{"paper":{"title":"On Entire Solutions of an Elliptic System Modeling Phase Separations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Henri Berestycki, Juncheng Wei, Kelei Wang, Susanna Terracini","submitted_at":"2012-04-04T19:08:40Z","abstract_excerpt":"We study the qualitative properties of a limiting elliptic system arising in phase separation for Bose-Einstein condensates with multiple states:\n\\Delta u=u v^2 in R^n,\n\\Delta v= v u^2 in R^n,\nu, v>0\\quad in R^n.\nWhen n=1, we prove uniqueness of the one-dimensional profile. In dimension 2, we prove that stable solutions with linear growth must be one-dimensional. Then we construct entire solutions in $\\R^2$ with polynomial growth $|x|^d$ for any positive integer $d \\geq 1$. For $d\\geq 2$, these solutions are not one-dimensional. The construction is also extended to multi-component elliptic sys"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.1038","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"}