Groundstates for Choquard equations with the upper critical exponent

Abstract: In this paper, an autonomous Choquard equation with the upper critical exponent is considered. By using the Poho\v{z}aev constraint method, the subcritical approximation method and the compactness lemma of Strauss, a groundstate solution in $H^1(\mathbb{R}^N)$ which is positive and radially symmetric is obtained. The result here extends and complements the earlier theorems.