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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1804.09612","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-04-25T15:03:36Z","cross_cats_sorted":[],"title_canon_sha256":"5b199ae70ea59e89194b21b6369fb9ced8c6d486141871f31381b3d849874063","abstract_canon_sha256":"b48b48d3fcaa96946d85d35c860689fb39c8df88afa6b64ed3a149067c605ae5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:17.351479Z","signature_b64":"HtUnt2LyW9QUr2+B2kP88MoeRiY6+FdyIEGgDNvekTsc7V0RhbVlmXvF117vdd/rRvYjFa9KdyA/xjL6hqRVBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"361dd774cf0aa3d9f6a6e2794e9e59ffdacd6cd5fee857f0ee213add55b2afbc","last_reissued_at":"2026-05-18T00:12:17.350658Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:17.350658Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\Delta_p u={\\rm div}\\, (|\\nabla u|^{p-2}\\nabla u)$. 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