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In this paper we consider a fractional $p$-Laplacian equation in the entirespace $\mathbb{R}^{N}$ with doubly critical singular nonlinearities involving alocal critical Sobolev term together with a nonlocal Choquard critical term;the problem also includes a homogeneous singular Hardy term. More precisely, wedeal with the problem \begin{align*}  \begin{cases}  (-\Delta)^{s}_{p,\theta} u  -\gamma \dfrac{|u|^{p-2}u}{|x|^{sp+ \theta}}  = \dfrac{|u|^{p^*_s(\beta,\theta)-2}u}%  {|x|^{\beta}}  + \left[  I_{\mu} \ast F_{\delta,\theta,\mu}(\cdot, u)  \right](x)f_{\delta,\theta,\mu}(x,u)  u \in \dot{W}^{s,p}_{\theta}(\mathbb{R}^N)  \end{cases} \end{align*} where $0 < s < 1$; $0 < \alpha, \,\beta < sp +\theta < N$; $0 < \mu < N$; $2\delta + \mu < N$; $\gamma < \gamma_{H}$ with thebest fractional Hardy constant $\gamma_{H}$; the Hardy-Sobolev and Stein-Weissupper critical fractional exponents are respectively defined by$p^*_s(\beta,\theta) := p(N-\beta)/(N-sp-\theta)$, and$p^\sharp_s(\delta,\theta,\mu) := p(N-\delta-\mu/2)/(N-sp-\theta)$. Moreover,$I_{\mu}(x) =|x|^{-\mu}$ is the Riesz potencial; $f_{\delta,\theta,\mu}(x,t) :=|x|^{-\delta} |t|^{p^{\sharp}_{s}(\delta,\theta,\mu)-2}t$  and $F_{\delta,\theta,\mu}(x,t) := |x|^{\delta}|t|^{p^{\sharp}_{s}(\delta,\theta,\mu)}$; and the term with convolutionintegral is known as Choquard type nonlinearity. To prove the main result wehave to show new embeddings involving the weighted Morrey spaces and a versionof the Caffarelli-Kohn-Nirenberg inequality. With the help of these newembedding results, we provide sufficient conditions under which a weaknontrivial solution to the problem exists via variational methods.

Fractional Sobolev-Chocard critical equation with Hardy term and weighted singularities