Nonlinear bound states on weakly homogeneous spaces

TL;DR

This paper establishes the existence and properties of ground state solutions for nonlinear Schrödinger equations on weakly homogeneous Riemannian manifolds, highlighting conditions for their existence, uniqueness, and decay.

Contribution

It introduces a framework for finding ground states on weakly homogeneous spaces and analyzes their smoothness, positivity, and decay, also identifying manifolds lacking such solutions.

Findings

Existence of ground state solutions on weakly homogeneous spaces

Conditions under which solutions are unique or non-existent

Decay and positivity properties of the solutions

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 Schr\"odinger equations. We examine two constrained minimization problems, which give rise to such solutions. One yields what we call $F_\lambda$-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.