{"paper":{"title":"Equations involving fractional Laplacian operator: Compactness and application","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jianfu Yang, Shusen Yan, Xiaohui Yu","submitted_at":"2015-03-03T00:32:05Z","abstract_excerpt":"In this paper, we consider the following problem involving fractional Laplacian operator: \\begin{equation}\\label{eq:0.1} (-\\Delta)^{\\alpha} u= |u|^{2^*_\\alpha-2-\\varepsilon}u + \\lambda u\\,\\, {\\rm in}\\,\\, \\Omega,\\quad u=0 \\,\\, {\\rm on}\\, \\, \\partial\\Omega, \\end{equation} where $\\Omega$ is a smooth bounded domain in $\\mathbb{R}^N$, $\\varepsilon\\in [0, 2^*_\\alpha-2)$, $0<\\alpha<1,\\, 2^*_\\alpha = \\frac {2N}{N-2\\alpha}$. We show that for any sequence of solutions $u_n$ of \\eqref{eq:0.1} corresponding to $\\varepsilon_n\\in [0, 2^*_\\alpha-2)$, satisfying $\\|u_n\\|_{H}\\le C$ in the Sobolev space $H$ def"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.00788","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"}