Fractional Kirchhoff Hardy problems with singular and critical Sobolev nonlinearities

Abstract

The paper deals with the following singular fractional problem

$$\begin{aligned} \left\{ \begin{array}{lll} M\left( \displaystyle \iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right) (-\Delta )^{s} u-\mu \displaystyle \frac{u}{|x|^{2s}}= \lambda f(x)u^{-\gamma }+ g(x){u^{2^*_s-1}}&{}\;\; \text {in}\; \Omega ,\\ u>0&{} \;\; \text {in}\; \Omega ,\\ u=0&{}\;\;\text {in}\;{\mathbb {R}}^N\setminus \Omega , \end{array}\right. \end{aligned}$$

where $\Omega \subset {\mathbb {R}}^N$ is an open bounded domain, with $0\in \Omega$, dimension $N>2s$ with $s\in (0,1)$, $2^*_s=2N/(N-2s)$ is the fractional critical Sobolev exponent, $\lambda$ and $\mu$ are positive parameters, exponent $\gamma \in (0,1)$, M models a Kirchhoff coefficient, f is a positive weight while g is a sign-changing function. The main feature and novelty of our problem is the combination of the critical Hardy and Sobolev nonlinearities with the bi-nonlocal framework and a singular nondifferentiable term. By exploiting the Nehari manifold approach, we provide the existence of at least two positive solutions.