{"paper":{"title":"Sharp non-existence results of prescribed L^2-norm solutions for some class of Schr\\\"odinger-Poisson and quasilinear equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Louis Jeanjean, Tingjian Luo","submitted_at":"2012-03-27T15:37:39Z","abstract_excerpt":"In this paper we study the existence of minimizers for $$ F(u) = \\1/2\\int_{\\R^3} |\\nabla u|^2 dx + 1/4\\int_{\\R^3}\\int_{\\R^3}\\frac{| u(x) |^2| u(y) |^2}{| x-y |}dxdy-\\frac{1}{p}\\int_{\\R^3}| u |^p dx$$ on the constraint $$S(c) = \\{u \\in H^1(\\R^3) : \\int_{\\R^3}|u|^2 dx = c \\},$$ where $c>0$ is a given parameter. In the range $p \\in [3, 10/3]$ we explicit a threshold value of $c>0$ separating existence and non-existence of minimizers. We also derive a non-existence result of critical points of $F(u)$ restricted to $S(c)$ when $c>0$ is sufficiently small. Finally, as a byproduct of our approaches, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.6002","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":""},"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"}