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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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