A Note on Liouville Type Theorem of Elliptic Inequality Δu + u σ ⩽ 0 on Riemannian Manifolds

Hui-Chun Zhang

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Alarcón, S., Garciá-Melián, J., Quaas, A.: Nonexistence of positive supersolutions to some nonlinear elliptic porblems. J. Math. Pures Appl. 99, 618–634 (2013)

Biduat-Veron, M.-F., Pohozaev, S.: Nonexistence results and estimates for some nonlinear elliptic problems. J. Anal. Math. 84, 1–49 (2001)

Caffarelli, L., Garofalo, N., Segala, F.: A gradient bound for entire solutions of quasi-linear equatios and its consequences. Comm. Pure Appl. Math. 47, 1457–1473 (1994)

Caristi, G., Mitidieri, E., Pohozaev, S.I.: Some Liouville theorems for quasilinear elliptic inequalities. Dokl. Math. 79(1), 118–124 (2009)

Cheng, S.Y., Yau, S.T.: Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math. 28(3), 333–354 (1975)

Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math. 34(4), 525–598 (1981)

Grigor ′yan, A.: On the existence of a Green function on a manifold (in Russian). Uspekhi Math. Nauk 38(1), 161–162 (1983). Engl. transl.: Russian Math. Surveys 38(1) (1983) 190–191

Grigor′yan, A.: On the existence of positive fundamental solution of the Laplace equation on Riemannian manifolds (in Russian). Maat. Sb. 128(3), 354–363 (1985). Engl. transl.: Math. USSR Sb. 56 (1987) 349–358

Grigor′yan, A., Kondratiev, V.A.: On the existence of positive solutions of semilinear elliptic inequalities on Riemannian manifolds Around the research of Vladimir Mazya II, International Mathematical Series (New York), vol. 12, pp 203–218. Springer, New York (2010)

Grigor′yan, A., Sun, Y.: On Nonnegative Solutions of the Inequality Δu + u σ ⩽ 0 on Riemnannian Manifolds. Comm. Pure Appl. Math. 67(8), 1336–1352 (2014)

Pohozaev, S.I.: Critical nonlinearities in partial differential equations. Milan J. Math. 77, 127–150 (2009)

Serrin, J.: Entire solutions of nonlinear Poisson equations. Proc. London Math. Soc. 24(3), 348–366 (1972)

Serrin, J., Zou, H.: Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta Math. 189, 79–142 (2002)

Varopoulos, N.: The Poisson kernel on positively curved manifolds. J. Funct. Anal. 44, 359–380 (1981)