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

Abstract

In this article, we deal with the following $p$-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity: $$M\left([u{]}_{s,A}^p\right)(-\Delta {)}_{p,A}^{s}u+V(x)|u{|}^{p-2}u=\lambda \left({\int }_{{\mathbb{R}}^N}\frac{|u{|}^{p_{\mu ,s}^{\ast }}}{|x-y{|}^{\mu }}\mathrm{d}y\right)|u{|}^{p_{\mu ,s}^{\ast }-2}u+k|u{|}^{q-2}u,x\in {\mathbb{R}}^N,$$ 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.

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