Lᵖloc Positivity Preservation and Liouville-Type Theorems

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