{"paper":{"title":"On positive solutions for $(p,q)$-Laplace equations with two parameters","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mieko Tanaka, Vladimir Bobkov","submitted_at":"2014-11-19T12:00:01Z","abstract_excerpt":"We study the existence and non-existence of positive solutions for the $(p,q)$-Laplace equation $-\\Delta_p u -\\Delta_q u = \\alpha |u|^{p-2} u + \\beta |u|^{q-2} u$, where $p \\neq q$, under the zero Dirichlet boundary condition in $\\Omega$. The main result of our research is the construction of a continuous curve in $(\\alpha,\\beta)$ plane, which becomes a threshold between the existence and non-existence of positive solutions. Furthermore, we provide the example of domains $\\Omega$ for which the corresponding first Dirichlet eigenvalue of $-\\Delta_p$ is not monotone w.r.t. $p > 1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.5192","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"}