Related papers: Gradient estimates for nonlinear elliptic equation…

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…

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…

Let $(M^N, g, e^{-f}dv)$ be a complete smooth metric measure space with $\infty$-Bakry-\'Emery Ricci tensor bounded from below. We derive elliptic gradient estimates for positive solutions of a weighted nonlinear parabolic equation…

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…

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, firstly, we study gradient estimates for positive solution of the following equation \begin{equation*} \Delta_\xi(u)-\partial_t u- q u =A(u),t\in (-\infty,\infty) \end{equation*} on metric measure space $…

In this paper, we study a solvability result for the nonlinear problem $$ \mbox {div } \left ( \vert \nabla_\omega u\vert^{p-2}\nabla_\omega u \right )+v(x) u^{q-1}+\mu u^{\gamma-1}=0, \quad z\in \Omega, \quad u \Big \vert_{\partial…

In this paper we consider the gradient estimates on positive solutions to the following elliptic (Lichnerowicz) equation defined on a complete Riemannian manifold $(M,\,g)$: $$\Delta v + \mu v + a v^{p+1} +b v^{-q+1} =0,$$ where $p\geq-1$,…

We prove a result of existence of positive solutions of the Dirichlet problem for $-\Delta_p u=\mathrm{w}(x)f(u,\nabla u)$ in a bounded domain $\Omega\subset\mathbb{R}^N$, where $\Delta_p$ is the $p$-Laplacian and $\mathrm{w}$ is a weight…

In this article we study local and global properties of positive solutions of $-\Delta_mu=|u|^{p-1}u+M|\nabla u|^q$ in a domain $\Omega$ of $\mathbb R^N$, with $m>1$, $p,q>0$ and $M\in\mathbb R$. Following some ideas used in…

Local and global pointwise gradient estimates are obtained for solutions to the quasilinear elliptic equation with measure data $-\operatorname{div}(A(x,\nabla u))=\mu$ in a bounded and possibly nonsmooth domain $\Omega$ in $\mathbb{R}^n$.…

In this paper, we present a series of Liouville-type theorems for a class of nonhomogeneous quasilinear elliptic equations featuring reactions that depend on the solution and its gradient. Specifically, we investigate equations of the form…

In this paper, we prove $L^q$-estimates for gradients of solutions to singular quasilinear elliptic equations with measure data $$-\operatorname{div}(A(x,\nabla u))=\mu,$$ in a bounded domain $\Omega\subset\mathbb{R}^{N}$, where $A(x,\nabla…

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 $c\neq 0$.

We establish local elliptic and parabolic gradient estimates for positive smooth solutions to a nonlinear parabolic equation on a smooth metric measure space. As applications, we determine various conditions on the equation's coefficients…

In this paper, we prove some qualitative properties for the positive solutions to some degenerate elliptic equation given by \[-\operatorname{div}(w|\nabla u|^{p-2}\nabla u)=f(x,u);\;\;w\in \mathcal{A}_p\] on smooth domain and for varying…

This article presents new local and global gradient estimates of Li-Yau type for positive solutions to a class of nonlinear elliptic equations on smooth metric measure spaces involving the Witten Laplacian. The estimates are derived under…

We establish gradient estimates of solutions to a class of nonlinear elliptic equations with measure data under Orlicz-type growth conditions. The growth is governed by the structural condition \[ 0<i_a\le t g'(t)/g(t)\le s_a<1. \] We…

In this paper we consider positive supersolutions of the nonlinear elliptic equation \[- \Delta u = \rho(x) f(u)|\nabla u|^p, \qquad \hfill \mbox{ in } \Omega,\] where $0\le p<1$, $ \Omega$ is an arbitrary domain (bounded or unbounded) in $…

This paper investigates the initial-boundary value problem for a nonlinear parabolic equation involving the $p$-Laplacian operator, nonlocal source terms, gradient absorption, and various nonlinearities: \[ \frac{\partial u}{\partial t} -…