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

Geometric properties of extremal domains for the $p$-Laplacian operator