Uniform Anti-maximum Principles

Consider a second or higher order elliptic partial differential equation u=λu+f on an open bounded domain Ω of n with homogeneous boundary conditions u=0. If there exists a simple eigenvalue for which the corresponding eigenfunction is positive and satisfies appropriate boundary estimates, then an anti-maximum principle holds. For positive f Lp(Ω) with p large enough there exists δf>0 such that for λ (λ1, λ1+δf) the solution is negative and for λ (λ1−δf, λ1) the solution is positive. We give conditions such that this sign reversing property is uniform: there is δ>0 such that for all positive f the solution u is negative for λ (λ1, λ1+δ) and positive for λ (λ1−δ, λ1). Two classes of higher order boundary value problems that satisfy these conditions will be given.