Semirelativistic Choquard equations with singular potentials and general nonlinearities arising from Hartree-Fock theory

Abstract

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 $\mu \to 0^+$.