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

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