Odd symmetry of least energy nodal solutions for the Choquard equation

We consider the Choquard equation (also known as the stationary Hartree equation or Schrödinger-Newton equation)−Δu+u=(Iα⁎|u|p)|u|p−2u. Here Iα stands for the Riesz potential of order α∈(0,N), and N−2N+α<1p≤12. We prove that least energy nodal solutions have an odd symmetry with respect to a hyperplane when α is either close to 0 or close to N.