# American Institute of Mathematical Sciences

September  2015, 14(5): 1743-1757. doi: 10.3934/cpaa.2015.14.1743

## On the uniqueness of nonnegative solutions of differential inequalities with gradient terms on Riemannian manifolds

 1 School of Mathematical Sciences and LPMC, Nankai University, 300071, Tianjin , China

Received  June 2014 Revised  April 2015 Published  June 2015

We investigate the uniqueness of nonnegative solutions to the following differential inequality \begin{eqnarray} div(A(x)|\nabla u|^{m-2}\nabla u)+V(x)u^{\sigma_1}|\nabla u|^{\sigma_2}\leq0, \tag{1} \end{eqnarray} on a noncompact complete Riemannian manifold, where $A, V$ are positive measurable functions, $m>1$, and $\sigma_1$, $\sigma_2\geq0$ are parameters such that $\sigma_1+\sigma_2>m-1$.
Our purpose is to establish the uniqueness of nonnegative solution to (1) via very natural geometric assumption on volume growth.
Citation: Yuhua Sun. On the uniqueness of nonnegative solutions of differential inequalities with gradient terms on Riemannian manifolds. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1743-1757. doi: 10.3934/cpaa.2015.14.1743
##### References:
 [1] S. Y. Cheng and S.-T Yau, Differential equations on Riemannian manifolds and their geometric applications,, \emph{Comm. Pure Appl. Math.}, 28 (1975), 333. Google Scholar [2] G. Caristi, L. D'Ambrosio and E. Mitidieri, Liouville Theorems for some nonlinear inequalities,, \emph{Proc. Steklov Inst. Math.}, 260 (2008), 90. doi: 10.1134/S0081543808010070. Google Scholar [3] G. Caristi, E. Mitidieri and Pokhozhaev, Some Liouville theorems for quasilinear elliptic inequalities,, \emph{Doklady Math.}, 79 (2009), 118. doi: 10.1134/S1064562409010360. Google Scholar [4] R. Filippucci, Nonexistence of positive weak solutions of elliptic inequalities,, \emph{Nonlinear Anal.}, 70 (2009), 2903. doi: 10.1016/j.na.2008.12.018. Google Scholar [5] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, \emph{Comm. Pure Appl. Math.}, 34 (1981), 525. doi: 10.1002/cpa.3160340406. Google Scholar [6] A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds,, \emph{Bull. Amer. Math. Soc.}, 36 (1999), 135. doi: 10.1090/S0273-0979-99-00776-4. Google Scholar [7] A. Grigor'yan and V. A. Kondratiev, On the existence of positive solutions of semi-linear elliptic inequalities on Riemannian manifolds,, \emph{International Mathematical Series}, 12 (2010), 203. doi: 10.1007/978-1-4419-1343-2_8. Google Scholar [8] A. Grigor'yan, The existence of positive fundamental solution of the Laplace equation on Riemannian manifolds,, \emph{Math. USSR Sb.}, 56 (1987), 349. Google Scholar [9] A. Grigor'yan and Y. Sun, On nonnegative of the inequality $\Delta u+u^{\sigma}\leq0$ on Riemannian manifolds,, \emph{Comm. Pure Appl. Math.}, 67 (2014), 1336. doi: 10.1002/cpa.21493. Google Scholar [10] I. Holopainen, Volume growth, Green's functions, and parabolicity of ends,, \emph{Duke Math. J.}, 97 (1999), 319. doi: 10.1215/S0012-7094-99-09714-4. Google Scholar [11] I. Holopainen, A sharp $L^q-$Liouville theorem for $p\text{-}$harmonic functions,, \emph{Israel J. Math.}, 115 (2000), 363. doi: 10.1007/BF02810597. Google Scholar [12] A. A. Kon'kov, Comparison theorems for second-order elliptic inequalities,, \emph{Nonlinear Anal.}, 59 (2004), 583. doi: 10.1016/j.na.2004.06.002. Google Scholar [13] E. Mitidieri and S. I. Pokhozhaev, Absence of global positive solutions of quasilinear elliptic inequalities,, \emph{Dokl. Akad. Nauk. (Russian)}, 359 (1998), 456. Google Scholar [14] E. Mitidieri and S. I. Pokhozhaev, Nonexistence of positive solutions for quasilinear elliptic problems on $\mathbbR^N$,, \emph{Proc. Steklov Inst. Math.}, 227 (1999), 186. Google Scholar [15] E. Mitidieri and S. I. Pokhozhaev, Absence of positive solutions for quasilinear elliptic problems in $\mathbbR^N$,, \emph{Tran. Math. Inst. Steklova. (Russian)}, 227 (1999), 192. Google Scholar [16] E. Mitidieri and S. I. Pokhozhaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities,, \emph{Tr. Math. Inst. Steklova (in Russian)}, 234 (2001), 1. Google Scholar [17] E. Mitidieri and S. I. Pokhozhaev, A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities (Nauka, Moscow, 2001),, \emph{Tr. Math. Inst. im., 234 (2001). Google Scholar [18] W. M. Ni and J. Serrin, Nonexistence theorems for quasilinear partial differential equations,, \emph{Proceedings of the conference commemorating the 1st centennial of the Circolo Matematico di Palermo (Italian)}, 8 (1985), 171. Google Scholar [19] W. M. Ni and J. Serrin, Existence and nonexistence theorems for ground states of quasilinear partial differential equations: the anomalous case,, \emph{Accad. Naz. Lincei}, 77 (1986), 231. Google Scholar [20] Y. Sun, Uniqueness result for non-negative solutions of semi-linear inequalities on Riemannian manifolds,, \emph{J. Math. Anal. Appl.}, 419 (2014), 643. doi: 10.1016/j.jmaa.2014.05.011. Google Scholar

show all references

##### References:
 [1] S. Y. Cheng and S.-T Yau, Differential equations on Riemannian manifolds and their geometric applications,, \emph{Comm. Pure Appl. Math.}, 28 (1975), 333. Google Scholar [2] G. Caristi, L. D'Ambrosio and E. Mitidieri, Liouville Theorems for some nonlinear inequalities,, \emph{Proc. Steklov Inst. Math.}, 260 (2008), 90. doi: 10.1134/S0081543808010070. Google Scholar [3] G. Caristi, E. Mitidieri and Pokhozhaev, Some Liouville theorems for quasilinear elliptic inequalities,, \emph{Doklady Math.}, 79 (2009), 118. doi: 10.1134/S1064562409010360. Google Scholar [4] R. Filippucci, Nonexistence of positive weak solutions of elliptic inequalities,, \emph{Nonlinear Anal.}, 70 (2009), 2903. doi: 10.1016/j.na.2008.12.018. Google Scholar [5] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, \emph{Comm. Pure Appl. Math.}, 34 (1981), 525. doi: 10.1002/cpa.3160340406. Google Scholar [6] A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds,, \emph{Bull. Amer. Math. Soc.}, 36 (1999), 135. doi: 10.1090/S0273-0979-99-00776-4. Google Scholar [7] A. Grigor'yan and V. A. Kondratiev, On the existence of positive solutions of semi-linear elliptic inequalities on Riemannian manifolds,, \emph{International Mathematical Series}, 12 (2010), 203. doi: 10.1007/978-1-4419-1343-2_8. Google Scholar [8] A. Grigor'yan, The existence of positive fundamental solution of the Laplace equation on Riemannian manifolds,, \emph{Math. USSR Sb.}, 56 (1987), 349. Google Scholar [9] A. Grigor'yan and Y. Sun, On nonnegative of the inequality $\Delta u+u^{\sigma}\leq0$ on Riemannian manifolds,, \emph{Comm. Pure Appl. Math.}, 67 (2014), 1336. doi: 10.1002/cpa.21493. Google Scholar [10] I. Holopainen, Volume growth, Green's functions, and parabolicity of ends,, \emph{Duke Math. J.}, 97 (1999), 319. doi: 10.1215/S0012-7094-99-09714-4. Google Scholar [11] I. Holopainen, A sharp $L^q-$Liouville theorem for $p\text{-}$harmonic functions,, \emph{Israel J. Math.}, 115 (2000), 363. doi: 10.1007/BF02810597. Google Scholar [12] A. A. Kon'kov, Comparison theorems for second-order elliptic inequalities,, \emph{Nonlinear Anal.}, 59 (2004), 583. doi: 10.1016/j.na.2004.06.002. Google Scholar [13] E. Mitidieri and S. I. Pokhozhaev, Absence of global positive solutions of quasilinear elliptic inequalities,, \emph{Dokl. Akad. Nauk. (Russian)}, 359 (1998), 456. Google Scholar [14] E. Mitidieri and S. I. Pokhozhaev, Nonexistence of positive solutions for quasilinear elliptic problems on $\mathbbR^N$,, \emph{Proc. Steklov Inst. Math.}, 227 (1999), 186. Google Scholar [15] E. Mitidieri and S. I. Pokhozhaev, Absence of positive solutions for quasilinear elliptic problems in $\mathbbR^N$,, \emph{Tran. Math. Inst. Steklova. (Russian)}, 227 (1999), 192. Google Scholar [16] E. Mitidieri and S. I. Pokhozhaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities,, \emph{Tr. Math. Inst. Steklova (in Russian)}, 234 (2001), 1. Google Scholar [17] E. Mitidieri and S. I. Pokhozhaev, A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities (Nauka, Moscow, 2001),, \emph{Tr. Math. Inst. im., 234 (2001). Google Scholar [18] W. M. Ni and J. Serrin, Nonexistence theorems for quasilinear partial differential equations,, \emph{Proceedings of the conference commemorating the 1st centennial of the Circolo Matematico di Palermo (Italian)}, 8 (1985), 171. Google Scholar [19] W. M. Ni and J. Serrin, Existence and nonexistence theorems for ground states of quasilinear partial differential equations: the anomalous case,, \emph{Accad. Naz. Lincei}, 77 (1986), 231. Google Scholar [20] Y. Sun, Uniqueness result for non-negative solutions of semi-linear inequalities on Riemannian manifolds,, \emph{J. Math. Anal. Appl.}, 419 (2014), 643. doi: 10.1016/j.jmaa.2014.05.011. Google Scholar
 [1] Keith Burns, Eugene Gutkin. Growth of the number of geodesics between points and insecurity for Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 403-413. doi: 10.3934/dcds.2008.21.403 [2] Yu-Zhao Wang. $\mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116 [3] YanYan Li, Tonia Ricciardi. A sharp Sobolev inequality on Riemannian manifolds. Communications on Pure & Applied Analysis, 2003, 2 (1) : 1-31. doi: 10.3934/cpaa.2003.2.1 [4] Rossella Bartolo. Periodic orbits on Riemannian manifolds with convex boundary. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 439-450. doi: 10.3934/dcds.1997.3.439 [5] Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas. Stability of boundary distance representation and reconstruction of Riemannian manifolds. Inverse Problems & Imaging, 2007, 1 (1) : 135-157. doi: 10.3934/ipi.2007.1.135 [6] David M. A. Stuart. Solitons on pseudo-Riemannian manifolds: stability and motion. Electronic Research Announcements, 2000, 6: 75-89. [7] Fei Liu, Jaume Llibre, Xiang Zhang. Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1097-1111. doi: 10.3934/dcds.2011.29.1097 [8] Eva Glasmachers, Gerhard Knieper, Carlos Ogouyandjou, Jan Philipp Schröder. Topological entropy of minimal geodesics and volume growth on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 75-91. doi: 10.3934/jmd.2014.8.75 [9] Patrizia Pucci, Marco Rigoli. Entire solutions of singular elliptic inequalities on complete manifolds. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 115-137. doi: 10.3934/dcds.2008.20.115 [10] Bo Guan, Heming Jiao. The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 701-714. doi: 10.3934/dcds.2016.36.701 [11] Anthony M. Bloch, Rohit Gupta, Ilya V. Kolmanovsky. Neighboring extremal optimal control for mechanical systems on Riemannian manifolds. Journal of Geometric Mechanics, 2016, 8 (3) : 257-272. doi: 10.3934/jgm.2016007 [12] Zhuoran Du, Baishun Lai. Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1407-1429. doi: 10.3934/dcds.2013.33.1407 [13] Mohammadreza Molaei. Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds. Electronic Research Announcements, 2018, 25: 8-15. doi: 10.3934/era.2018.25.002 [14] Weisong Dong, Tingting Wang, Gejun Bao. A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1769-1780. doi: 10.3934/cpaa.2016013 [15] Simone Fiori. Synchronization of first-order autonomous oscillators on Riemannian manifolds. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1725-1741. doi: 10.3934/dcdsb.2018233 [16] Marcelo Nogueira, Mahendra Panthee. On the Schrödinger-Debye system in compact Riemannian manifolds. Communications on Pure & Applied Analysis, 2020, 19 (1) : 425-453. doi: 10.3934/cpaa.2020022 [17] Rafael de la Llave, Jason D. Mireles James. Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4321-4360. doi: 10.3934/dcds.2012.32.4321 [18] Wolfgang Walter. Nonlinear parabolic differential equations and inequalities. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 451-468. doi: 10.3934/dcds.2002.8.451 [19] Radu Saghin. Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3789-3801. doi: 10.3934/dcds.2014.34.3789 [20] Robert J. Kipka, Yuri S. Ledyaev. Optimal control of differential inclusions on manifolds. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4455-4475. doi: 10.3934/dcds.2015.35.4455

2018 Impact Factor: 0.925