{"paper":{"title":"Liouville type theorems for stable solutions of elliptic system involving the Grushin operator","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Foued Mtiri","submitted_at":"2020-12-20T21:05:31Z","abstract_excerpt":"We examine the degenerate elliptic system\n  $$-\\Delta_{s} u = v^p, \\quad -\\Delta_{s} v= u^\\theta, \\quad u,v>0 \\quad\\mbox{in }\\; \\mathbb{R}^N=\\mathbb{R}^{N_1}\\times \\mathbb{R}^{N_2}, \\quad\\mbox{where }\\;\\;\\;\\; s \\geq 0\\;\\; \\mbox{and} \\;\\;p,\\theta >0.$$ We prove that the system has no smooth stable solution provided $p,\\theta >0$ and $N_s< 2 + \\alpha + \\beta,$ where\n  $$\\alpha = \\frac{2(p+1)}{p\\theta - 1} \\quad\\mbox{and} \\quad \\beta = \\frac{2(\\theta +1)}{p\\theta - 1}.$$\n  This result is an extension of some result in \\cite{ MY}. In particular, we establish a new the integral estimate for $u$ and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2012.11023","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/2012.11023/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"}