Concentration and multiplicity of semiclassical states for the double phase problems with Choquard term

In this paper, we investigate the following singularly perturbed double phase problem in [Formula: see text] with nonlocal Choquard reaction and competing potentials: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] is a small parameter, [Formula: see text] and [Formula: see text] are continuous real functions. Under some natural assumptions, using variational methods together with Nehari manifold argument, energy comparisons and compactness analysis, we prove the existence and concentration of semiclassical ground state solutions. Moreover, based on the Lusternik–Schnirelmann category theory, we establish a multiplicity result which depends on the topology property of the set where [Formula: see text] attains its global minimum and [Formula: see text] attains its global maximum.