Signed and sign-changing solutions to the nonlinear Choquard problem with upper critical exponent

Abstract

We consider a class of Choquard problem set in Ω \Omega which is a symmetric bounded smooth domain of R N \mathbb {R}^{N} ( N ≥ 3 N\geq 3 ) with the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. Assume that Ω \Omega is invariant under the action of a group G G of orthogonal transformations, up to some range which depends on the symmetries, we prove the existence of infinitely many nontrivial G G -invariant solutions to this problem, one of them is positive and the rest are sign-changing. Moreover we show the gradient estimate of solutions is closely related to the best Hardy-Littlewood-Sobolev constant. The main results extend and complement the earlier works in the literature.