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We show the existence of at least one solution for above equations for $\\lambda=0$. For $\\lambda>0$ small enough"},"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":"1907.03200","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-07-06T22:33:55Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"729ae27e78b195415cb7ed53781412cf61aed15540f744d1d1caa35d2158d0f9","abstract_canon_sha256":"05ff2f63075349da4463792e3fe92d4a32a821d0982bdcebb99abab01a9e1ae7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:35.196644Z","signature_b64":"sp2Hs2JjOgIKIJJf7nZMo/SECoHtStz096hR6ZcUe+h1v11kwoMMC/bSwiTX+Fat2Vt/57X4jMLoCYf4H0ZhBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"72e4892cb8aae3fbdd6fa8e7d0691c04f14acaf6b1c3d9ddf57386fe16b77895","last_reissued_at":"2026-05-17T23:39:35.195854Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:35.195854Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Solutions for fourth-order Kirchhoff type elliptic equations involving concave-convex nonlinearities in $\\mathbb{R}^{N}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.DS","authors_text":"Dong-Lun Wu, Fengying Li","submitted_at":"2019-07-06T22:33:55Z","abstract_excerpt":"In this paper, we show the existence and multiplicity of solutions for the following fourth-order Kirchhoff type elliptic equations \\begin{eqnarray*} \\Delta^{2}u-M(\\|\\nabla u\\|_{2}^{2})\\Delta u+V(x)u=f(x,u),\\ \\ \\ \\ \\ x\\in \\mathbb{R}^{N}, \\end{eqnarray*} where $M(t):\\mathbb{R}\\rightarrow\\mathbb{R}$ is the Kirchhoff function, $f(x,u)=\\lambda k(x,u)+ h(x,u)$, $\\lambda\\geq0$, $k(x,u)$ is of sublinear growth and $h(x,u)$ satisfies some general 3-superlinear growth conditions at infinity. We show the existence of at least one solution for above equations for $\\lambda=0$. 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