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More precisely, we prove the existence of a saddle-shaped solution in every even dimension $2m$, as well as its monotonicity properties, asymptotic behaviour, and instability in dimensions $2m=4$ and $2m=6$. These results are relevant in connection with the analog for fractional equations of a conjecture"},"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":"1107.2306","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-07-12T14:46:27Z","cross_cats_sorted":[],"title_canon_sha256":"2ed7dda0e12fe848e587d947e7826e0b80667953100e625be175be64f7d1d6f1","abstract_canon_sha256":"b78d659f96b9d0b1fcde2c11eed408e02f9a1678c6d7844faeb20f3b36de9a96"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:24.826133Z","signature_b64":"vkwmlsYdP4MRcyIUukBmsHNR9Cr+IBnUMVUkWHvJ0TpoMXYRtMp6C+NRdvGMayoxaTgXTS2Knl4L93NB2PArCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"51e903f4e1b81f729c7c3bf34d8c3e2763c5a327aef356c4f214214c724c7978","last_reissued_at":"2026-05-18T04:18:24.825530Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:24.825530Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Saddle-shaped solutions of bistable elliptic equations involving the half-Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Eleonora Cinti","submitted_at":"2011-07-12T14:46:27Z","abstract_excerpt":"We establish existence and qualitative properties of saddle-shaped solutions of the elliptic fractional equation $(-\\Delta)^{1/2}u=f(u)$ in all the space $\\re^{2m}$, where $f$ is of bistable type. 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