{"paper":{"title":"Some remarks on a minimization problem associated to a fourth order nonlinear Schr\\\"odinger equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Antonio J. Fern\\'andez, Louis Jeanjean, Nabile Boussa\\\"id","submitted_at":"2019-10-29T10:29:23Z","abstract_excerpt":"Let $\\gamma > 0\\,$, $\\beta > 0\\,$, $\\alpha > 0$ and $0 < \\sigma N < 4$. In the present paper, we study, for $c > 0$ given, the constrained minimization problem \\begin{equation*} \\label{MinL2fixed} m(c):=\\inf_{u\\in S (c) }E(u), \\end{equation*} where \\begin{equation*} E (u):=\\frac{\\gamma}{2}\\int_{\\mathbb{R}^N}|\\Delta u|^2\\, dx -\\frac{\\beta}{2}\\int_{\\mathbb{R}^N}|\\nabla u|^2\\, dx-\\frac{\\alpha}{2\\sigma+2}\\int_{\\mathbb{R}^N}|u|^{2\\sigma+2}\\, dx, \\end{equation*} and \\begin{equation*} S(c):=\\left\\{u\\in H^2(\\mathbb{R}^N):\\int_{\\mathbb{R}^N}|u|^{2}\\, dx=c\\right\\}. \\end{equation*} The aim of our study i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1910.13177","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1910.13177/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}