On the instability of positive solution of an elliptic equation
G. A. Afrouzi, Sayyed Hashem Rasouli · International Mathematical Forum · 2007
We study the stability of positive stationary solutions of { −Δu(x) = λf(x, u), x ∈ Ω, Bu = 0, x ∈ ∂Ω, where Ω is a bounded domain 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 fuu(x, u)> 0 for all fixed x ∈ Ω (u ∈ R+), fx(0, 0) = 0, f(x, u) 0 for u> β for some β> 0 (for all fixed x ∈ Ω). We provide a simple proof to establish that every positive stationary solution is linearly unstable.