{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:FM6GCTDR37RGQZ45POV24QD4YS","short_pith_number":"pith:FM6GCTDR","schema_version":"1.0","canonical_sha256":"2b3c614c71dfe268679d7babae407cc48ec2e5e6edc6b71d615a27d5e1a11c05","source":{"kind":"arxiv","id":"1802.02089","version":1},"attestation_state":"computed","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 $1 0$, $q \\in [1,2)$, $B_1=B_1(0)$ 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