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