# 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.

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## Acknowledgements

A. Fiscella is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica “G. Severi" (INdAM). A. Fiscella realized the manuscript within the auspices of the INdAM-GNAMPA project titled Equazioni alle derivate parziali: problemi e modelli (Prot_20191219-143223-545), of the FAPESP Project titled Operators with non standard growth (2019/23917-3), of the FAPESP Thematic Project titled Systems and partial differential equations (2019/02512-5) and of the CNPq Project titled Variational methods for singular fractional problems (3787749185990982).

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Correspondence to Alessio   Fiscella.