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

Abstract

Abstract In this paper, we consider the following critical nonlocal problem: \left\{\begin{aligned} &\displaystyle M\bigg{(}\iint_{\mathbb{R}^{2N}}\frac{% \lvert u(x)-u(y)\rvert^{2}}{\lvert x-y\rvert^{N+2s}}\,dx\,dy\biggr{)}(-\Delta)% ^{s}u=\frac{\lambda}{u^{\gamma}}+u^{2^{*}_{s}-1}&&\displaystyle\phantom{}\text% {in }\Omega,\\ \displaystyle u&\displaystyle>0&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}^{% N}\setminus\Omega,\end{aligned}\right. where Ω 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, {\gamma\in(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 solutions.