## A fractional Kirchhoff problem involving a singular term and a critical nonlinearity.(English)Zbl 1419.35035

Summary: In this paper, we consider the following critical nonlocal problem:
$\begin{cases} M\bigg(\iint_{\mathbb{R}^{2N}}\frac{ \vert u(x)-u(y)\vert^{2}}{\vert x-y\vert^{N+2s}}\,dx\,dy\bigg)(-\Delta)^{s}u=\frac{\lambda}{u^{\gamma}}+u^{2^{*}_{s}-1}\quad&\text{in }\Omega, \\ u>0&\text{in }\Omega, \\ u=0&\text{in }\mathbb{R}^{N}\setminus\Omega,\end{cases}$
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, $$\gamma\in(0,1)$$ 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.

### MSC:

 35J60 Nonlinear elliptic equations 35R11 Fractional partial differential equations 35A15 Variational methods applied to PDEs
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### References:

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