On the $p$-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity

In this article, we deal with the following $p$-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity: $$M([u{]}_{s,A}^p)(-\mathrm{\Delta }{)}_{p,A}^{s}u+V(x)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u=\lambda \left({\int }_{{\mathbb{R}}^N}\frac{\mathrm{\mid }u{\mathrm{\mid }}^{p_{\mu ,s}^{\ast }}}{\mathrm{\mid }x-y{\mathrm{\mid }}^{\mu }}\mathrm{d}y\right)\mathrm{\mid }u{\mathrm{\mid }}^{p_{\mu ,s}^{\ast }-2}u+k\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u,x\in {\mathbb{R}}^N,$$ where $0<s<1<p$, $ps<N$, $p<q<2p_{s,\mu }^{\ast }$, $0<\mu <N$, $\lambda$ and $k$ are some positive parameters, $p_{s,\mu }^{\ast }=\frac{pN-p\frac{\mu }{2}}{N-ps}$ is the critical exponent with respect to the Hardy-Littlewood-Sobolev inequality, and functions $V$, $M$ satisfy the suitable conditions. By proving the compactness results with the help of the fractional version of concentration compactness principle, we establish the existence of nontrivial solutions to this problem.