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Odd symmetry of least energy nodal solutions for the Choquard equation. (English) Zbl 1377.35011

Summary: We consider the Choquard equation (also known as the stationary Hartree equation or Schrödinger-Newton equation) \[ -\Delta u+u=(I_\alpha \ast |u|^p)|u|^{p-2}u. \] Here \(I_{\alpha}\) stands for the Riesz potential of order \(\alpha \in (0,N)\), and \(\frac{N-2}{N+\alpha}< \frac {1}{p} \leq \frac {1}{2}\). 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 \(N\).

MSC:

35B06 Symmetries, invariants, etc. in context of PDEs
35J61 Semilinear elliptic equations
35R09 Integro-partial differential equations
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