{"paper":{"title":"Normalized bound states for the nonlinear Schrodinger equation in bounded domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Dario Pierotti, Gianmaria Verzini","submitted_at":"2016-07-15T14:19:53Z","abstract_excerpt":"Given $\\rho>0$, we study the elliptic problem \\[ \\text{find } (U,\\lambda)\\in H^1_0(\\Omega)\\times \\mathbb{R} \\text{ such that } \\begin{cases} -\\Delta U+\\lambda U=|U|^{p-1}U \\int_{\\Omega} U^2\\, dx=\\rho, \\end{cases} \\] where $\\Omega\\subset\\mathbb{R}^N$ is a bounded domain and $p>1$ is Sobolev-subcritical, searching for conditions (about $\\rho$, $N$ and $p$) for the existence of solutions. By the Gagliardo-Nirenberg inequality it follows that, when $p$ is $L^2$-subcritical, i.e. $1<p\\leq1+4/N$, the problem admits solution for every $\\rho>0$. In the $L^2$-critical and supercritical case, i.e. when "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.04520","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"}