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 pp-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity: M([u]s,Ap)(−Δ)p,Asu+V(x)∣u∣p−2u=λ∫RN∣u∣pμ,s*∣x−y∣μdy∣u∣pμ,s*−2u+k∣u∣q−2u,x∈RN,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<p0\lt s\lt 1\lt p, ps<Nps\lt N, p<q<2ps,μ*p\lt q\lt 2{p}_{s,\mu }^{* }, 0<μ<N0\lt \mu \lt N, λ\lambda , and kk are some positive parameters, ps,μ*=pN−pμ2N−ps{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 VV and MM 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.

Keywords

• hardy-littlewood-sobolev nonlinearity
• schrödinger-kirchhoff equations
• variational methods
• electromagnetic fields
• 35j10
• 35b99
• 35j60
• 47g20