{"paper":{"title":"Limit Behavior of Mass Critical Hartree Minimization Problems with Steep Potential Wells","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Yong Luo, Yujin Guo, Zhi-qiang Wang","submitted_at":"2018-03-27T07:37:27Z","abstract_excerpt":"We consider minimizers of the following mass critical Hartree minimization problem:\n  \\[ e_\\lambda(N):=\\underset{\\{u\\in H^1(R^d),\\,\\|u\\|^2_2=N\\}}{\\inf} E_\\lambda(u),\\,\\ d\\ge 3, \\] where the Hartree energy functional $E_\\lambda(u)$ is defined by \\[\n  E_\\lambda(u):=\\int_{R ^d}|\\nabla u(x)|^2dx+\\lambda \\int_{R ^d}g(x)u^2(x)dx-\\frac{1}{2} \\int_{R ^d}\\int_{R ^d} \\frac{u^2(x)u^2(y)}{|x-y|^2}dxdy,\\,\\ \\lambda>0,\\] and the steep potential $g(x)$ satisfies $0=g(0)=\\inf _{R^d}g(x)\\le g(x)\\le 1$ and $1-g(x)\\in L^{\\frac{d}{2}}(R^d)$. We prove that there exists a constant $N^*>0$, independent of $\\lambda g("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.09936","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"}