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

Abstract

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 $({\lambda }_1,{\lambda }_2)$.

17 pages, 1 figure. A reference added, minor typos corrected, minor textual changes incorporated according to referee's suggestions. Published in Journal of Differential Equations