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S\\'anchez","submitted_at":"2011-06-29T23:29:40Z","abstract_excerpt":"We study the effect of lower order perturbations in the existence of positive solutions to the following critical elliptic problem involving the fractional Laplacian: \n(-\\Delta)^{\\alpha/2}u=\\lambda u^q+u^{\\frac{N+\\alpha}{N-\\alpha}}, \\quad u>0 &\\quad in \\Omega, \nu=0&\\quad on \\partial\\Omega,$$\nwhere $\\Omega\\subset\\mathbb{R}^N$ is a smooth bounded domain, $N\\ge1$, $\\lambda>0$, $0<q<\\frac{N+\\alpha}{N-\\alpha}$, $0<\\alpha<\\min\\{N,2\\}$. For suitable conditions on $\\alpha$ depending on $q$, we prove: In the case $q<1$, there exist at least two solutions for every $0<\\lambda<\\Lambda$ and some $\\Lambda>"},"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":"1106.6081","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-06-29T23:29:40Z","cross_cats_sorted":[],"title_canon_sha256":"282a98209cb0698f071e1fb2e76959447aaf88fd0291f6832cad5e20ebbec445","abstract_canon_sha256":"7b94d84c94a2d806db019bc3eedd11ada84f470e01ed90dd92295f103345b06b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:17:14.702180Z","signature_b64":"l4T9jCViHuBU5rD5nV0rIWCd9s3LxdtOc2WVMfrdX3wLEvBv/Ck5vVdQYVHNUX/0g0CEEzwMysrmp13Q+UqtAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"944fbacc28e0f4175dcb49af6bd690e7d62a8f9f107c3cef50dfaf67d8346dd0","last_reissued_at":"2026-05-18T04:17:14.701527Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:17:14.701527Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On some Critical Problems for the Fractional Laplacian Operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"A. de Pablo, B. Barrios, E. Colorado, U. S\\'anchez","submitted_at":"2011-06-29T23:29:40Z","abstract_excerpt":"We study the effect of lower order perturbations in the existence of positive solutions to the following critical elliptic problem involving the fractional Laplacian: \n(-\\Delta)^{\\alpha/2}u=\\lambda u^q+u^{\\frac{N+\\alpha}{N-\\alpha}}, \\quad u>0 &\\quad in \\Omega, \nu=0&\\quad on \\partial\\Omega,$$\nwhere $\\Omega\\subset\\mathbb{R}^N$ is a smooth bounded domain, $N\\ge1$, $\\lambda>0$, $0<q<\\frac{N+\\alpha}{N-\\alpha}$, $0<\\alpha<\\min\\{N,2\\}$. 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