Instability of nonnegative solutions for a class of semilinear elliptic boundary value problems

We consider the boundary value problem −Δu(x) = λf(u(x)), x ∈ Ω, Bu(x) = 0, x ∈ ∂Ω, where Ω is a bounded region in Rn with smooth boundary Bu(x) = αh(x)u + (1 − α)∂u∂n where α ∈ [0,1], h: ∂Ω → R+ with h = 1 when α = 1, λ > 0, f is a smooth function such that ƒ″(u) > 0 for u > 0, ƒ(u) < 0 for u ∈ (0,β) and ƒ(u) > 0 for u > β for some β > 0. We provide a simple proof to establish that every non-trivial nonnegative solution is unstable.