{"paper":{"title":"Multiple standing waves for the nonlinear Helmholtz equation concentrating in the high frequency limit","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gilles Ev\\'equoz","submitted_at":"2016-08-16T09:59:27Z","abstract_excerpt":"This paper studies for large frequency number $k>0$ the existence and multiplicity of solutions of the semilinear problem $$ -\\Delta u -k^2 u=Q(x)|u|^{p-2}u\\quad\\text{ in }\\mathbb{R}^N, \\quad N\\geq 2. $$ The exponent $p$ is subcritical and the coefficient $Q$ is continuous, nonnegative and satisfies the condition $$ \\limsup_{|x|\\to\\infty}Q(x)<\\sup_{x\\in\\mathbb{R}^N}Q(x). $$ In the limit $k\\to\\infty$, sequences of solutions associated to ground states of a dual equation are shown to concentrate, after rescaling, at global maximum points of the function $Q$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.04534","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"}