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We prove the existence results in two cases: First, the nonresonance case, where $(\\alpha,\\beta)$ is not an eleme"},"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":"1607.07584","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-07-26T08:28:23Z","cross_cats_sorted":[],"title_canon_sha256":"78e77a425135143354d2517605114fb4370ece9589bd5e3cd4f005d85e8e9809","abstract_canon_sha256":"8524d43418be2254411640b2cc4fe420a27595c4b91a6bc231311d5f24822f29"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:28.227415Z","signature_b64":"J2BXnbeTsyyZ7Wk1dnuFufsBr0zFi0IcTC0cW0b2XmnpZL7TEMMBQpB9ZsyifHzolSil00uTZQc6Y6GpA66OAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ce8775c1a4bb358663a98998e461a73f79dc0353fc27e71cda9c27cb52de2610","last_reissued_at":"2026-05-18T01:10:28.226898Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:28.226898Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the solvability of resonance problems for nonlocal elliptic equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Sarika Goyal","submitted_at":"2016-07-26T08:28:23Z","abstract_excerpt":"In this article, we consider the following problem: $$ \\quad \\left\\{ \\begin{array}{lr} \\quad (-\\Delta)^s u = \\alpha u^+ -\\beta u^{-} + f(u) + h \\; \\text{in}\\;\\Omega \\quad \\quad \\quad \\quad u =0 \\; \\text{on}\\; \\mathbb{R}^n\\setminus \\Omega, \\end{array} \\right. $$ where $\\Omega\\subset \\mathbb{R}^n$ is a bounded domain with Lipschitz boundary, $n> 2s$, $0<s<1$, $(\\alpha, \\beta) \\in \\mathbb{R}^2$, $f: \\mathbb{R}\\to \\mathbb{R}$ is a bounded and continuous function and $h\\in L^2(\\Omega)$. 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