Saddle solutions for the Choquard equation II

Abstract

We study nodal solutions of saddle type for the Choquard equation −Δu+u=(Iα∗|u|p)|u|p−2uinRNwhere N≥3 and 2≤p<N+αN−2. The function Iα refers to the Riesz potential with order α∈(0,N). In a noncompact setting, we construct saddle solutions whose nodal domains are of conical shapes demonstrating polyhedral symmetry configurations in RN. More precisely, when N≥4, for given integers ℓ1,ℓ2≥2, this equation admits saddle solutions with its 4ℓ1ℓ2 nodal domains meeting at the origin. These among other saddle solutions not only reveal the nonlocal feature of the Choquard equation but also further complete the variational framework of constructing sign-changing solutions for the Choquard equation initiated in Ghimenti and Van Schaftingen (2016) and further developed in Xia and Wang (2019).