{"paper":{"title":"Bifurcation and segregation in quadratic two-populations Mean Field Games systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gianmaria Verzini, Marco Cirant","submitted_at":"2015-11-30T15:22:02Z","abstract_excerpt":"We search for non-constant normalized solutions to the semilinear elliptic system \\[ \\begin{cases} - \\nu \\Delta v_i + g_i(v_j^2) v_i = \\lambda_i v_i,\\quad v_i>0 & \\text{in $\\Omega$} \\\\ \\partial_n v_i = 0 & \\text{on $\\partial \\Omega$}\\\\ \\int_\\Omega v_i^2\\,dx = 1, & 1\\leq i,j\\leq 2, \\quad j\\neq i, \\end{cases} \\] where $\\nu>0$, $\\Omega \\subset \\mathbb{R}^N$ is smooth and bounded, the functions $g_i$ are positive and increasing, and both the functions $v_i$ and the parameters $\\lambda_i$ are unknown.\n  This system is obtained, via the Hopf-Cole transformation, from a two-populations ergodic Mean F"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.09343","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"}