{"paper":{"title":"Nonlinear equations with gradient natural growth and distributional data, with applications to a Schr\\\"odinger type equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Karthik Adimurthi, Nguyen Cong Phuc","submitted_at":"2018-04-25T15:03:36Z","abstract_excerpt":"We obtain necessary and sufficient conditions with sharp constants on the distribution $\\sigma$ for the existence of a globally finite energy solution to the quasilinear equation with a gradient source term of natural growth of the form $-\\Delta_p u = |\\nabla u|^p + \\sigma$ in a bounded open set $\\Omega\\subset \\mathbb{R}^n$. Here $\\Delta_p$, $p>1$, is the standard $p$-Laplacian operator defined by $\\Delta_p u={\\rm div}\\, (|\\nabla u|^{p-2}\\nabla u)$. The class of solutions that we are interested in consists of functions $u\\in W^{1,p}_0(\\Omega)$ such that $e^{{\\mu} u}\\in W^{1,p}_0(\\Omega)$ for s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.09612","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"}