{"paper":{"title":"On $\\big(p_{1}(x), p_{2}(x)\\big)$-Laplace Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Duchao Liu, Jinghua Yao, Xiaoyan Wang","submitted_at":"2012-05-09T01:22:20Z","abstract_excerpt":"In this paper, we investigate the $(p_{1}(x), p_{2}(x))$-Laplace operator, the properties of the corresponding integral functional and weak solutions to the related differential equations. We show that the integral functional admits a derivative of type $(S_+)$ which induces a homeomorphism between duality space pairs. As applications of the above results, we gave some existence results of the $(p_{1}(x), p_{2}(x))$-Laplace equation\n-\\text{div}(|\\nabla u|^{p_{1}(x)-2}\\nabla u)-\\text{div}(|\\nabla u|^{p_{2}(x)-2}\\nabla u)=f(x,u)\nin a bounded smooth domain $\\Omega\\subset\\mathbb{R}^{N}$ with Diric"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.1854","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"}