A critical fractional Choquard–Kirchhoff problem with magnetic field

Abstract

In this paper, we are interested in a fractional Choquard–Kirchhoff-type problem involving an external magnetic potential and a critical nonlinearity [Formula: see text] [Formula: see text] where [Formula: see text] with [Formula: see text], [Formula: see text] is the Kirchhoff function, [Formula: see text] is the magnetic potential, [Formula: see text] is the fractional magnetic operator, [Formula: see text] is a continuous function, [Formula: see text], [Formula: see text] is a parameter, [Formula: see text] and [Formula: see text] is the critical exponent of fractional Sobolev space. We first establish a fractional version of the concentration-compactness principle with magnetic field. Then, together with the mountain pass theorem, we obtain the existence of nontrivial radial solutions for the above problem in non-degenerate and degenerate cases.