Picone’s identity on hyperbolic space and its applications

We establish Picone’s identity for p-Laplace operator and biharmonic operator on hyperbolic space. We consider both, the ball model $${\mathbb {B}}^N$$ and the half space model $$\mathbb {H}^N.$$ As an application of Picone’s identity, we show that the principal eigenvalue $$\lambda _1$$ of $$- \Delta _{p, \mathbb {H}}$$ is simple. We also obtain a Hardy type inequality on hyperbolic space.