{"paper":{"title":"Normalized solutions to the mixed dispersion nonlinear Schr\\\"odinger equation in the mass critical and supercritical regime","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Denis Bonheure, Jean-Baptiste Casteras, Louis Jeanjean, Tianxiang Gou","submitted_at":"2018-02-26T09:39:06Z","abstract_excerpt":"In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schr\\\"odinger equation $$ \\gamma \\Delta ^2 u -\\Delta u + \\alpha u=|u|^{2 \\sigma} u, \\quad u \\in H^2(\\R^N), $$ under the constraint $$ \\int_{\\R^N}|u|^2 \\, dx =c>0. $$ We assume $\\gamma >0, N \\geq 1, 4 \\leq \\sigma N < \\frac{4N}{(N-4)^+}$, whereas the parameter $\\alpha \\in \\R$ will appear as a Lagrange multiplier. Given $c \\in \\R^+$, we consider several questions including the existence of ground states, of positive solutions and the multiplicity of radial solutions. We also discuss the stability of the standing "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.09217","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"}