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

TL;DR

This paper investigates the existence and behavior of solutions to a generalized semirelativistic Choquard equation with singular potentials and nonlinearities, extending previous work and analyzing the impact of parameters on ground state solutions.

Contribution

It extends prior results to more general nonlinearities and potentials, providing new existence criteria and asymptotic analysis for ground states as  approaches zero.

Findings

Existence of ground states depends on potential conditions.

Asymptotic behavior of solutions as  ^+ is characterized.

Extended results to broader nonlinearities and potentials.

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 $\mu \to 0^+$.