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

Abstract

This paper intend to study the following critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields in R3{{\mathbb{R}}}^{3}: ε2sM([u]s,A2)(−Δ)Asu+V(x)u+(∣x∣2t−3∗∣u∣2)u=f(x,∣u∣2)u+∣u∣2s∗−2u,x∈R3.{\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.

Keywords

• fractional schrödinger-kirchhoff-poisson equations
• fractional magnetic operator
• critical nonlinearity
• variational methods
• 35a15
• 35b99
• 35j60
• 47g20