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By using constrained variational methods, we proved that there is $a^*>0$, which can be given explicitly, such that problem (\\ref{P}) has a ground state $u$ with $\\|u\\|_{L^p}=1$ for some $\\mu \\in \\mathbb{R}$ and all $a\\in [0,a^*)"},"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":"1605.08206","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-05-26T09:32:01Z","cross_cats_sorted":[],"title_canon_sha256":"539f379781ae9c18e51a97c5ff980bbc52c26c26a604a94a7cfb80bea8cf3b8a","abstract_canon_sha256":"722d37bf1247d9d43eff91bfb40503b995081126f3c43d34540363a37033454a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:02:52.068114Z","signature_b64":"uEzL09a2A7PkSynp97x7cl1vsWiFulw7s8VJSIyZ3LQeM/CWlY6NYWvAZzZdFvcyosChRucifQzexm/5J4ruAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6db3819c7dde73b1bce43f8ae2c9ffb663ee44e0bd7808ec994d6cfd505fa18f","last_reissued_at":"2026-05-18T01:02:52.067658Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:02:52.067658Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Eigenvalue problem for a p-Laplacian equation with trapping potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Huan-Song Zhou, Long-Jiang Gu, Xiaoyu Zeng","submitted_at":"2016-05-26T09:32:01Z","abstract_excerpt":"Consider the following eigenvalue problem of p-Laplacian equation \\begin{equation}\\label{P}\n  -\\Delta_{p}u+V(x)|u|^{p-2}u=\\mu|u|^{p-2}u+a| u|^{s-2}u, x\\in \\mathbb{R}^{n}, \\tag{P} \\end{equation} where $a\\geq0$, $p\\in (1,n)$ and $\\mu\\in\\mathbb{R}$. $V(x)$ is a trapping type potential, e.g., $\\inf\\limits_{x \\in \\mathbb{R}^n}V(x)< \\lim\\limits_{|x|\\rightarrow+\\infty}V(x)$. 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