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The anti-maximum principle for the homogeneous Dirichlet problem to − Δ p u = λ | u | p − 2 u + f (x) with positive f ∈ L ∞ (Ω) states the existence of a critical value λ f > λ 1 such that any solution of this problem with λ ∈ (λ 1 , 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 in (λ 1 , λ 2). [ABSTRACT FROM AUTHOR]