Abstract
In this paper, we mainly study the fractional elliptic equation:
\(\left(a+b\int\limits_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx\right)(-\Delta)^{s}u+u=(|x|^{-\mu}*|u|^{p})|u|^{p-2}u,\quad x\in\mathbb{R}^{3},\)
where \(\mu\in(0,3)\) , \(s\in(0,1)\) , \(2-\frac{\mu}{3}<p<\frac{6-\mu}{3-2s}\) , \(a,b>0\) . For this equation, we will discuss it in two case. For \(s\in(0,\frac{3}{4}]\) , we prove the existence of solutions by establishing an equivalent system. For \(s\in(\frac{3}{4},1)\) , we use the symmetric mountain pass lemma to prove that the equation has infinitely many solutions.
