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 –2 Du = λf ( u ) in Ω, u = 0 on ∂Ω, p > 1, Ω ⊂ R N a 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 ), $ \underline {\lim} _{u \to 0} + {{f(u)} \over {u^{p-1}}} = \alpha^*, \alpha^* > 0 $ , 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 → ∞.