{"paper":{"title":"Quasilinear elliptic equations and weighted Sobolev-Poincar\\'{e} inequalities with distributional weights","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Benjamin J. Jaye, Igor E. Verbitsky, Vladimir G. Maz'ya","submitted_at":"2012-04-13T18:03:55Z","abstract_excerpt":"We introduce a class of weak solutions to the quasilinear equation $-\\Delta_p u = \\sigma |u|^{p-2}u$ in an open set $\\Omega\\subset\\mathbf{R}^n$. Here $p>1$, and $\\Delta_p u$ is the $p$-Laplacian operator. Our notion of solution is tailored to general distributional coefficients $\\sigma$ satisfying a certain weighted Sobolev-Poincare inequality. We also study weak solutions of the closely related equation $-\\Delta_p v = (p-1)|\\nabla v|^p + \\sigma$, under the same conditions on $\\sigma$. Our results for this latter equation will allow us to characterize the class of distributions $\\sigma$ which "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.3063","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":""},"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"}