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The study of radial stationary states leads to the nonlocal problem: $$ - \\Delta u(x) + \\left(\\omega + \\frac{h^2(|x|)}{|x|^2} + \\int_{|x|}^{+\\infty} \\frac{h(s)}{s} u^2(s)\\, ds \\right) u(x) = |u(x)|^{p-1}u(x), $$ where $$ h(r)= \\frac{1}{2}\\int_0^{r} s u^2(s) \\, ds. $$\n  This problem is the Euler-Lagrange equation of a certain energy functional. In this paper the study of the global behavior of such functional is completed. We show that for $p\\in(1,3)$, the functional may be bound"},"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":"1306.2051","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-06-09T19:04:53Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"67dd4621feef34fe08d9a164f4aa778c6b0cd313a5f4cf4584f2e81830a15679","abstract_canon_sha256":"a561f776f94fbcf4b719fc93f85561a9ce828ceb403505edac8ae3d05f1c77d6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:21:22.562181Z","signature_b64":"NpZfC5NaqoBaBf4cVPD8cUvKla8tWgZKpwbW8xKQO6T18+wC6MdbheInE4uvU4BUSQQf5gQn4SLmgJCLNp3SBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ffee600bce95ed2f0ae253093f60ee4656628517b3195bff4342fa6713dcec4","last_reissued_at":"2026-05-18T03:21:22.561580Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:21:22.561580Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Variational Analysis of a Gauged Nonlinear Schr\\\"odinger Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Alessio Pomponio, David Ruiz","submitted_at":"2013-06-09T19:04:53Z","abstract_excerpt":"This paper is motivated by a gauged Schr\\\"odinger equation in dimension 2 including the so-called Chern-Simons term. The study of radial stationary states leads to the nonlocal problem: $$ - \\Delta u(x) + \\left(\\omega + \\frac{h^2(|x|)}{|x|^2} + \\int_{|x|}^{+\\infty} \\frac{h(s)}{s} u^2(s)\\, ds \\right) u(x) = |u(x)|^{p-1}u(x), $$ where $$ h(r)= \\frac{1}{2}\\int_0^{r} s u^2(s) \\, ds. $$\n  This problem is the Euler-Lagrange equation of a certain energy functional. In this paper the study of the global behavior of such functional is completed. 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