On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields

Abstract

Abstract This paper intend to study the following critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields in R 3 {{\mathbb{R}}}^{3} : ε 2 s M ( [ u ] s , A 2 ) ( − Δ ) A s u + V ( x ) u + ( ∣ x ∣ 2 t − 3 ∗ ∣ u ∣ 2 ) u = f ( x , ∣ u ∣ 2 ) u + ∣ u ∣ 2 s ∗ − 2 u , x ∈ R 3 . {\varepsilon }^{2s}{\mathfrak{M}}\left({\left[u]}_{s,A}^{2}){\left(-\Delta )}_{A}^{s}u+V\left(x)u+\left(| x\hspace{-0.25em}{| }^{2t-3}\ast | u\hspace{-0.25em}{| }^{2})u=f\left(x,| u\hspace{-0.25em}{| }^{2})u+| u\hspace{-0.25em}{| }^{{2}_{s}^{\ast }-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{3}. Under suitable assumptions, together with the concentration compactness principle and variational method, we prove that the existence and multiplicity of semiclassical solutions for above problem as ε → 0 \varepsilon \to 0 .