{"paper":{"title":"A semilinear equation involving the fractional Laplacian in $\\mathbb{R}^n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Yan Li","submitted_at":"2015-03-07T22:51:17Z","abstract_excerpt":"In this paper, we consider the semilinear equation involving the fractional Laplacian in the Euclidian space $\\mathbb{R}^n$: \\begin{equation} (-\\Delta)^{\\alpha/2} u(x) = f(x_n) \\,u^p(x), \\quad x \\in \\mathbb{R}^n \\label{n26} \\end{equation} in the subcritical case with $1<p<\\frac{n+\\alpha}{n-\\alpha}$. Instead of carrying out direct investigations on pseudo-differential equation (\\ref{n26}), we first seek its equivalent form in an integral equation as below: \\begin{equation} u(x)=\\int_{\\mathbb{R}^n}G_{\\infty}(x,y)\\,f(y_n)\\, u^{p}(y)\\,dy, \\label{n27} \\end{equation} where $ G_{\\infty}(x,y)$ is the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.02224","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"}