{"paper":{"title":"Arbitrary many positive solutions for a nonlinear problem involving the fractional Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jinguo Zhang, Xiaochun Liu","submitted_at":"2014-12-29T01:55:52Z","abstract_excerpt":"We establish the existence and multiplicity of positive solutions to the problems involving the fractional Laplacian: \\begin{equation*} \\left\\{\\begin{array}{lll} &(-\\Delta)^{s}u=\\lambda u^{p}+f(u),\\,\\,u>0 \\quad &\\mbox{in}\\,\\,\\Omega,\\\\ &u=0\\quad &\\mbox{in}\\,\\,\\mathbb{R}^{N}\\setminus\\Omega,\\\\ \\end{array}\\right. \\end{equation*} where $\\Omega\\subset \\mathbb{R}^{N}$ $(N\\geq 2)$ is a bounded smooth domain, $s\\in (0,1)$, $p>0$, $\\lambda\\in \\mathbb{R}$ and $(-\\Delta)^{s}$ stands for the fractional Laplacian. When $f$ oscillates near the origin or at infinity, via the variational argument we prove that"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.8241","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"}