Llocp Positivity Preservation and Liouville-Type Theorems

On a complete Riemannian manifold (M, g), we consider Llocp distributional solutions of 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.