Antimaximum principle for elliptic problems with weight

This paper is concerned with the antimaximum principle for the linear problem with weight $-Delta u = lambda m(x) u +h(x)$, under Dirichlet or Neumann boundary conditions.We investigate the following three questions: Toys Where exactly can this principle hold? If it holds, does Ironing Board it hold uniformly or not? If it holds uniformly, what is the exact interval of uniformity? We will in particular obtain a variational characterization of this interval of uniformity.