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

The anti-maximum principle for the homogeneous Dirichlet problem to -Delta(p)u =lambda|u|(p-2)u + f(x) with positive f is an element of L-infinity (Omega) states the existence of a critical value lambda(f) > lambda(1) such that any solution of this problem with lambda is an element of(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)). (c) 2020 Elsevier Inc. All rights reserved.