Ground states and semiclassical states of nonlinear Choquard equations involving Hardy-Littlewood-Sobolev critical growth

Abstract

We are concerned with the existence of ground states for nonlinear Choquard equations involving a critical nonlinearity in the sense of Hardy-Littlewood-Sobolev. Our result complements previous results by Moroz and Van Schaftingen where the subcritical case was considered. Then, we focus on the existence of semi-classical states and by using a truncation argument approach, we establish the existence and concentration of single peak solutions concentrating around minima of the Schr\"odinger potential, as the Planck constant goes to zero. The result is robust in the sense that the nonlinearity is not required to satisfy monotonicity conditions nor the Ambrosetti-Rabinowitz condition.