({\\mathbb R})$ is an odd, monotone increasing function. 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