{"paper":{"title":"Schr\\\"odinger-Poisson equations with singular potentials in $R^3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Huan-Song Zhou, Yongsheng Jiang","submitted_at":"2012-04-09T08:25:57Z","abstract_excerpt":"The existence and $L^{\\infty}$ estimate of positive solutions are discussed for the following Schr\\\"{o}dinger-Poisson system\n{ll}\n-\\Delta u +(\\lambda+\\frac{1}{|y|^\\alpha})u+\\phi (x) u =|u|^{p-1}u, x=(y,z)\\in \\mathbb{R}^2\\times\\mathbb{R},\n-\\Delta\\phi = u^2,\\ \\lim\\limits_{|x|\\rightarrow +\\infty}\\phi(x)=0, \\hfill y=(x_1,x_2) \\in\n\\mathbb{R}^2 with |y|=\\sqrt{x_1^2+x_2^2},\nwhere $\\lambda\\geqslant0$, $\\alpha\\in[0,8)$ and $\\max\\{2,\\frac{2+\\alpha}{2}\\}<p<5$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.1825","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"}