{"paper":{"title":"Normalized solutions for a fourth-order Schr\\\"{o}dinger equation with positive second-order dispersion coefficient","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Tao Yang, Xiao Luo","submitted_at":"2019-08-07T10:42:05Z","abstract_excerpt":"We are concerned with the existence and asymptotic properties of solutions to the following fourth-order Schr\\\"{o}dinger equation \\begin{equation}\\label{1} {\\Delta}^{2}u+\\mu \\Delta u-{\\lambda}u={|u|}^{p-2}u, ~~~~x \\in \\R^{N}\\\\ \\end{equation} under the normalized constraint\n  $$\\int_{{\\mathbb{R}^N}} {{u}^2}=a^2,$$ where $N\\!\\geq\\!2$, $a,\\mu\\!>\\!0$, $2+\\frac{8}{N}\\!<\\!p\\!<\\! 4^{*}\\!=\\!\\frac{2N}{(N-4)^{+}}$ and $\\lambda\\in\\R$ appears as a Lagrange multiplier. Since the second-order dispersion term affects the structure of the corresponding energy functional $$ E_{\\mu}(u)=\\frac{1}{2}{||\\Delta u||}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.03079","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/1908.03079/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"}