{"paper":{"title":"Multiplicity of solutions to a class of degenerate elliptic equations in both sub-critical and critical cases","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kaushik Bal, Sanjit Biswas","submitted_at":"2024-12-06T06:00:35Z","abstract_excerpt":"Given a smooth, bounded domain $\\Omega\\subset\\mathbb{R}^N$, we establish the existence of two non-trivial, non-negative solutions to the semilinear degenerate elliptic equation \\begin{align*}\n  \\left. \\begin{array}{l}\n  -\\Delta_\\lambda u=\\mu g(z)|u|^{r-1}u+h(z)|u|^{s-1}u \\;\\text{in}\\; \\Omega\n  u\\in H^{1,\\lambda}_0(\\Omega)\n  \\end{array}\\right\\}\n  \\end{align*} where $\\Delta_\\lambda=\\Delta_x+|x|^{2\\lambda}\\Delta_y$ denotes the Grushin Laplacian Operator, $z=(x,y)\\in\\Omega$, $N=n+m;\\, n,\\, m\\geq 1$, $\\lambda>0$, $0\\leq r<1<s<2^*_\\lambda-1$ and $\\mu$ is a positive parameter. The functions $g$ and $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.04794","kind":"arxiv","version":2},"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/2412.04794/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"}