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

Abstract

Abstract In this article, we deal with the following p p -fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity: M ( [ u ] s , A p ) ( − Δ ) p , A s u + V ( x ) ∣ u ∣ p − 2 u = λ ∫ R N ∣ u ∣ p μ , s * ∣ x − y ∣ μ d y ∣ u ∣ p μ , s * − 2 u + k ∣ u ∣ q − 2 u , x ∈ R N , M({\left[u]}_{s,A}^{p}){\left(-\Delta )}_{p,A}^{s}u+V\left(x){| u| }^{p-2}u=\lambda \left(\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{{| u| }^{{p}_{\mu ,s}^{* }}}{{| x-y| }^{\mu }}{\rm{d}}y\right){| u| }^{{p}_{\mu ,s}^{* }-2}u+k{| u| }^{q-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N}, where 0 < s < 1 < p 0\lt s\lt 1\lt p , p s < N ps\lt N , p < q < 2 p s , μ * p\lt q\lt 2{p}_{s,\mu }^{* } , 0 < μ < N 0\lt \mu \lt N , λ \lambda , and k k are some positive parameters, p s , μ * = p N − p μ 2 N − p s {p}_{s,\mu }^{* }=\frac{pN-p\frac{\mu }{2}}{N-ps} is the critical exponent with respect to the Hardy-Littlewood-Sobolev inequality, and functions V V and M M satisfy the suitable conditions. By proving the compactness results using the fractional version of concentration compactness principle, we establish the existence of nontrivial solutions to this problem.