Related papers: Gradient estimates for singular $p$-Laplace type e…

We establish a priori regularity estimates for viscosity solutions of degenerate fully nonlinear elliptic equations with integrable right-hand sides. When the nonhomogeneous term belongs to $L^p$ with $p>n$, we prove optimal interior…

We prove a local higher integrability result for the gradient of a weak solution to parabolic double-phase systems of $p$-Laplace type when $\tfrac{2n}{n+2}< p\le2$. The result is based on a reverse H\"older inequality in intrinsic…

We consider second-order elliptic equations in a half space with leading coefficients measurable in a tangential direction. We prove the $W^2_p$-estimate and solvability for the Dirichlet problem when $p\in (1,2]$, and for the Neumann…

We obtain the global weighted $W^{1,p}$ estimates for weak solutions of nonlinear elliptic equations over Reifenberg flat domains. Where nonlinearity $A(x,z,\xi)$ is assumed to be local uniform continuous in $z$ and have small BMO semi-norm…

In this article, the problems to be studied are the following \leqnomode \begin{equation*} \label{p} \left\{\begin{array}{ll} (-\Delta )_p^s u \pm \dfrac{|u|^{p-2}u}{|x|^{sp}} = \lambda f(x,u) & \quad \mbox{in }\ \Omega\\[0.3cm] u= 0 &…

We investigate the problem $$ \left\{ \begin{array}{ll} -\Delta_p u = g(u)|\nabla u|^p + f(x,u) \ & \mbox{in} \ \ \Omega, \ \ \\ u>0 \ &\mbox{in} \ \ \Omega, \ \ u = 0 \ &\mbox{on} \ \ \partial\Omega, \end{array} \right. \leqno{(P)} $$ in a…

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…

Lorentz and Lorentz-Morrey estimates are obtained for gradients of very weak solutions to quasilinear equations of the form $$\text{div}\,\mathcal{A}(x, \nabla u)=\text{div}\, |{\bf f}|^{p-2}{\bf f},$$ where $\text{div}\,\mathcal{A}(x,…

We derive a priori second order estimates for fully nonlinear elliptic equations which depend on the gradients of solutions in critical ways on Hermitian manifolds. The global estimates we obtained apply to an equation arising from a…

Global weighted $L^{p}$-estimates are obtained for the gradient of solutions to a class of linear singular, degenerate elliptic Dirichlet boundary value problems over a bounded non-smooth domain. The coefficient matrix is symmetric,…

We derive global gradient estimates for $W^{1,p}_0(\Omega)$-weak solutions to quasilinear elliptic equations of the form $$ \mathrm{div\,}\mathbf{a}(x,u,Du)=\mathrm{div\,}(|F|^{p-2}F) $$ over $n$-dimensional Reifenberg flat domains. The…

In this paper, positive solutions to the Laplace equation with 1-dimensional circular singularities are investigated. First, we establish $L^p$ integrability estimates for such solutions $u$ near the singularities, in comparison with…

In this article our main concern is to prove the quantitative unique estimates for the $p$-Laplace equation, $1<p<\infty$, with a locally Lipschitz drift in the plane. To be more precise, let $u\in W^{1,p}_{loc}(\mathbb{R}^2)$ be a…

We provide global gradient estimates for solutions to a general type of nonlinear parabolic equations, possibly in a Riemannian geometry setting. Our result is new in comparison with the existing ones in the literature, in light of the…

We extend the so-called universal potential estimates of the Kuusi-Mingione type (J.Funct. Anal. 2012) to the singular case $1<p\leq 2-1/n$ for the quasilinear equation with measure data \begin{equation*} -\operatorname{div}(A(x,\nabla…

We consider quasilinear parabolic equations with measurable coefficients when the right-hand side is a signed Radon measure with finite total mass, having $p$-Laplace type: $$u_t - \textrm{div} \, \mathbf{a}(Du,x,t) = \mu \quad \textrm{in}…

We study positive solutions of equation (E) $-\Delta u + u^p|\nabla u|^q= 0$ ($0<p$, $0\leq q\leq 2$, $p+q>1$) and other related equations in a smooth bounded domain $\Omega \subset {\mathbb R}^N$. We show that if $N(p+q-1)<p+1$ then, for…

We establish the local H\"older regularity of the spatial gradient of bounded weak solutions $u\colon E_T\to\R^k$ to the non-linear system of parabolic type \begin{equation*} \partial_tu-\Div\Big(…

In this paper we study $W^{1,p}$ global regularity estimates for solutions of $\Delta u = f$ on Riemannian manifolds. Under integral (lower) bounds on the Ricci tensor we prove the validity of $L^p$-gradient estimates of the form $|| \nabla…

In this paper, we study quasilinear elliptic equations with the nonlinearity modelled after the $p(x)$-Laplacian on nonsmooth domains and obtain sharp Calder\'on-Zygmund type estimates in the variable exponent setting. In a recent work of…