Nonlinear bound states on weakly homogeneous spaces

2012/03/16 by Hans Christianson, Jeremy Marzuola, Christianson, Hans +7
Mathematics · #35J61 #58J05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP #msc:35J61 #msc:58J05

paper · pdf · doi:10.48550/arxiv.1203.3612

49 pages

arxiv created 2012/03/16 · openalex publication_date 2012/03/16 · arxiv updated 2012/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove the existence of ground state solutions for a class of nonlinear elliptic equations, arising in the production of standing wave solutions to an associated family of nonlinear Schrödinger equations. We examine two constrained minimization problems, which give rise to such solutions. One yields what we call F_λ-minimizers, the other energy minimizers. We produce such ground state solutions on a class of Riemannian manifolds called weakly homogeneous spaces, and establish smoothness, positivity, and decay properties. We also identify classes of Riemannian manifolds with no such minimizers, and classes for which essential uniqueness of positive solutions to the associated elliptic PDE fails.

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