{"paper":{"title":"Dual variational methods and nonvanishing for the nonlinear Helmholtz equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gilles Evequoz, Tobias Weth","submitted_at":"2014-02-12T22:58:21Z","abstract_excerpt":"We set up a dual variational framework to detect real standing wave solutions of the nonlinear Helmholtz equation $$ -\\Delta u-k^2 u =Q(x)|u|^{p-2}u,\\qquad u \\in W^{2,p}(\\mathbb{R}^N) $$ with $N\\geq 3$, $\\frac{2(N+1)}{(N-1)}< p<\\frac{2N}{N-2}$ and nonnegative $Q \\in L^\\infty(\\mathbb{R}^N)$. We prove the existence of nontrivial solutions for periodic $Q$ as well as in the case where $Q(x)\\to 0$ as $|x|\\to\\infty$. In the periodic case, a key ingredient of the approach is a new nonvanishing theorem related to an associated integral equation. The solutions we study are superpositions of outgoing a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.3003","kind":"arxiv","version":2},"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"}