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 $0<s<1$; $0<\alpha ,\beta <sp+\theta <N$; $0<\mu <N$; $2\delta +\mu <N$; $\gamma <{\gamma }_H$ with the best fractional Hardy constant ${\gamma }_H$; the Hardy-Sobolev and Stein-Weiss upper critical fractional exponents are respectively defined by $p_s^{\ast }(\beta ,\theta ):=p(N-\beta )/(N-sp-\theta )$, and $p_s^{♯}(\delta ,\theta ,\mu ):=p(N-\delta -\mu /2)/(N-sp-\theta )$. Moreover, $I_{\mu }(x)=|x{|}^{-\mu }$ is the Riesz potencial; $f_{\delta ,\theta ,\mu }(x,t):=|x{|}^{-\delta }|t{|}^{p_s^{♯}(\delta ,\theta ,\mu )-2}t$ and $F_{\delta ,\theta ,\mu }(x,t):=|x{|}^{\delta }|t{|}^{p_s^{♯}(\delta ,\theta ,\mu )}$; 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.

40 pages, original research article