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

Abstract

In this paper we consider a fractional $p$-Laplacian equation in the entire space $\mathbb{R}^{N}$ with doubly critical singular nonlinearities involving a local critical Sobolev term together with a nonlocal Choquard critical term; the problem also includes a homogeneous singular Hardy term. More precisely, we deal 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 the best fractional Hardy constant $\gamma_{H}$; the Hardy-Sobolev and Stein-Weiss upper 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 convolution integral is known as Choquard type nonlinearity. To prove the main result we have to show new embeddings involving the weighted Morrey spaces and a version of the Caffarelli-Kohn-Nirenberg inequality. With the help of these new embedding results, we provide sufficient conditions under which a weak nontrivial solution to the problem exists via variational methods.