A variational problem associated to a hyperbolic Caffarelli--Kohn--Nirenberg inequality

We prove a Caffarelli--Kohn--Nirenberg inequality in the hyperbolic space. For a semilinear elliptic equation involving the associated weighted Laplace--Beltrami operator, we establish variationally the existence of positive radial solutions in the subcritical regime. We also show a non-existence result in star-shaped domains when the exponent is supercritical.