Lᵖloc positivity preservation and Liouville-type theorems

This is the authors' abstract. We don't add key points for this paper.

On a complete Riemannian manifold $(M,g)$, we consider Lᵖloc distributional solutions of the the differential inequality -Δu + λu ≥ 0 with $λ>0$ a locally bounded function that may decay to $0$ at infinity. Under suitable growth conditions on the Lᵖ norm of u over geodesic balls, we obtain that any such solution must be nonnegative. This is a kind of generalized Lᵖ-preservation property that can be read as a Liouville type property for nonnegative subsolutiuons of the equation Δu ≥ λu. An application of the analytic results to Lᵖ growth estimates of the extrinsic distance of complete minimal submanifolds is also given.