{"paper":{"title":"Bound states for logarithmic Schrodinger equations with potentials unbounded below","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chengxiang Zhang, Xu Zhang","submitted_at":"2019-05-16T12:26:07Z","abstract_excerpt":"We study the existence and concentration behavior of the bound states for the following logarithmic Schr\\\"odinger equation \\begin{equation*} \\begin{cases} -\\varepsilon^2\\Delta v+V(x)v=v\\log v^2 \\ \\ &\\text {in}\\ \\ \\mathbb R^N,\\\\ v(x)\\to 0 \\ \\ &\\text {as}\\ \\ |x|\\to\\infty, \\end{cases} \\end{equation*} where $N\\geq 1$, $\\varepsilon>0$ is a small parameter, and $V$ may be unbounded below at infinity with a speed of at most quadratic strength. We show that around various types of local topological critical points of the potential function, positive bound state solutions exist and concentrate as $\\var"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.06687","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":""},"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"}