Fractional Choquard-Kirchhoff problems with critical nonlinearity and Hardy potential

Abstract

In this paper, we investigate the following fractional p-Kirchhoff type problem

$$\begin{aligned} \left\{ \begin{array}{ll} \left( a+b[u]_{s,p}^{p(\theta -1)}\right) (-\Delta )^s_pu = \Big ({\mathcal {I}}_\mu *|u|^q\Big )|u|^{q-2}u+\frac{|u|^{p_{\alpha }^*-2}u}{|x|^\alpha },\ u>0, &{}\text{ in }\ \Omega ,\\ u=0, \ &{} \mathrm{in}\ {\mathbb {R}}^N\backslash \Omega , \end{array} \right. \end{aligned}$$

where $[u]_{s,p}^{p}=\displaystyle \iint _{{\mathbb {R}}^{2N}} \frac{|u(x) - u(y)|^{p}}{|x - y|^{N+ps}}\, dxdy$, $\Omega$ is a bounded smooth domain of ${\mathbb {R}}^N$ containing 0 with Lipschitz boundary, $(-\Delta )_{p}^{s}$ denotes the fractional p-Laplacian, $0\le \alpha<ps<N$ with $s\in (0,1)$, $p>1$, $a\ge 0$, $b>0$, $1<\theta \le p_\alpha ^*/ p$, $p_\alpha ^*=\frac{(N-\alpha )p}{N-ps}$ is the fractional critical Hardy-Sobolev exponent, ${\mathcal {I}}_\mu (x)=|x|^{-\mu }$ is the Riesz potential of order $\mu \in (0,\min \{N,2ps\})$, $q\in (1, Np/(N-ps))$ satisfies some restrictions. By the concentration-compactness principle and mountain pass theorem, we obtain the existence of a positive weak solution for the above problem as q satisfies suitable ranges.