Estimates on the spectral interval of validity of the anti-maximum principle

Abstract

The anti-maximum principle for the homogeneous Dirichlet problem to −∆p = λ|u| p−2 u + f (x) with positive f ∈ L ∞ (Ω) states the existence of a critical value λ f > λ1 such that any solution of this problem with λ ∈ (λ1, λ f ) is strictly negative. In this paper, we give a variational upper bound for λ f and study its properties. As an important supplementary result, we investigate the branch of ground state solutions of the considered boundary value problem on (λ1, λ2).