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

Abstract

In this paper we consider the following critical nonlocal problem $$ \left\{\begin{array}{ll} M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right)(-Δ)^s u = \displaystyle\fracλ{u^γ}+u^{2^*_s-1}&\quad\mbox{in } Ω,\\ u>0&\quad\mbox{in } Ω,\\ u=0&\quad\mbox{in } \mathbb{R}^N\setminusΩ, \end{array}\right. $$ where $\Omega$ is an open bounded subset of ${\mathbb{R}}^N$ with continuous boundary, dimension $N>2s$ with parameter $s\in (0,1)$, $2_s^{\ast }=2N/(N-2s)$ 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.