Multiple solutions for a critical Kirchhoff–Choquard type equation involving fractional p-Laplacian in RN

We are concerned with a critical Kirchhoff–Choquard type equation involving the fractional $p$-Laplacian in $R^N$ $M\left({\Vert u\Vert }^p\right)\left({(-\Delta )}_p^{s}u+V\left(x\right){\left|u\right|}^{p-2}u\right)=\frac{\lambda }{2p_{\mu ,s}^{\ast }}{p}_s^{\ast }\left(\frac{\mu }{q_2}\right)\left(K_{\mu }\ast {\left|u\right|}^{p_s^{\ast }\left(\frac{\mu }{q_1}\right)}\right){\left|u\right|}^{p_s^{\ast }\left(\frac{\mu }{q_2}\right)-2}u+\frac{\lambda }{2p_{\mu ,s}^{\ast }}{p}_s^{\ast }\left(\frac{\mu }{q_1}\right)\left(K_{\mu }\ast {\left|u\right|}^{p_s^{\ast }\left(\frac{\mu }{q_2}\right)}\right){\left|u\right|}^{p_s^{\ast }\left(\frac{\mu }{q_1}\right)-2}u+b\frac{f(x,u)}{{\left|x\right|}^{\alpha }}.$Based on a new version of the concentration compactness alternative, we obtain the multiplicity and concentration property of solutions for the problem when $f$ ($x,u$) = $h$ ($x$)${\left|u\right|}^{q-1}u$. For $q\in$ ($p,p_s^{\ast }$ ($\alpha$)], we establish desired results by applying Benci’s pseudo-index theory and some new techniques, where the much more difficult doubly critical case of $q=p_s^{\ast }$ ($\alpha$) is considered, which is a hard problem and has rarely been studied previously. Moreover, for the case of $q=p$ , we prove the existence of infinitely many solutions for the problem above by means of the truncation function method and Krasnoselskii’s genus theory. For the case of $q\in$ ($1,p$), we also obtain a result on the infinitely many solutions in a general setting in view of the dual fountain theorem.