{"paper":{"title":"Solutions to sublinear elliptic equations with finite generalized energy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Adisak Seesanea, Igor E. Verbitsky","submitted_at":"2018-04-24T20:51:05Z","abstract_excerpt":"We give necessary and sufficient conditions for the existence of a positive solution with zero boundary values to the elliptic equation \\[ \\mathcal{L}u = \\sigma u^{q} + \\mu \\quad \\text{in} \\;\\; \\Omega, \\] in the sublinear case $0<q<1$, with finite generalized energy: $\\mathbb{E}_{\\gamma}[u]:=\\int_{\\Omega} |\\nabla u|^{2} u^{\\gamma-1}dx<\\infty$, for $\\gamma >0$. In this case $u \\in L^{\\gamma+q}(\\Omega, \\sigma)\\cap L^{\\gamma}(\\Omega, \\mu)$, where $\\gamma=1$ corresponds to finite energy solutions.\n  Here $\\mathcal{L} u:= -\\,\\text{div}(\\mathcal{A}\\nabla u)$ is a linear uniformly elliptic operator w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.09255","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"}