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

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