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

Abstract

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 ⊂ RNa 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 uλ with sup Ω uλ < 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 : uλ(x) = a} ≠ ∅︁ exists. Moreover, the asymptotic behaviour of the flat core is studied as λ → ∞.