$$L^p_{loc}$$ Positivity Preservation and Liouville-Type Theorems

Abstract

AbstractOn a complete Riemannian manifold (M, g), we consider $$L^{p}_{loc}$$ L loc p distributional solutions of the differential inequality $$-\Delta u + \lambda u \ge 0$$ - Δ u + λ u ≥ 0 with $$\lambda >0$$ λ > 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 $$\Delta u \ge \lambda u$$ Δ u ≥ λ 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.