Related papers: Sharp regularity estimates for quasi-linear ellipt…

In this work, we establish universal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations with oblique boundary conditions, whose general model is given by $$ \left\{ \begin{array}{rcl} F(D^2u,x) &=& f(x) \quad…

We prove sharp regularity estimates for solutions of obstacle type problems driven by a class of degenerate fully nonlinear operators; more specifically, we consider viscosity solutions of \[ |D u|^\gamma F(x, D^2u) = f(x)\chi_{\{u>\phi\}}…

We give a survey of nonlinear potential estimates and their applications obtained recently for positive solutions to sublinear problems of the type \[ u = \mathbf{G}(\sigma u^q) + f \quad \textrm{in} \,\, \Omega, \] where $0 < q < 1$,…

In this article we study the quasi-linear equation \[ \left\{ \begin{aligned} \mathrm{div}\, \mathcal A(x,u,\nabla u)&=\mathcal B(x,u,\nabla u)&&\text{in }\Omega,\\ u\in H^{1,p}_{loc}&(\Omega;wdx) \end{aligned} \right. \] where $\mathcal A$…

This paper investigates the relation between the boundary geometric properties and the boundary regularity of the solutions of elliptic equations. We prove by a new unified method the pointwise boundary H\"{o}lder regularity under proper…

The aim of this paper is to present the global estimate for gradient of renormalized solutions to the following quasilinear elliptic problem: \begin{align*} \begin{cases} -div(A(x,\nabla u)) &= \mu \quad \text{in} \ \ \Omega, \\ u &=0 \quad…

The authors of this paper deal with the existence and regularities of weak solutions to the homogenous $\hbox{Dirichlet}$ boundary value problem for the equation $-\hbox{div}(|\nabla u|^{p-2}\nabla u)+|u|^{p-2}u=\frac{f(x)}{u^{\alpha}}$.…

In this paper, we establish the existence of a solution for a class of quasilinear equations characterized by the prototype: \begin{equation} \left\{\begin{aligned} -\operatorname{div}(\vartheta_\alpha|\nabla u|^{p-2} \nabla…

We study local boundedness for subsolutions of nonlinear nonuniformly elliptic equations whose prototype is given by $\nabla \cdot (\lambda |\nabla u|^{p-2}\nabla u)=0$, where the variable coefficient $0\leq\lambda$ and its inverse…

We prove sharp asymptotic estimates for the gradient of positive solutions to certain nonlinear $p$-Laplace equations in Euclidean space by showing symmetry and uniqueness of positive solutions to associated limiting problems.

This paper investigates the regularity of Lipschitz solutions $u$ to the general two-dimensional equation $\text{div}(G(Du))=0$ with highly degenerate ellipticity. Just assuming strict monotonicity of the field $G$ and heavily relying on…

In this work, we establish sharp and improved regularity estimates for viscosity solutions of Hardy-H\'{e}non-type equations with possibly singular weights and strong absorption governed by the $\infty$-Laplacian $$ \Delta_{\infty} u(x) =…

We study the monotonicity and one-dimensional symmetry of positive solutions to the problem $-\Delta_p u = f(u)$ in $\mathbb{R}^N_+$ under zero Dirichlet boundary condition, where $p>1$ and $f:(0,+\infty)\to\mathbb{R}$ is a locally…

We investigate nonnegative solutions of indefinite elliptic problems which enjoy the dead core phenomenon. Our model is the subhomogeneous problem $$ -\Delta_p u = (a^+(x) - \mu a^-(x))|u|^{q-2}u, \quad u \in W_0^{1,p}(\Omega), $$ where…

This paper studies the Sobolev regularity estimates for weak solutions of a class of degenerate, and singular quasi-linear elliptic problems of the form $\text{div}[\mathbf{A}(x,u, \nabla u)]= \text{div}[\mathbf{F}]$ with non-homogeneous…

We give necessary and sufficient conditions for the existence of weak solutions to the model equation $$-\Delta_p u=\sigma \, u^q \quad \text{on} \, \, \, \R^n,$$ in the case $0<q<p-1$, where $\sigma\ge 0$ is an arbitrary locally integrable…

We introduce a new class of quasi-linear parabolic equations involving nonhomogeneous degeneracy or/and singularity $$ \partial_t u=[|D u|^q+a(x,t)|D u|^s]\left(\Delta u+(p-2)\left\langle D^2 u\frac{D u}{|D u|},\frac{D u}{|D…

It has been well known that if $\Omega$ is a bounded $C^1$-domain in $\R^n,\ n \ge 2$, then for every Radon measure $f$ on $\Omega$ with finite total variation, there exists a unique weak solution $u\in W_0^{1,1}(\Omega )$ of the Poisson…

In this paper we are concerned with higher regularity properties of the elliptic system \[ \Delta\mathbf{u}= |\mathbf{u}|^{q-1}\mathbf{u}\chi_{\{|\mathbf{u}|>0\}},\qquad\mathbf{u}=(u^1,\dots,u^m) \] for $0\leq q<1$. We show analyticity of…

We study positive solutions to the problem $-\Delta_p u + \vartheta |\nabla u|^q = \frac{1}{u^\gamma} + f(u)$ in $\mathbb{R}^N_+$ with the zero Dirichlet boundary condition, where $p>1$, $\gamma>0$, $0<q\le p$, $\vartheta\ge0$ and…