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, Comm. Pure Appl. Math., 28 (1975), 333-354.  Google Scholar

[2]

G. Caristi, L. D'Ambrosio and E. Mitidieri, Liouville Theorems for some nonlinear inequalities, Proc. Steklov Inst. Math., 260 (2008), 90-111. doi: 10.1134/S0081543808010070.  Google Scholar

[3]

G. Caristi, E. Mitidieri and Pokhozhaev, Some Liouville theorems for quasilinear elliptic inequalities, Doklady Math., 79 (2009), 118-124. doi: 10.1134/S1064562409010360.  Google Scholar

[4]

R. Filippucci, Nonexistence of positive weak solutions of elliptic inequalities, Nonlinear Anal., 70 (2009), 2903-2916. 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, Comm. Pure Appl. Math., 34 (1981), 525-598. 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, Bull. Amer. Math. Soc., 36 (1999), 135-249. 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, International Mathematical Series, 12 (2010), 203-218. 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, Math. USSR Sb., 56 (1987), 349-358, (Translated from Russian Matem. Sbornik, 128 (1985), 354-363.  Google Scholar

[9]

A. Grigor'yan and Y. Sun, On nonnegative of the inequality $\Delta u+u^{\sigma}\leq0$ on Riemannian manifolds, Comm. Pure Appl. Math., 67 (2014), 1336-1352. doi: 10.1002/cpa.21493.  Google Scholar

[10]

I. Holopainen, Volume growth, Green's functions, and parabolicity of ends, Duke Math. J., 97 (1999), 319-346. doi: 10.1215/S0012-7094-99-09714-4.  Google Scholar

[11]

I. Holopainen, A sharp $L^q-$Liouville theorem for $p\text{-}$harmonic functions, Israel J. Math., 115 (2000), 363-379. doi: 10.1007/BF02810597.  Google Scholar

[12]

A. A. Kon'kov, Comparison theorems for second-order elliptic inequalities, Nonlinear Anal., 59 (2004), 583-608. 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, Dokl. Akad. Nauk. (Russian), 359 (1998), 456-460.  Google Scholar

[14]

E. Mitidieri and S. I. Pokhozhaev, Nonexistence of positive solutions for quasilinear elliptic problems on $\mathbbR^N$, Proc. Steklov Inst. Math., 227 (1999), 186-216.  Google Scholar

[15]

E. Mitidieri and S. I. Pokhozhaev, Absence of positive solutions for quasilinear elliptic problems in $\mathbbR^N$, Tran. Math. Inst. Steklova. (Russian), (1999), 192-222; translation in Proc. Steklov Inst. Math., 227 (1999), 186-216.  Google Scholar

[16]

E. Mitidieri and S. I. Pokhozhaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Math. Inst. Steklova (in Russian), 234 (2001), 1-384. Engl. transl. Proc. Steklov Inst. Math., 234 (2001), 1-362.  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), Tr. Math. Inst. im., V. A. Steklova, Ross. Akad. Nauk, 234 (2001).  Google Scholar

[18]

W. M. Ni and J. Serrin, Nonexistence theorems for quasilinear partial differential equations, Proceedings of the conference commemorating the 1st centennial of the Circolo Matematico di Palermo (Italian), (Palermo, 1984). Rend. Circ. Mat. Palermo (2) Suppl. 8 (1985), 171-185.  Google Scholar

[19]

W. M. Ni and J. Serrin, Existence and nonexistence theorems for ground states of quasilinear partial differential equations: the anomalous case, Accad. Naz. Lincei, 77 (1986), 231-257. Google Scholar

[20]

Y. Sun, Uniqueness result for non-negative solutions of semi-linear inequalities on Riemannian manifolds, J. Math. Anal. Appl., 419 (2014), 643-661. 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, Comm. Pure Appl. Math., 28 (1975), 333-354.  Google Scholar

[2]

G. Caristi, L. D'Ambrosio and E. Mitidieri, Liouville Theorems for some nonlinear inequalities, Proc. Steklov Inst. Math., 260 (2008), 90-111. doi: 10.1134/S0081543808010070.  Google Scholar

[3]

G. Caristi, E. Mitidieri and Pokhozhaev, Some Liouville theorems for quasilinear elliptic inequalities, Doklady Math., 79 (2009), 118-124. doi: 10.1134/S1064562409010360.  Google Scholar

[4]

R. Filippucci, Nonexistence of positive weak solutions of elliptic inequalities, Nonlinear Anal., 70 (2009), 2903-2916. 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, Comm. Pure Appl. Math., 34 (1981), 525-598. 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, Bull. Amer. Math. Soc., 36 (1999), 135-249. 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, International Mathematical Series, 12 (2010), 203-218. 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, Math. USSR Sb., 56 (1987), 349-358, (Translated from Russian Matem. Sbornik, 128 (1985), 354-363.  Google Scholar

[9]

A. Grigor'yan and Y. Sun, On nonnegative of the inequality $\Delta u+u^{\sigma}\leq0$ on Riemannian manifolds, Comm. Pure Appl. Math., 67 (2014), 1336-1352. doi: 10.1002/cpa.21493.  Google Scholar

[10]

I. Holopainen, Volume growth, Green's functions, and parabolicity of ends, Duke Math. J., 97 (1999), 319-346. doi: 10.1215/S0012-7094-99-09714-4.  Google Scholar

[11]

I. Holopainen, A sharp $L^q-$Liouville theorem for $p\text{-}$harmonic functions, Israel J. Math., 115 (2000), 363-379. doi: 10.1007/BF02810597.  Google Scholar

[12]

A. A. Kon'kov, Comparison theorems for second-order elliptic inequalities, Nonlinear Anal., 59 (2004), 583-608. 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, Dokl. Akad. Nauk. (Russian), 359 (1998), 456-460.  Google Scholar

[14]

E. Mitidieri and S. I. Pokhozhaev, Nonexistence of positive solutions for quasilinear elliptic problems on $\mathbbR^N$, Proc. Steklov Inst. Math., 227 (1999), 186-216.  Google Scholar

[15]

E. Mitidieri and S. I. Pokhozhaev, Absence of positive solutions for quasilinear elliptic problems in $\mathbbR^N$, Tran. Math. Inst. Steklova. (Russian), (1999), 192-222; translation in Proc. Steklov Inst. Math., 227 (1999), 186-216.  Google Scholar

[16]

E. Mitidieri and S. I. Pokhozhaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Math. Inst. Steklova (in Russian), 234 (2001), 1-384. Engl. transl. Proc. Steklov Inst. Math., 234 (2001), 1-362.  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), Tr. Math. Inst. im., V. A. Steklova, Ross. Akad. Nauk, 234 (2001).  Google Scholar

[18]

W. M. Ni and J. Serrin, Nonexistence theorems for quasilinear partial differential equations, Proceedings of the conference commemorating the 1st centennial of the Circolo Matematico di Palermo (Italian), (Palermo, 1984). Rend. Circ. Mat. Palermo (2) Suppl. 8 (1985), 171-185.  Google Scholar

[19]

W. M. Ni and J. Serrin, Existence and nonexistence theorems for ground states of quasilinear partial differential equations: the anomalous case, Accad. Naz. Lincei, 77 (1986), 231-257. Google Scholar

[20]

Y. Sun, Uniqueness result for non-negative solutions of semi-linear inequalities on Riemannian manifolds, J. Math. Anal. Appl., 419 (2014), 643-661. doi: 10.1016/j.jmaa.2014.05.011.  Google Scholar

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