Existence and uniqueness of bound states with prescribed number of nodes for nonlinear dispersive equations on spherically symmetric manifolds

Abstract

Standing waves $e^{iωt}u(x)$ of nonlinear dispersive equations, such as the nonlinear Schrödinger and Klein--Gordon equations, have a profile $u$ that solves a nonlinear elliptic ODE of the form $Δu+f(u)=0$. We prove the existence of radial bound states with a prescribed number of nodal surfaces for this equation, on a large class of spherically symmetric manifolds and nonlinearities $f$. Under mild assumptions, we prove uniqueness of such bound states for the model nonlinearity $f(u)=|u|^{p-1}u-u$. Both existence and uniqueness results apply to some manifolds with a singularity at the origin.