{"paper":{"title":"Normalized solutions to Schr\\\"{o}dinger systems with linear and nonlinear couplings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Zhaoyang Yun, Zhitao Zhang","submitted_at":"2021-04-09T02:21:47Z","abstract_excerpt":"In this paper, we study important Schr\\\"{o}dinger systems with linear and nonlinear couplings \\begin{equation}\\label{eq:diricichlet} \\begin{cases} -\\Delta u_1-\\lambda_1 u_1=\\mu_1 |u_1|^{p_1-2}u_1+r_1\\beta |u_1|^{r_1-2}u_1|u_2|^{r_2}+\\kappa (x)u_2~\\hbox{in}~\\mathbb{R}^N,\\\\ -\\Delta u_2-\\lambda_2 u_2=\\mu_2 |u_2|^{p_2-2}u_2+r_2\\beta |u_1|^{r_1}|u_2|^{r_2-2}u_2+\\kappa (x)u_1~ \\hbox{in}~\\mathbb{R}^N,\\\\ u_1\\in H^1(\\mathbb{R}^N), u_2\\in H^1(\\mathbb{R}^N),\\nonumber \\end{cases} \\end{equation} with the condition $$\\int_{\\mathbb{R}^N} u_1^2=a_1^2, \\int_{\\mathbb{R}^N} u_2^2=a_2^2,$$ where $N\\geq 2$, $\\mu_1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2104.04158","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2104.04158/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}