{"paper":{"title":"Real solutions to the nonlinear Helmholtz equation with local nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gilles Evequoz, Tobias Weth","submitted_at":"2013-02-03T20:20:11Z","abstract_excerpt":"In this paper, we study real solutions of the nonlinear Helmholtz equation $$ - \\Delta u - k^2 u = f(x,u),\\qquad x\\in \\R^N $$ satisfying the asymptotic conditions $$ u(x)=O(|x|^{\\frac{1-N}{2}}) \\quad \\text{and} \\quad \\frac{\\partial^2 u}{\\partial r^2}(x)+k^2 u(x)) =o(|x|^{\\frac{1-N}{2}}) \\qquad \\text{as $r=|x| \\to \\infty$.} $$ We develop the variational framework to prove the existence of nontrivial solutions for compactly supported nonlinearities without any symmetry assumptions. In addition, we consider the radial case in which, for a larger class of nonlinearities, infinitely many solutions "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.0530","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"}