{"paper":{"title":"On existence and phase separation of solitary waves for nonlinear Schr\\\"odinger systems modelling simultaneous cooperation and competition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Nicola Soave","submitted_at":"2013-10-31T13:28:16Z","abstract_excerpt":"We study the existence of positive bound states for the nonlinear elliptic system \\[ \\begin{cases} - \\Delta u_i + \\lambda_i u_i = \\sum_{j=1}^d \\beta_{ij} u_j^2 u_i & \\text{in $\\Omega$} \\\\ u_1 =\\cdots = u_d=0 & \\text{on $\\partial \\Omega$}, \\end{cases} \\] where $d \\ge 2$, $\\beta_{ij}= \\beta_{ji}$, $\\beta_{ii},\\lambda_i >0$, and $\\Omega$ is either a bounded domain of $\\mathbb{R}^N$, or $\\Omega=\\mathbb{R}^N$, with $N \\le 3$. In light of its applicability in several physical contexts, the problem has been intensively studied in recent years, and several results concerning existence, multiplicity an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.8492","kind":"arxiv","version":3},"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"}