  In this paper, we explore the geometric properties of unbounded extremal
domains for the $p$-Laplacian operator in both Euclidean and hyperbolic spaces.
Assuming that the nonlinearity grows at least as the nonlinearity of the
eigenvalue problem, we prove that these domains exhibit remarkable geometric
properties and cannot be arbitrarily wide. In two dimensions, we prove that
such domains with connected complements must necessarily be balls. In the
hyperbolic space, we highlight the constraints on extremal domains and the
geometry of their asymptotic boundaries.
