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