Nodal solutions for a fractional Choquard equation

Abstract

In this paper, we study the existence of nodal solutions for the following fractional Choquard equation(−△)αu+u=∫RN|u(z)|p|x−z|μdz|u(x)|p−2u(x),x∈RN, where 0<μ<2α<N and 2N−μN−1<p<2N−μN−2α with α∈(12,1). In view of the nonlocality of the fractional Laplacian operator and the so-called Hartree term ∫RN|u(z)|p|x−z|μdz|u(x)|p−2u(x), the corresponding variational functional has entirely different properties with respect to the Laplacian case. For any k∈N, we prove that the problem possesses at least a radially symmetrical solution which changes sign k times.