$L^p_{loc}$ positivity preservation and Liouville-type theorems

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