The study of nonlinear eigenvalue problems for quasilinear operators on unbounded domains involving the p-Laplacian is motivated by various applications. We refer only to fluid mechanics, to mathematical models of the torsional creep, and to non-linear field equations arising in quantum mechanics. For instance, in fluid mechanics, the shear stress and the velocity gradient ∇pu of certain fluids obey a relation of the form (x) = a(x)∇pu(x), where ∇pu ≔∣u∣p−2∇u and p > 1 is an arbitrary real number. The case p = 2 (respectively, p < 2, p > 2) corresponds to a Newtonian (respectively, pseudo-plastic, dilatant) fluid. Then the resulting equations of motion involve div (a∇pu), which reduces to a∆pu = a div (∆pu), provided that a is a constant.

The p-Laplace operator also appears in the study of flows through porous media (p = 3/2, [209]) or glacial sliding (p ∈ (1, 4 /3], [176]). We refer to Aronsson and Janfalk [13] for the mathematical treatment of the Hele-Shaw flow of “power-law fluids.” The concept of Hele-Shaw flow corresponds to a flow between two closely spaced parallel plates, where the gap between the plates is small compared with the dimension of the plates.