In this paper we consider the gradient estimates on positive solutions to the following elliptic (Lichnerowicz) equation defined on a complete Riemannian manifold $(M,\,g)$: $$Δv + μv + a v^{p+1} +b v^{-q+1} =0,$$ where $p\geq-1$, $q\geq1$, $μ$, $a$ and $b$ are real constants.
In the case $μ\geq0$ and $b\geq0$ or $μ<0$ , $a>0$ and $b>0$ ($μ$ has a lower bound), we employ the Nash-Moser iteration technique to obtain some refined gradient estimates of the solutions to the above equation, if $(M,\,g)$ satisfies $Ric \geq -(n-1)κ$ , where $n\geq3$ is the dimension of $M$ and $κ$ is a nonnegative constant, and $μ$ , $a$ , $b$ , $p$ and $q$ satisfy some technique conditions. By the obtained gradient estimates we also derive some Liouville type theorems for the above equation under some suitable geometric and analysis conditions. As applications, we can derive some Cheng-Yau's type gradient estimates for solutions to the $n$-dimensional Einstein-scalar field Lichnerowicz equation where $n\geq3$.
In this paper, first we study carefully the positive solutions to $Δu+λ_{1}u\ln u +λ_{2}u^{b+1}=0$ defined on a complete noncompact Riemannian manifold $(M, g)$ with $Ric(g)\geq -Kg$, which can be regarded as Lichnerowicz-type equations, and obtain the gradient estimates of positive solutions to these equations which do not depend on the bounds of the solutions and the Laplacian of the distance function on $(M, g)$. Then, we extend our techniques to a class of more general semilinear elliptic equations $Δu(x)+uh(\ln u)=0$ and obtain some similar results under some suitable analysis conditions …
In this paper, we generalize Liouville type theorems for some semilinear partial differential inequalities to sub-Riemannian manifolds satisfying a nonnegative generalized curvature-dimension inequality introduced by Baudoin and Garofalo in [5]. In particular, our results apply to all Sasakian manifolds with nonnegative horizontal Webster-Tanaka-Ricci curvature. The key ingredient is to construct a class of "good" cut-off functions. We also provide some upper bounds for lifespan to parabolic and hyperbolic inequalities.
Liouville type theorems for some $(p,q)$-Laplace equations with gradient dependent reaction on Riemannian manifolds
School of Mathematics and Information Sciences, Guangzhou University（广州大学数学与信息科学学院）；State Key Laboratory of Mathematical Sciences (SKLMS), Academy of Mathematics and Systems Science, and School of Mathematical Sciences, UCAS（中国科学院数学与系统科学研究院，中国科学院大学数学科学学院，数学科学国家重点实验室）
This note studies local integral gradient bounds for distributional solutions of a large class of partial differential inequalities with diffusion in divergence form and power-like first-order terms. The applications of these estimates are two-fold. First, we show the (sharp) global Hölder regularity of distributional semi-solutions to this class of diffusive PDEs with first-order terms having supernatural growth and right-hand side in a suitable Morrey class posed on a bounded and regular open set $Ω$. Second, we provide a new proof of entire Liouville properties for inequalities with superli…
We investigate the nonexistence and existence of nontrivial positive solutions to $Δ_m u+u^p|\nabla u|^q\leq0$ on noncompact geodesically complete Riemannian manifolds, where $m>1$, and $(p,q)\in \mathbb{R}^2$. According to classification of $(p, q)$, we establish different volume growth conditions to obtain Liouville's theorems for the above quasilinear differential inequalities, and we also show these volume growth conditions are sharp in most cases. Moreover, the results are completely new for $(p, q)$ of negative pair, even in the Euclidean space.
Stability in the anisotropic Calderón problem for Painlevé-Liouville Riemannian manifolds
Université Marie et Louis Pasteur, CNRS（玛丽和路易·巴斯德大学，法国国家科学研究中心）；McGill University（麦吉尔大学）；Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629（让·勒雷数学实验室，CNRS联合研究单位6629）
