A fractional Kirchhoff problem involving a singular term and a critical nonlinearity

In this paper we consider the following critical nonlocal problem \left\{\begin{array}{ll} M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right)(-\Delta)^s u = \displaystyle\frac{\lambda}{u^\gamma}+u^{2^*_s-1}&\quad\mbox{in } \Omega,\\ u>0&\quad\mbox{in } \Omega,\\ u=0&\quad\mbox{in } \mathbb{R}^N\setminus\Omega, \end{array}\right. where $\Omega$ is an open bounded subset of $\mathbb R^N$ with continuous boundary, dimension $N>2s$ with parameter $s\in (0,1)$, $2^*_s=2N/(N-2s)$ is the fractional critical Sobolev exponent, $\lambda>0$ is a real parameter, exponent $\gamma\in(0,1)$, $M$ models a Kirchhoff type coefficient, while $(-\Delta)^s$ is the fractional Laplace operator. In particular, we cover the delicate degenerate case, that is when the Kirchhoff function $M$ is zero at zero. By combining variational methods with an appropriate truncation argument, we provide the existence of two solutions

A fractional Kirchhoff problem involving a singular term and a critical nonlinearity

In this paper, we consider the following critical nonlocal problem: { M( integral(R)integral(2N) vertical bar u(x) - u(y)vertical bar(2)/vertical bar x - y vertical bar(N+2s) dxdy)(-Delta)(s)u = lambda/u(gamma) + u(2s)*(-1) in Omega, u > 0 in Omega, u = 0 in R-N \ Omega, where Omega is an open bounded subset of R-N with continuous boundary, dimension N > 2s with parameter s is an element of (0, 1), 2(s)* = 2N/(N - 2s) is the fractional critical Sobolev exponent, lambda > 0 is a real parameter, gamma is an element of (0, 1) and M models a Kirchhoff-type coefficient, while (-Delta)(s) is the fractional Laplace operator. In particular, we cover the delicate degenerate case, that is, when the Kirchhoff function M is zero at zero. By combining variational methods with an appropriate truncation argument, we provide the existence of two solutions81645660COORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL DE NÍVEL SUPERIOR - CAPES33003017003P

Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity

This paper deals with the existence and the asymptotic behavior of non-negative solutions for a class of stationary Kirchhoff problems driven by a fractional integro-differential operator $\mathcal L_K$ and involving a critical nonlinearity. The main feature, as well as the main difficulty, of the analysis is the fact that the Kirchhoff function $M$ can be zero at zero, that is the problem is degenerate. The adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature

Gevrey regularity for integro-differential operators

We prove for some singular kernels $K(x,y)$ that viscosity solutions of the integro-differential equation $\int_{\mathbb{R}^n} \left[u(x+y)+u(x-y)-2u(x)\right]\,K(x,y)dy=f(x)$ locally belong to some Gevrey class if so does $f$. The fractional Laplacian equation is included in this framework as a special case.Comment: 15 page