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

The anti-maximum principle for the homogeneous Dirichlet problem to $-Δ_p u = λ|u|^{p-2}u + f(x)$ with positive $f \in L^\infty(Ω)$ states the existence of a critical value $λ_f > λ_1$ such that any solution of this problem with $λ\in (λ_1, λ_f)$ 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 $(λ_1,λ_2)$.