Semirelativistic Choquard equations with singular potentials and general nonlinearities arising from Hartree–Fock theory. (April 2022) - Bernini, Federico; Bieganowski, Bartosz; Secchi, Simone
Abstract: We are interested in a general Choquard equation − Δ + m 2 u − m u + V ( x ) u − μ | x | u = ∫ R N F ( y, u ( y ) ) | x − y | N − α d y f ( x, u ) − K ( x ) | u | q − 2 u 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 μ = 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 μ → 0 + .