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We show that, as $\\alpha \\to \\infty$, the Morse index of nontrivial radial solutions of this problem (positive or sign-changing) tends to infinity. This result is new even for the corr"},"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":"1803.02712","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-03-07T15:21:59Z","cross_cats_sorted":[],"title_canon_sha256":"7dae669dc5f580204e4979b7b7c4bd8f98132e56ce25ae683770da09bc611eb0","abstract_canon_sha256":"17dc0d2530d30bb673b2cde604380c9ba00f17023843a5a2ad53949e0982c2b7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:21:49.442141Z","signature_b64":"jYcEHdrp1s0NLD4ZVMoRKiBC1wiNTlqwEwqvfezZCZqp1foiia/hRgn6HzKPkOcoj2MEs/DdCWH0gaJomKSxBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1244803b94bf965c75b974ab4a2e6e45fca0a7ebc54bb9ea6afa63a3025c5679","last_reissued_at":"2026-05-18T00:21:49.441510Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:21:49.441510Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Symmetry breaking via Morse index for equations and systems of H\\'enon-Schr\\\"odinger type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Tobias Weth, Zhenluo Lou, Zhitao Zhang","submitted_at":"2018-03-07T15:21:59Z","abstract_excerpt":"We consider the Dirichlet problem for the Schr\\\"odinger-H\\'enon system $$ -\\Delta u + \\mu_1 u = |x|^{\\alpha}\\partial_u F(u,v),\\quad \\qquad\n  -\\Delta v + \\mu_2 v = |x|^{\\alpha}\\partial_v F(u,v) $$ in the unit ball $\\Omega \\subset \\mathbb{R}^N, N\\geq 2$, where $\\alpha>-1$ is a parameter and $F: \\mathbb{R}^2 \\to \\mathbb{R}$ is a $p$-homogeneous $C^2$-function for some $p>2$ with $F(u,v)>0$ for $(u,v) \\not = (0,0)$. We show that, as $\\alpha \\to \\infty$, the Morse index of nontrivial radial solutions of this problem (positive or sign-changing) tends to infinity. 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