Odd symmetry of least energy nodal solutions for the Choquard equation

We consider the Choquard equation (also known as the stationary Hartree equation or Schrodinger Newton equation) -Delta u+u = (I-alpha*|u|(p))|u|(p-2)u. Here I-alpha stands for the Riesz potential of order alpha is an element of (0, N), and N-2/N+alpha < 1/p <= 1/2. We prove that least energy nodal solutions have an odd symmetry with respect to a hyperplane when alpha is either close to 0 or close to N. (C) 2017 Elsevier Inc. All rights reserved.