Existence of solutions for critical Choquard problem with singular coefficients

Abstract

In this paper, we investigate the following fractional Choquard type equation: \[ (- Δ)_p^s\, u = λ\frac{|u|^{r-2}u}{|x|^α}\,+γ\big(\int_Ω\frac{|u|^q}{|x-y|^μ}dy\big) |u|^{q-2}u \ \ \text{in } Ω,\ \ u = 0 \ \text{in } \R^N \setminus Ω, \] where $\hat{I}\text{c◯}$ is a bounded domain in ${\mathbb{R}}^N$ with Lipschitz boundary, $p>1$, $0<s<1$, $N>sp$, $0\le \hat{I}\pm \le sp$, $0<\hat{I}¼<N$,$\hat{I}»,{\hat{I}}^3>0$, $p\le r\le p_{\hat{I}}^{\ast }\pm$, $p\le 2q\le 2p_{\hat{I}¼,s}^{\ast }$, $p_{\hat{I}}{\pm }^{\ast }=\frac{(N-\hat{I}\pm )p}{N-sp}$ and $p_{\hat{I}¼,s}^{\ast }=\frac{(N-\frac{\hat{I}}{¼}2)p}{N-sp}$ are the fractional critical Hardy-Sobolev and the critical exponents in the sense of Hardy-Littlewood-Sobolev inequality, respectively. Under some suitable assumptions, positive and sign-changing solutions are obtained.