Odd symmetry of least energy nodal solutions for the Choquard equation

Ruiz, David;Van Schaftingen, Jean
(2018) Journal of Differential Equations — Vol. 264, p. 1231-1262 (2018)

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Abstract
We consider the Choquard equation (also known as 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.
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Ruiz, D., & Van Schaftingen, J. (2018). Odd symmetry of least energy nodal solutions for the Choquard equation. Journal of Differential Equations, 264, 1231-1262. https://doi.org/10.1016/j.jde.2017.09.034 (Original work published 2018)