Uniqueness and Flat Core of Positive Solutions for Quasilinear Elliptic Eigenvalue Problems in General Smooth Domains

The structure of positive solutions to the quasilinear elliptic problems –div(|Du|^p–2Du = λf(u) in Ω, u = 0 on ∂Ω, p > 1, Ω ⊂ R^Na bounded smooth domain, is precisely studied when λ is sufficiently large, for a class of logistic‐type nonlinearities f(u) satisfying that f(0) = f(a) = 0, a > 0, f(u) > 0 for u ∈ (0,a), , while u = a is a zero point of f with order ω. It is shown that if ω ≥ p – 1, the problem has a unique positive solution u_λ with sup _Ω u_λ < a, which develops a boundary layer near ∂Ω. It is shown that if 0 < ω < p – 1, the problem also has a unique positive solution u _λ, but the flat core {x ∈ Ω : u_λ(x) = a} ≠ ∅︁ exists. Moreover, the asymptotic behaviour of the flat core is studied as λ → ∞.