Multiplicity of solutions to a nonlocal Choquard equation involving fractional magnetic operators and critical exponent

Abstract

In this article, we study the multiplicity of solutions to a nonlocal fractional Choquard equation involving an external magnetic potential and critical exponent, namely, $$\displaylines{ (a+b[u]_{s,A}^2)(-\Delta)_A^su+V(x)u =\int_{\mathbb{R}^N}\frac{|u(y)|^{2_{\mu,s}^*}}{|x-y|^{\mu}}dy|u|^{2_{\mu,s}^*-2}u +\lambda h(x)|u|^{p-2}u\quad \text{in }\mathbb{R}^N, \cr [u]_{s,A}=\Big(\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^N} \frac{|u(x)-e^{i(x-y)\cdot A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{N+2s}}\,dx\,dy\Big) ^{1/2} }$$ where $a\geq 0$, b>0, $00$ is a parameter, $2_{\mu,s}^*=\frac{2N-\mu}{N-2s}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality and $20.

Keywords

• Choquard equation
• fractional magnetic operator
• variational method
• critical exponent