Existence of solutions for critical Choquard problem with singular coefficients

Abstract

In this paper, we investigate the following fractional Choquard type equation: \[ (- \Delta)_p^s\, u = \lambda\frac{|u|^{r-2}u}{|x|^\alpha}\,+\gamma \big(\int_\Omega \frac{|u|^q}{|x-y|^\mu}dy\big) |u|^{q-2}u \ \ \text{in } \Omega,\ \ u = 0 \ \text{in } \R^N \setminus \Omega, \] where $\Omega$ is a bounded domain in $\R^N$ with Lipschitz boundary, $p>1$, $0<s<1$, $N>sp$, $0\leq\alpha\leq sp$, $0<\mu<N$,$\lambda, \gamma>0$, $p\leq r\leq p^*_\alpha$, $p\leq 2q\leq 2p_{\mu,s}^*$, $p_\alpha^*=\frac{(N-\alpha)p}{N-sp}$ and $p_{\mu,s}^*=\frac{(N-\frac{\mu}{2})p}{N-sp}$ 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.