We are interested in the general Choquard equation \begin{ multline*} \sqrt{ \strut -\Delta + m^2} \ u - mu + V(x)u - \frac{ \mu}{ |x|} u = \left( \int_{ \mathbb{ R}^N} \frac{ F(y,u(y))}{ |x-y|^{ N-\alpha}} \, dy \right) f(x,u) - K (x) |u|^{ q-2}u \end{ multline*} under suitable assumptions on the bounded potential \(V\) and on the nonlinearity \(f\). Our analysis extends recent results by the second and third author on the problem with $\mu = 0$ and pure-power nonlinearity $f(x,u)=|u|^{ p-2}u$. We show that, under appropriate assumptions on the potential, whether the ground state does exist or not. Finally, we study the asymptotic behaviour of ground states as $\mu \to 0^+$.
Categories: math.AP