摘要：
In this paper, we consider gradient estimates for positive solutions to the following equation Delta vu + au(p) log u = 0 on complete noncompact Riemannian manifold with k-dimensional Bakry- mery Ricci curvature bounded from below. Using the Bochner formula and the Cauchy inequality, we obtain upper bounds of vertical bar del u vertical bar with respect to the lower bound of the Bakry- mery Ricci curvature.
摘要：
In this paper, we discuss the lower diameter estimate for a class of compact generalized quasi-Einstein manifolds which are closely related to the conformal geometry. Using the Bochner formula and the Hopf maximum principle, we get a gradient estimate for the potential function of the manifold. Based on the gradient estimate, we get the lower diameter estimate for this class of generalized quasi-Einstein manifolds.
作者机构：
[Deng, Yi-Hua] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Peoples R China.
通讯机构：
[Deng, Yi-Hua] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Peoples R China.
关键词：
p-Laplace equation;variational methods;infinitely many solutions;nonnegative solution
摘要：
In this paper, we study a class of p-Laplace equations. Using variational methods, we prove that there are two solutions and one of these solutions is nonnegative. Using recurrence method, we prove that there are infinitely many solutions to this class of equations.
摘要：
In this note, we get a necessary and sufficient condition such that the scalar curvature of generalized m-quasi-Einstein manifold with m = 1 is constant. In particular, we discuss a class of generalized quasi-Einstein manifolds which are more general than (m, rho)-quasi-Einstein manifolds and prove that these manifolds with dimension four are either Einstein or locally conformally flat under some suitable conditions. (C) 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
摘要：
We prove that if M is a complete noncompact Riemannian manifold whose Ricci tensor is cyclic parallel and whose scalar curvature is nonpositive, then M is Einstein, provided the Sobolev constant is positive and an integral inequality is satisfied.
期刊：
BULLETIN OF THE IRANIAN MATHEMATICAL SOCIETY,2014年40(5):1087-1095 ISSN：1017-060X
通讯作者：
Deng, Y. H.
作者机构：
[Deng, YH] Hengyang Normal Univ, Dept Math & Computat Sci, Hengyang, Peoples R China.;[Deng, Y. H.] Hengyang Normal Univ, Dept Math & Computat Sci, POB 421002, Hengyang, Peoples R China.
通讯机构：
[Deng, Y. H.] H;Hengyang Normal Univ, Dept Math & Computat Sci, POB 421002, Hengyang, Peoples R China.
关键词：
Ground state solution;Minimization problem;p-Laplace equation;The Schwartz symmetrization process
期刊：
TURKISH JOURNAL OF MATHEMATICS,2014年38(6):985-993 ISSN：1300-0098
通讯作者：
Deng Yihua
作者机构：
[Deng Yihua] Hengyang Normal Univ, Dept Math & Comp Sci, Hengyang, Hunan, Peoples R China.
通讯机构：
[Deng Yihua] H;Hengyang Normal Univ, Dept Math & Comp Sci, Hengyang, Hunan, Peoples R China.
关键词：
Gradient estimates;porous medium type equation;smooth metric measure space;positive solution;Gradient estimates;porous medium type equation;smooth metric measure space;positive solution
摘要：
The porous medium equation arises in different applications to model diffusive phenomena. In this paper, we obtain several gradient estimates for some porous medium type equations on smooth metric measure space with N-Bakry-Emery Ricci tensor bounded from below. In particular, we improve and generalize some current gradient estimates for the porous medium equations.
