{"paper":{"title":"The nodal set of solutions to some elliptic problems: sublinear equations, and unstable two-phase membrane problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Nicola Soave, Susanna Terracini","submitted_at":"2018-02-06T17:43:48Z","abstract_excerpt":"We are concerned with the nodal set of solutions to equations of the form \\begin{equation*} -\\Delta u = \\lambda_+ \\left(u^+\\right)^{q-1} - \\lambda_- \\left(u^-\\right)^{q-1} \\quad \\text{in $B_1$} \\end{equation*} where $\\lambda_+,\\lambda_- > 0$, $q \\in [1,2)$, $B_1=B_1(0)$ is the unit ball in $\\mathbb{R}^N$, $N \\ge 2$, and $u^+:= \\max\\{u,0\\}$, $u^-:= \\max\\{-u,0\\}$ are the positive and the negative part of $u$, respectively. This class includes, the \\emph{unstable two-phase membrane problem} ($q=1$), as well as \\emph{sublinear} equations for $1<q<2$.\n  We prove the following main results: (a) the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.02089","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"}