Existence of Solutions to the Logarithmic Choquard Equations in High Dimensions

In this paper, we consider the following logarithmic Choquard equation $$\begin{aligned} - \,\Delta u + a(x)u + \lambda (\log |\cdot |*|u|{^2})u = b|u|^{p-2}u, ~~\ \text{ in }~\ {\mathbb {R}}^n, \end{aligned}$$where $$n\ge 3, \lambda >0, 2 2$$, this equation has been studied extensively. In this paper, we prove the existence of a mountain-pass solution and a ground state solution for $$n\ge 3$$ by using the variational method.