{"paper":{"title":"Semilinear elliptic equations with the pseudo-relativistic operator on a bounded domain","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jinmyoung Seok, Woocheol Choi, Younghun Hong","submitted_at":"2016-06-15T18:19:08Z","abstract_excerpt":"We study the Dirichlet problem for the semilinear equations involving the pseudo-relativistic operator on a bounded domain, (\\sqrt{-\\Delta + m^2} - m)u =|u|^{p-1}u \\quad \\textrm{in}~\\Omega, with the Dirichlet boundary condition $u=0$ on $\\partial \\Omega$. Here, $p \\in (1,\\infty)$ and the operator $(\\sqrt{-\\Delta + m^2} - m)$ is defined in terms of spectral decomposition. In this paper, we investigate existence and nonexistence of a nontrivial solution, depending on the choice of $p$, $m$ and $\\Omega$. Precisely, we show that $(i)$ if $p$ is not $H^1$ subcritical ($p \\geq \\frac{n+2}{n-2}$) and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.04892","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"}