Choquard Equations with Mixed Potential

In this paper, we study the following class of nonlinear Choquard equation, $$-\mathrm{\Delta }u+a(z)u=K(u)f(u)\text{in}{\mathbb{R}}^N,$$ where ${\mathbb{R}}^N={\mathbb{R}}^L\times {\mathbb{R}}^M$, $L\ge 2$, $K(u)=\mathrm{\mid }\mathrm{.}{\mathrm{\mid }}^{-\gamma }\ast F(u)$, $\gamma \in (0,N)$, $a$ is a continuous real function and $F$ is the primitive function of $f$. Under some suitable assumptions mixed on the potential $a$. We prove existence of a nontrivial solution for the above equation.