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

The anti-maximum principle for the homogeneous Dirichlet problem to $-{\mathrm{\Delta }}_{p}u=\lambda \mathrm{\mid }u{\mathrm{\mid }}^{p-2}u+f(x)$ with positive $f \in L^\infty(\Omega)$statestheexistenceofacriticalvalue$\lambda_f > \lambda_1$suchthatanysolutionofthisproblemwith$\lambda \in (\lambda_1, \lambda_f)$ 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 $({\lambda }_1,{\lambda }_2)$.