Related papers: Certain fractional Laplacian equations that do not…

In this work we analyze the existence of solutions to the fractional quasilinear problem, $$ (P) \left\{ \begin{array}{rcll} u_t+(-\Delta )^s u &=&|\nabla u|^{\alpha}+ f &\inn \Omega_T\equiv\Omega\times (0,T),\\ u(x,t)&=&0 &…

In this paper, we consider the fractional elliptic equation \begin{align*} \left\{\begin{aligned} &(-\Delta)^s u-\mu\frac{u}{|x|^{2s}} = \frac{|u|^{2_s^\ast (\alpha)-2}u}{|x|^{\alpha}} + f(x,u), && \mbox{in} \ \Omega,\\ &u=0, && \mbox{in} \…

We study the regularity properties of the solutions to the nonlinear equation with fractional diffusion $$ \partial_tu+(-\Delta)^{\sigma/2}\varphi(u)=0, $$ posed for $x\in \mathbb{R}^N$, $t>0$, with $0<\sigma<2$, $N\ge1$. If the…

In this contribution, we study a class of doubly nonlinear elliptic equations with bounded, merely integrable right-hand side on the whole space $\mathbb{R}^N$. The equation is driven by the fractional Laplacian $(-\Delta)^{\frac{s}{2}}$…

We study the existence of non-zero positive solutions of a class of systems of differential equations driven by fractional powers of the Laplacian. Our approach is based on the notion of fixed point index, and allows us to deal with…

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…

The present paper studies the non-local fractional analogue of the famous paper of Brezis and Nirenberg in [4]. Namely, we focus on the following model, $$\begin{align*}\left(\mathcal{P}\right) \begin{cases} \left(-\Delta\right)^s u-\lambda…

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,\…

This paper works on the direct method of moving spheres and establishes a Liouville-type theorem for the fractional elliptic equation \[ (-\Delta)^{\alpha/2} u =f(u) ~~~~~~ \text{in } \mathbb{R}^{n} \] with general non-linearity. One of the…

Inspired by some experimental (numerical) works on fractional diffusion PDEs, we develop a rigorous framework to prove that solutions to the fractional Navier-Stokes equations, which involve the fractional Laplacian operator…

In this paper, we study the existence and nonexistence of solutions for the following Kirchhoff-type fractional $(p\text{-}q)$-Laplacian problem: \begin{equation*} \begin{cases} M\left([u]^p_{p,s_1}\right)(-\Delta)^{s_1}_p u +…

In this paper, first a class of fractional differential equations are obtained by using the fractional variational principles. We find a fractional Lagrangian $L(x(t)$, where $_a^cD_t^\alpha x(t))$ and $0<\alpha< 1$, such that the following…

We face a rigidity problem for the fractional $p$-Laplace operator to extend to this new framework some tools useful for the linear case. It is known that $(-\Delta)^s(1-|x|^{2})^s_+$ and $-\Delta_p(1-|x|^{\frac{p}{p-1}})$ are constant…

We formulate a numerical method to solve the porous medium type equation with fractional diffusion \[\frac{\partial u}{\partial t}+(-\Delta)^{1/2} (u^m)=0.\] The problem is posed in $x\in \mathbb{R}^N$, $m\geq 1$ and with nonnegative…

In this paper, we study the existence and uniqueness of positive solutions for the following nonlinear fractional elliptic equation: \begin{eqnarray*} (-\Delta)^\alpha u=\lambda a(x)u-b(x)u^p&{\rm in}\,\,\R^N, \end{eqnarray*} where $…

We study positive solutions to the fractional semi-linear elliptic equation $$ (- \Delta)^\sigma u = K(x) u^\frac{n + 2 \sigma}{n - 2 \sigma} ~~~~~~ in ~ B_2 \setminus \{ 0 \} $$ with an isolated singularity at the origin, where $K$ is a…

The fractional Laplacian $(-\Delta)^{\alpha/2}$ is a non-local operator which depends on the parameter $\alpha$ and recovers the usual Laplacian as $\alpha \to 2$. A numerical method for the fractional Laplacian is proposed, based on the…

We describe a duality method to prove both existence and uniqueness of solutions to nonlocal problems like $$ (-\Delta)^s v = \mu \quad \text{in}\ \mathbb{R}^N, $$ with vanishing conditions at infinity. Here $\mu$ is a bounded Radon measure…

In this article, we prove the existence of solutions to a nonlinear nonlocal elliptic problem with a singualrity and a discontinuous critical nonlinearity which is given as follows. \begin{align} \begin{split}\label{main_prob}…

In this paper, we establish Liouville type theorems for stable solutions on the whole space $\mathbb R^N$ to the fractional elliptic equation $$(-\Delta)^su=f(u)$$ where the nonlinearity is nondecreasing and convex. We also obtain a…