Related papers: Gradient estimates and Liouville theorems for Lich…

In this paper we consider the gradient estimates on positive solutions to the following elliptic equation defined on a complete Riemannian manifold $(M,\,g)$: $$\Delta v+v^r-v^s= 0,$$ where $r$ and $s$ are two real constants. When$(M,\,g)$…

In this paper, first we study carefully the positive solutions to $\Delta u+\lambda_{1}u\ln u +\lambda_{2}u^{b+1}=0$ defined on a complete noncompact Riemannian manifold $(M, g)$ with $Ric(g)\geq -Kg$, which can be regarded as…

In this paper, we use Nash-Moser iteration method to study the local and global behaviours of positive solutions to the nonlinear elliptic equation $\Delta_pv +av^{q}=0$ defined on a complete Riemannian manifolds $(M,g)$ where $p>1$, $a$…

In this paper, we use the Saloff-Coste Sobolev inequality and Nash-Moser iteration method to study the local and global behaviors of positive solutions to the nonlinear elliptic equation $\Delta_pv+a(v+b)^q=0$ defined on a complete…

In this paper the Nash-Moser iteration method is used to study the gradient estimates of solutions to the quasilinear elliptic equation $\Delta_p u-|\nabla u|^q+b(x)|u|^{r-1}u=0$ defined on a complete Riemannian manifold $(M,g)$. When…

In this short note, we use a unified method to consider the gradient estimates of the positive solution to the following nonlinear elliptic equation $\Delta u + au^{p+1}=0$ defined on a complete noncompact Riemannian manifold $(M, g)$ where…

Let $(M^n,g)$ be an n-dimensional complete Riemannian manifold. We consider gradient estimates and Liouville type theorems for positive solutions to the following nonlinear elliptic equation: $$\Delta u+au\log u=0,$$ where $a$ is a nonzero…

In this paper, we combine Bochner formula, Saloff-Coste's Sobolev inequality and the Nash-Moser iteration method to study the local and global behaviors of solutions to the nonlinear elliptic equation $\Delta_pu+\Delta_qu+h(u,|\nabla…

In this paper, we provide a new routine to employ the Nash-Moser iteration technique to analyze the local and global properties of positive solutions to the equation $$\Delta_pv + a|\nabla v|^qv^r =0$$ on a complete Riemannian manifold with…

In this paper, we employ the Nash-Moser iteration technique to study local and global properties of positive solutions to the equation $$\Delta_pv+bv^q+cv^r =0$$ on complete Riemannian manifolds with Ricci curvature bounded from below,…

In this paper, we consider the gradient estimates of the positive solutions to the following equation defined on a complete Riemannian manifold $(M, g)$ $$\Delta u + au(\log u)^{p}+bu=0,$$ where $a, b\in \mathbb{R}$ and $p$ is a rational…

In this paper we establish gradient estimates for positive solutions to the nonlinear elliptic equation $$\Delta_{V}u^{m}+\mu(x)u+p(x)u^{\alpha}=0 , \quad m>1$$on any smooth metric measure space whose $k$-Bakry-\'{E}mery curvature is…

In this short note, we consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a complete Riemannian manifold: $$\Delta u+cu^{\alpha}=0,$$ where $c, \alpha$ are two real constants and $c\neq 0$.

In this paper, combining Nash-Moser iteration and Sallof-Coste type Sobolev ineualities, we establish fundamental and concise $C^0$ and $C^1$ estimates for solutions to a class of nonlinear elliptic equations of the form $$\Delta…

This paper studies a class of $p$-Laplace equations with cubic polynomial nonlinearity \[ \Delta_p v + (v-a_1)(v-a_2)(v-a_3) = 0 \] on complete Riemannian manifolds $M$ with lower Ricci curvature bounds, where $a_1 < a_2 < a_3$ are real…

In this paper, we derive the gradient estimates for the positive solutions of the equation $\Delta_b u + au^{p+1} = 0$ on complete noncompact pseudo-Hermitian manifolds, where $a > 0$ and $p \leq 0$ or $a < 0$ and $p > 0$ are two constants.…

In this paper, we consider the weighted $p$-Laplacian equation $$ \Delta_{p,f}u+au^{\sigma}\ln u=0$$ defined on a complete smooth metric measure space under the conditon that the $m$-Bakry-\'{E}mery Ricci curvature has a lower bound, where…

Let $(N, g)$ be a complete noncompact Riemannian manifold with Ricci curvature bounded from below. In this paper, we study the gradient estimates of positive solutions to a class of nonlinear elliptic equations $$\Delta u(x)+a(x)u(x)\log…

We study the elliptic version of doubly nonlinear diffusion equations on a complete Riemannian manifold $(M,g)$. Through the combination of a special nonlinear transformation and the standard Nash-Moser iteration procedure, some Cheng-Yau…

Given a complete, smooth metric measure space $(M,g,e^{-f}dv)$ with the Bakry-\'Emery Ricci curvature bounded from below, various gradient estimates for solutions of the following general $f$-heat equations $$ u_t=\Delta_f u+au\log u+bu…