{"paper":{"title":"Free boundary problems arising in the theory of maximal solutions of equations with exponential nonlinearities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Angela Pistoia, Giusi Vaira, Michal Kowalczyk, Piotr Rybka","submitted_at":"2018-08-01T01:03:19Z","abstract_excerpt":"We consider equations of the form $\\Delta u +\\lambda^2 V(x)e^{\\,u}=\\rho$ in various two dimensional settings. We assume that $V>0$ is a given function, $\\lambda>0$ is a small parameter and $\\rho=\\mathcal O(1)$ or $\\rho\\to +\\infty$ as $\\lambda\\to 0$. In a recent paper we prove the existence of the maximal solutions for a particular choice $V\\equiv 1$, $\\rho=0$ when the problem is posed in doubly connected domains under Dirichlet boundary conditions. We related the maximal solutions with a novel free boundary problem. The purpose of this note is to derive the corresponding free boundary problems"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.00125","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"}