{"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. This result is new even for the corr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.02712","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}