Related papers: Classification of solutions to equations involving…

In this article, we investigate the H\"{o}lder regularity of the fractional $p$-Laplace equation of the form $(-\Delta_p)^s u=f$ where $p>1, s\in (0, 1)$ and $f\in L^\infty_{\rm loc}(\Omega)$. Specifically, we prove that $u\in C^{0,…

We prove existence of solutions to a nonlinear degenerate elliptic equation of the form \[ \begin{cases} -\Delta_{1} u+ \frac{|D u|}{(1-u)^{\gamma}}=g & \mbox{in $\Omega$,}\\ u=0 \hfill & \mbox{on $\partial\Omega$,} \end{cases} \] in a…

We propose a new asymptotic expansion for the fractional $p$-Laplacian with precise computations of the errors. Our approximation is shown to hold in the whole range $p\in(1,\infty)$ and $s\in(0,1)$, with errors that do not degenerate as…

We deal with the higher-order fractional Laplacians by two methods: the integral method and the system method. The former depends on the integral equation equivalent to the differential equation. The latter works directly on the…

In this article we are interested in the following fractional $p$-Laplacian equation in $\mathbb{R}^n$ \begin{eqnarray*} &(-\Delta)_{p}^{\alpha}u + V(x)u^{p-2}u = f(x,u) \mbox{ in } \mathbb{R}^{n}, \end{eqnarray*} where $p\geq 2$, $0< s <…

We show an example of a non-symmetric integro-differential equation of order $\alpha$, for $\alpha \in (0,1)$, for which H\"older estimates do not hold even though the kernels are comparable to the fractional Laplacian.

In this paper, we investigate the following $D^{1,p}$-critical quasi-linear H\'enon equation involving $p$-Laplacian \begin{equation*}\label{00} \left\{ \begin{aligned} &-\Delta_p u=|x|^{\alpha}u^{p_\al^*-1}, & x\in \R^N, \\ &u>0, & x\in…

In this paper, we consider nonlinear equations involving the fractional p-Laplacian $$ (-\lap)_p^s u(x)) \equiv C_{n,s,p} PV \int_{\mathbb{R}^n} \frac{|u(x)-u(y)|^{p-2}[u(x)-u(y)]}{|x-z|^{n+ps}} dz= f(x,u).$$ We prove a {\em maximum…

Let $0<\alpha<2$ be any real number. In this paper, we investigate the following semilinear equations involving the fractional Laplacian \begin{equation}(-\bigtriangleup)^{\alpha/2} u(x)=f(u),\end{equation} on $\mathbb{R}^n$ and…

In this article, we prove existence results of positive solutions for the following nonlinear elliptic problem with gradient terms: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-\Delta)^\alpha u=f(x,u,v,\nabla u, \nabla v) &{\rm…

We establish the optimal regularity of solutions to the Neumann problem for the fractional Laplacian, $(-\Delta)^s u=h$ in $\Omega$, with the external condition $\mathcal N^s u=0$ in $\Omega^c$. For this, a key point is to establish a 1D…

Our purpose of this paper is to investigate positive solutions of the elliptic equation with regional fractional Laplacian $$ ( - \Delta )_{B_1}^s u +u= h(x,u) \quad {\rm in} \ \, B_1,\qquad u\in C_0(B_1), $$ where $( - \Delta )_{B_1}^s$…

The aim of this paper is to study radial symmetry and monotonicity properties for positive solution of elliptic equations involving the fractional Laplacian. We first consider the semi-linear Dirichlet problem (-\Delta)^{\alpha} u=f(u)+g,\…

In this paper, we study the following class of fractional Hamiltonian systems: \begin{eqnarray*} \begin{aligned}\displaystyle \left\{ \arraycolsep=1.5pt \begin{array}{ll} (-\Delta)^{\frac{1}{2}} u + u = \Big(I_{\mu_{1}}\ast G(v)\Big)g(v) \…

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 study the effect of lower order perturbations in the existence of positive solutions to the following critical elliptic problem involving the fractional Laplacian: (-\Delta)^{\alpha/2}u=\lambda u^q+u^{\frac{N+\alpha}{N-\alpha}}, \quad…

We study the fractional laplacian problem (-\Delta)^s u &=& u^p -\epsilon u^q \quad\text{in }\quad \Omega, u &\in& H^s(\Omega)\cap L^{q+1}(\Omega),u &>&0 \quad\text{in }\quad \Omega, u&=&0 \quad\text{in}\quad \mathbb{R}^N\setminus\Omega,…

We introduce a notion of fractional Laplacian for functions which grow more than linearly at infinity. In such case, the operator is not defined in the classical sense: nevertheless, we can give an ad-hoc definition which can be useful for…

We prove a nonlocal, nonlinear commutator estimate concerning the transfer of derivatives onto testfunctions. For the fractional $p$-Laplace operator it implies that solutions to certain degenerate nonlocal equations are higher…

We consider a semipositone problem involving the fractional $p$ Laplace operator of the form \begin{equation*} \begin{aligned} (-\Delta)_p^s u &=\mu( u^{r}-1) \text{ in } \Omega,\\ u &>0 \text{ in }\Omega,\\ u &=0 \text{ on }\Omega^{c},…