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The equation (E2) generalizes a equation considered by Aubin, where he has considered, a compact Riemannian manifold $(M,g)$, the differential equation ($p=2$)  \\Delta u + a(x)u = \\lambda f(u,x), (E1) where $a(x)$ is a $C^{\\infty}$ function defined on $M$ and $f(u,x)$ is a $C^{\\infty}$ function defined on $\\mathbb{R}\\times M$. We show that the equation (E2) has so"},"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":"1611.02909","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-11-09T12:56:00Z","cross_cats_sorted":[],"title_canon_sha256":"f100bb59b0a179e216ca9c92d5284bd2bb430bea50fd66ebebfdb51047b67fcb","abstract_canon_sha256":"3b5d13b85934f6c42253598b2d142c8ce374c529e3346b9930bbd7f46d850eaf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:59:43.201045Z","signature_b64":"H16i+CH/tXFaprNq9aEYk+zxp5LriaWV4Rxpw4aFm5YywRXy35sEwearAlKaUYS9EfxaFwYZKdkwco2ZCP0ZDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5165111161d5b86607b7c07aaa2ba8196f83b278b7af22490505266c4249e92b","last_reissued_at":"2026-05-18T00:59:43.200563Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:59:43.200563Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the study of a class of non-linear differential equations on compact Riemannian Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Carlos Silva, Marcelo Souza, Romildo Pina","submitted_at":"2016-11-09T12:56:00Z","abstract_excerpt":"We study the existence of solutions of the non-linear differential equations on the compact Riemannian manifolds $(M^n,g), n\\geq 2$, \\Delta_p u + a(x)u^{p-1} = \\lambda f(u,x), (E2) where $\\Delta_p$ is the $p-$laplacian, with $1<p<n$. The equation (E2) generalizes a equation considered by Aubin, where he has considered, a compact Riemannian manifold $(M,g)$, the differential equation ($p=2$)  \\Delta u + a(x)u = \\lambda f(u,x), (E1) where $a(x)$ is a $C^{\\infty}$ function defined on $M$ and $f(u,x)$ is a $C^{\\infty}$ function defined on $\\mathbb{R}\\times M$. 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