{"paper":{"title":"Semitrivial vs. fully nontrivial ground states in cooperative cubic Schr\\\"odinger systems with $d\\ge3$ equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Filipe Oliveira, Hugo Tavares, Sim\\~ao Correia","submitted_at":"2015-08-07T19:14:42Z","abstract_excerpt":"In this work we consider the weakly coupled Schr\\\"odinger cubic system \\[ \\begin{cases} \\displaystyle -\\Delta u_i+\\lambda_i u_i= \\mu_i u_i^{3}+ u_i\\sum_{j\\neq i}b_{ij} u_j^2 \\\\ u_i\\in H^1(\\mathbb{R}^N;\\mathbb{R}), \\quad i=1,\\ldots, d, \\end{cases} \\] where $1\\leq N\\leq 3$, $\\lambda_i,\\mu_i >0$ and $b_{ij}=b_{ji}>0$ for $i\\neq j$. This system admits semitrivial solutions, that is solutions $\\mathbf{u}=(u_1,\\ldots, u_d)$ with null components. We provide optimal qualitative conditions on the parameters $\\lambda_i,\\mu_i$ and $b_{ij}$ under which the ground state solutions have all components nontri"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.01783","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"}