## 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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### References:

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