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} (-Δ)^{s}_{p,θ} u -γ\dfrac{|u|^{p-2}u}{|x|^{sp+ θ}} = \dfrac{|u|^{p^*_s(β,θ)-2}u}% {|x|^β} + \left[ I_μ \ast F_{δ,θ,μ}(\cdot, u) \right](x)f_{δ,θ,μ}(x,u) u \in \dot{W}^{s,p}_θ(\mathbb{R}^N) \end{cases} \end{align*} where $0 < s < 1$; $0 < α, \,β< sp + θ< N$; $0 < μ< N$; $2δ+ μ< N$; $γ< γ_{H}$ with the best fractional Hardy constant $γ_{H}$; the Hardy-Sobolev and Stein-Weiss upper critical fractional exponents are respectively defined by $p^*_s(β,θ) := p(N-β)/(N-sp-θ)$, and $p^\sharp_s(δ,θ,μ) := p(N-δ-μ/2)/(N-sp-θ)$. Moreover, $I_μ(x) =|x|^{-μ}$ is the Riesz potencial; $f_{δ,θ,μ}(x,t) := |x|^{-δ} |t|^{p^{\sharp}_{s}(δ,θ,μ)-2}t$ and $F_{δ,θ,μ}(x,t) := |x|^δ |t|^{p^{\sharp}_{s}(δ,θ,μ)}$; 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.