  The anti-maximum principle for the homogeneous Dirichlet problem to
$-\Delta_p u = \lambda |u|^{p-2}u + f(x)$ with positive $f \in
L^\infty(\Omega)$ states the existence of a critical value $\lambda_f >
\lambda_1$ such that any solution of this problem with $\lambda \in (\lambda_1,
\lambda_f)$ is strictly negative. In this paper, we give a variational upper
bound for $\lambda_f$ and study its properties. As an important supplementary
result, we investigate the branch of ground state solutions of the considered
boundary value problem in $(\lambda_1,\lambda_2)$.
