{"paper":{"title":"Symmetry breaking and Morse index of solutions of nonlinear elliptic problems in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Francesca Gladiali, Massimo Grossi, S\\'ergio Neves","submitted_at":"2013-08-02T14:50:42Z","abstract_excerpt":"In this paper we study the problem\n  -\\Delta u =\\left(\\frac{2+\\alpha}{2}\\right)^2\\abs{x}^{\\alpha}f(\\lambda,u), & \\hbox{in}B_1 \\\\ u > 0, & \\hbox{in}B_1 u = 0, & \\hbox{on} \\partial B_1 where $B_1$ is the unit ball of $\\R^2$, $f$ is a smooth nonlinearity and $\\a$, $\\l$ are real numbers with $\\a>0$. From a careful study of the linearized operator we compute the Morse index of some radial solutions to \\eqref{i0}. Moreover, using the bifurcation theory, we prove the existence of branches of nonradial solutions for suitable values of the positive parameter $\\l$. The case $f(\\lambda,u)=\\l e^u$ provide"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.0519","kind":"arxiv","version":3},"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"}