Uniqueness of the ground state of the NLS on $\mathbb{H}^d$ via analytical and topological methods

Abstract

We give an analytical and topological proof of the uniqueness of the ground state of the nonlinear Schrödinger equation defined on the Hyperbolic space ${ℍ}^d$ when the power type nonlinearity has $H^1\left({ℍ}^d\right)$-subcritical exponent ($1<p<1+4∕\left(d-2\right)$ for $d\ge 3$ and $1<p<+\infty$ for $d=2$) and the phase $\lambda$ is positive. Differently from what it is available in the literature, we use the polar model of ${ℍ}^d$ and we do not take advantage of the dual Euclidean problem. Our proof of uniqueness uses the shooting method, some new monotonicity formulas and the geometry of the potential energy.