The semirelativistic Choquard equation with a local nonlinear term

We propose an existence result for the semirelativistic Choquard equation with a local nonlinearity in ${\mathbb{R}}^N$ \begin{equation*} \sqrt{\strut -\Delta + m^2} u - mu + V(x)u = \left( \int_{$\mathbb{R}$^N} \frac{|u(y)|^p}{|x-y|^{N-\alpha}} , dy \right) |u|^{p-2}u - \Gamma (x) |u|^{q-2}u, \end{equation*} where $m>0$ and the potential $V$ is decomposed as the sum of a ${\mathbb{Z}}^N$-periodic term and of a bounded term that decays at infinity. The result is proved by variational methods applied to an auxiliary problem in the half-space ${\mathbb{R}}_{+}^{N+1}$.