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The strong instability of standing waves with positive energy was proven by Ohta and Yamaguchi (2015). In this paper, we improve the previous result, that is, we prove that if $\\partial_\\lambda^2S_\\omega(\\phi_\\omega^\\lambda)|_{\\lambda=1}\\le0$, the standing wave is strongly unstable, where $S_\\omega$ is the action, and $\\phi_\\omega^\\lambda(x)\\mathrel{\\math"},"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":"1806.01639","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-06-05T12:35:44Z","cross_cats_sorted":[],"title_canon_sha256":"91b874f9a63c9a39d06738c71bb0554dd62c8cd7c31a702ac8a05de877e4bc41","abstract_canon_sha256":"1eb05139957289416ac3eb552672b5c09e2735356d9f3f0914d759e085805b77"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:12.185792Z","signature_b64":"4psekC1TUc7O2kHfvFqsdKEhQ1KJRi4tNiXSadasWmRP4PAHUQ6pH18PPRUsGicz/3WxLl7WuxtA1Mv9ndnoDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bdd672569addcae40a3374e06ba306ec2d2af9a91fa3592368102b65375a19bd","last_reissued_at":"2026-05-18T00:14:12.185361Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:12.185361Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Strong instability of standing waves with negative energy for double power nonlinear Schr\\\"odinger equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Masahito Ohta, Noriyoshi Fukaya","submitted_at":"2018-06-05T12:35:44Z","abstract_excerpt":"We study the strong instability of ground-state standing waves $e^{i\\omega t}\\phi_\\omega(x)$ for $N$-dimensional nonlinear Schr\\\"odinger equations with double power nonlinearity. 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