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 -Δ+ m^2} \ u - mu + V(x)u - \fracμ{|x|} u = \left( \int_{\mathbb{R}^N} \frac{F(y,u(y))}{|x-y|^{N-α}} \, 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 $\mu \to 0^{+}$.

References in corpus (3)

• Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation
• Effective Dynamics for Boson Stars
• Solutions to a nonlinear Maxwell equation with two competing nonlinearities in ${\mathbb{R}}^3$

Cited by in corpus (1)

• Asymptotic decay of solutions for sublinear fractional Choquard equations