Localized nodal solutions for system of critical Choquard equations

Abstract

In this paper, we study the system of critical Choquard equations ɛ2Δvj−a(x)vj+∑i=1kβijɛ−α(∫RN|vi(y)|2α∗dy|x−y|N−α)|vj|2α∗−2vj+λj|vj|q−2vj=0,x∈RN,vj(x)→0as|x|→∞,j=1,…,k,where N≥3, λj>0,j=1,…,k,(N−4)+<α<N,2α∗=N+αN−2, max{2,2∗−1}<q<2∗=2NN−2, ɛ>0 is a small parameter and the potential function a is bounded and positive. By the truncation method and the method of invariant sets of descending flow, we establish for small ɛ the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function a.