Related papers: Gradient estimates for the fractional $p$-Poisson …

We prove a Hopf's lemma in the point-wise sense. The essential technique is to prove $(-\Delta)^s_p u(x)$ is uniformly bounded in the unit ball $B_1\subset\mathbb{R}^n$, where $u(x)=(1-|x|^2)^s_{+}$. Also we study the global H\"older…

We prove the existence of an unbounded branch of solutions to the non-linear non-local equation $$ (-\Delta)^s_p u=\lambda |u|^{p-2}u + f(x,u,\lambda) \quad\text{in}\quad \Omega,\quad u=0 \quad\text{in}\quad \mathbb{R}^n\setminus\Omega, $$…

We consider energy solutions of the inhomogeneous parabolic $p$-Laplacien system $\partial_t u-\text{div}(|D u|^{p-2}D u)=-\text{div} g$. We show in the case $p\geq 2$ that if the right hand side $g$ is locally in $L^\infty(\text{BMO})$,…

For positive $p$-harmonic functions on Riemannian manifolds, we derive a gradient estimate and Harnack inequality with constants depending only on the lower bound of the Ricci curvature, the dimension $n$, $p$ and the radius of the ball on…

We deal with a wide class of nonlinear nonlocal equations led by integro-differential operators of order $(s,p)$, with summability exponent $p \in (1,\infty)$ and differentiability exponent $s\in (0,1)$, whose prototype is the fractional…

In this paper, we establish $L_p$ estimates and solvability for time fractional divergence form parabolic equations in the whole space when leading coefficients are merely measurable in one spatial variable and locally have small mean…

In this paper, we establish gradient bounds for $p(\cdot)$-harmonic differential forms subject to a Coulomb-type gauge condition. For variable exponents satisfying the log-H\"older continuity assumption, we derive higher integrability…

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…

We deal with a class of equations driven by nonlocal, possibly degenerate, integro-differential operators of differentiability order $s\in (0,1)$ and summability growth $p>1$, whose model is the fractional $p$-Laplacian with measurable…

In this article, we deal with the global regularity of weak solutions to a class of problems involving the fractional $(p,q)$-Laplacian, denoted by $(-\Delta)^{s_1}_{p}+(-\Delta)^{s_2}_{q}$, for $s_2, s_1\in (0,1)$ and $1<p,q<\infty$. We…

We study the behavior of weak solutions to the singular quasilinear elliptic problem $-\Delta_p u + \vartheta |\nabla u|^q = \frac{1}{u^\gamma} + f(u)$, in a bounded domain with the Dirichlet boundary condition, where $p>1$, $\gamma>0$,…

We study quasilinear evolutionary partial integro-differential equations of second order which include time fractional $p$-Laplace equations of time order less than one. By means of suitable energy estimates and De Giorgi's iteration…

Under a condition of the Calderon type on $\varphi$, we show that a homeomorphism $f$ of finite distortion in $W^{1,\varphi}_{\rm loc}$ and, in particular, $f\in W^{1,q}_{\rm loc}$ for $q>n-1$ in ${\Bbb R}^n$, $n\ge 3$, is a lower…

This paper studies global a priori gradient estimates for divergence-type equations patterned over the $p$-Laplacian with first-order terms having polynomial growth with respect to the gradient, under suitable integrability assumptions on…

This note is a continuation of the work \cite{CaoXiangYan2014}. We study the following quasilinear elliptic equations \[ -\Delta_{p}u-\frac{\mu}{|x|^{p}}|u|^{p-2}u=Q(x)|u|^{\frac{Np}{N-p}-2}u,\quad\, x\in\mathbb{R}^{N}, \] where…

In this article, first we address the regularity of weak solution for a class of $p$-fractional Choquard equations: \begin{equation*} \;\;\; \left.\begin{array}{rl}…

In this paper, we investigate the existence and uniqueness of a non-trivial solution for a class of nonlocal equations involving the fractional $p$-Laplacian operator defined on compact Riemannian manifold, namely,…

We review recent progress in the fractional Calder\'on problem, where one tries to determine an unknown coefficient in a fractional Schr\"odinger equation from exterior measurements of solutions. This equation enjoys remarkable uniqueness…

For a smooth bounded domain $\Omega$ and $p \geq q \geq 2$, we establish quantified versions of the classical Friedrichs inequality $\|\nabla u\|_p^p - \lambda_1 \|u\|_q^p \geq 0$, $u \in W_0^{1,p}(\Omega)$, where $\lambda_1$ is a…

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…