\(L^p_{loc}\) Positivity Preservation and Liouville-Type Theorems

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