{"paper":{"title":"Global well-posedness, scattering and blow-up for the energy-critical, Schr\\\"odinger equation with indefinite potential in the radial case","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jun Wang, Zhaoyang Yin","submitted_at":"2024-06-22T11:44:41Z","abstract_excerpt":"In this paper, we study the well-posedness theory and the scattering asymptotics for the energy-critical, Schr\\\"odinger equation with indefinite potential \\begin{equation*}\n  \\left\\{\\begin{array}{l} i \\partial_t u+\\Delta u-V(x)u +|u|^{\\frac{4}{N-2}}u=0,\\ (x, t) \\in \\mathbb{R}^N \\times \\mathbb{R}, \\\\ \\left.u\\right|_{t=0}=u_0 \\in H ^1(\\mathbb{R}^N), \\end{array}\\right. \\end{equation*} where $V(x):\\mathbb{R}^N\\rightarrow \\mathbb{R}$ is indefinite and satisfies appropriate conditions. Using contraction mapping method and concentration compactness argument, we obtain the well-posedness theory in pro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.01588","kind":"arxiv","version":1},"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/2407.01588/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"}