  We first give a sufficient condition, issued from pluripotential theory, for
an unbounded domain in the complex Euclidean space $\mathbb C^n$ to be
Kobayashi hyperbolic. Then, we construct an example of a rigid pseudoconvex
domain in $\mathbb C^3$ that is Kobayashi hyperbolic and has a nonempty core.
In particular, this domain is not biholomorphic to a bounded domain in $\mathbb
C^3$ and the mentioned above sufficient condition for Kobayashi hyperbolicity
is not necessary.
