{"paper":{"title":"Nonlinear scalar field equations with $L^2$ constraint: Mountain pass and symmetric mountain pass approaches","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jun Hirata, Kazunaga Tanaka","submitted_at":"2018-03-14T05:29:28Z","abstract_excerpt":"We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in ${\\mathbb R}^N$ ($N\\geq 2$):\n  $$ (*)_m \\left\\{\n  \\eqalign{\n  -&\\Delta u = g(u) -\\mu u \\quad \\hbox{in}\\ {\\mathbb R}^N, \\cr\n  &\\| u\\|_{L^2({\\mathbb R}^N)} = m, \\cr\n  &u \\in H^1({\\mathbb R}^N), \\cr} \\right.\n  $$ where $g(\\xi)\\in C({\\mathbb R},{\\mathbb R})$, $m>0$ is a given constant and $\\mu\\in {\\mathbb R}$ is a Lagrange multiplier.\n  We introduce a new approach using a Lagrange formulation of the problem $(*)_m$. We develop a new deformation argument under a new version of the Palais-Smal"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.05139","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"}