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The exponents satisfy $2<p<2+\\frac{8}{d+n}<q<4^*=\\frac{2(d+n)}{d+n-4}$, so that the nonlinearity is a combination of a mass subcritical and a mass "},"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":"2410.00032","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-09-20T09:14:04Z","cross_cats_sorted":[],"title_canon_sha256":"bbf5ea15c2e67470cb5c2a7e14fd48e05aba6e7c60a9c5332428c53633eae1d1","abstract_canon_sha256":"99a7bb23968fb331f61c09dc972961f92ca30fc126408db08443f236471ef8dd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:13:58.495097Z","signature_b64":"DQeN73WbnQRl01+9fg9UPlMg9DeDC5KdxSPEnYORFS7xDOGJWL0CeAgAXFaICEtA6p2VPrOlaHfHMmWm8wJ1Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"469f9430f5a03234b886b5149238bbe3733649f11e121333a29037139c5143cf","last_reissued_at":"2026-07-05T09:13:58.494705Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:13:58.494705Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Normalized solutions and stability for biharmonic Schr\\\"odinger equation with potential on waveguide manifold","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jun Wang, Zhaoyang Yin","submitted_at":"2024-09-20T09:14:04Z","abstract_excerpt":"In this paper, we study the following biharmonic Schr\\\"odinger equation with potential and mixed nonlinearities \\begin{equation*}\n  \\left\\{\\begin{array}{ll}\\Delta^2 u +V(x,y)u+\\lambda u =\\mu|u|^{p-2}u+|u|^{q-2}u,\\ (x, y) \\in \\Omega_r \\times \\mathbb{T}^n, \\\\ \\int_{\\Omega_r\\times\\mathbb{T}^n}u^2dxdy=\\Theta,\\end{array} \\right. \\end{equation*} where $\\Omega_r \\subset \\mathbb{R}^d$ is an open bounded convex domain, $r>0$ is large and $\\mu\\in\\mathbb{R}$. 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