Anti-maximum principles for indefinite-weight elliptic problems

This paper is concerned with anti-maximum principles (AMPs) for indefinite-weight elliptic problems. We consider the equation (P-lambdaW)u(lambda) = f not greater than or equal to 0, where P is a second order, linear, subcritical, elliptic operator defined on a noncompact, Riemannian manifold Omega, and W is an indefinite-weight function which is a, 'small' perturbation of the operator P in Omega. There exists a generalized positive principal eigenvalue lambda (+) such that the AMP holds above lambda (+). More precisely, our AMP reads roughly that if the functions f and u(lambda) do not grow too fast, then there exists is an element of < 0, which may depend on f, such that u(lambda) < 0, for all lambda is an element of (lambda (+), lambda (+) + is an element of). AMPs are proved also when P is critical even in the singular case.