MR3221843
Liang ZHAO
Liang ZHAO
Liouville Theorem for Lichnerowicz Equation on Complete Noncompact Manifolds
Funkcialaj Ekvacioj. Serio Internacia
57
2014
163--172
http://fe.math.kobe-u.ac.jp/FE/FullPapers/57-1/57_163.pdf
http://www.ams.org/mathscinet-getitem?mr=MR3221843
In this paper, we study the gradient estimates for positive solutions to the following Lichnerowicz equation $\triangle_{f}u+hu=Au^{p}+Bu^{-q}$, where $h$, $p$, $q$, $A$, $B$ are real constants and $p>1$, $q>1$. Moreover, by these gradient estimates, we can get some very interesting Liouville theorems.
Gradient estimates, Positive solutions, Liouville theorem.
Primary 53J05, Secondary 58J35.
57-163
2014
Liouville Theorem for Lichnerowicz Equation on Complete Noncompact Manifolds
Liang ZHAO
Liang ZHAO
1
Chen, L.; Chen, W.-Y.
Gradient estimates for a nonlinear parabolic equation on complete non-compact Riemannian manifolds
Ann. Global Anal. Geom.
35
2009
397-404
MR2506242
2
Chen, L.; Chen, W.-Y.
Gradient estimates for positive smooth $f$-harmonic functions
Acta Math. Sci. Ser. B Engl. Ed.
30
2010
1614-1618
MR2778630
3
Cheng, S.-Y.; Yau, S.-T.
Differential equations on Riemannian manifolds and their geometric applications
Comm. Pure Appl. Math.
28
1975
333-354
MR0385749
4
Huang, G.-Y.; Li, H.-Z.. Gradient estimates and entropy formulae of porous medium and fast diffusion equations for the Witten Laplacian, preprint 2012. Available at http://arxiv.org/abs/1203.5482
5
Huang, G.-Y.; Ma, B.-Q.
Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds
Arch. Math. (Basel)
94
2010
265-275
MR2602453
6
Li, X.-D.
Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds
J. Math. Pures Appl. (9)
84
2005
1295-1361
MR2170766
7
Ma, L.
Gradient estimates for a simple elliptic equation on complete non-compact Riemannian manifolds
J. Funct. Anal.
241
2006
374-382
MR2264255
8
Qian, Z.-M.
A comparison theorem for an elliptic operator
Potential Analysis
8
1998
137-142
MR1618434
9
Song, X.-F.; Zhao, L.
Gradient estimates for the elliptic and parabolic Lichnerowicz equations on compact manifolds
Z. Angew. Math. Phys.
61
2010
655-662
MR2673328
10
Wu, J.-Y.
Li-Yau type estimates for a nonlinear parabolic equation on complete manifolds
J. Math. Anal. Appl.
369
2010
400-407
MR2643878
11
Zhang, J.; Ma, B.-Q.
Gradient estimates for a nonlinear equation $\triangle_{f}u+cu^{-\alpha}=0$ on complete noncompact manifolds
Commun. Math.
19
2011
73-84
MR2855392
12
Zhu, X.-B.
Hamilton's gradient estimates and Liouville theorems for fast diffusion equations on noncompact Riemannian manifolds
Proc. Amer. Math. Soc.
139
2011
1637-1644
MR2763753