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STABLE SOLUTIONS TO THE STATIC CHOQUARD EQUATION
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 102 / Issue 3 / December 2020
- Published online by Cambridge University Press:
- 10 June 2020, pp. 471-478
- Print publication:
- December 2020
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- Article
- Export citation
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This paper is concerned with the static Choquard equation
where
$$\begin{eqnarray}-\unicode[STIX]{x1D6E5}u=\bigg(\frac{1}{|x|^{N-\unicode[STIX]{x1D6FC}}}\ast |u|^{p}\bigg)|u|^{p-2}u\quad \text{in }\mathbb{R}^{N},\end{eqnarray}$$
$N,p>2$ and 
$\max \{0,N-4\}<\unicode[STIX]{x1D6FC}<N$. We prove that if 
$u\in C^{1}(\mathbb{R}^{N})$ is a stable weak solution of the equation, then 
$u\equiv 0$. This phenomenon is quite different from that of the local Lane–Emden equation, where such a result only holds for low exponents in high dimensions. Our result is the first Liouville theorem for Choquard-type equations with supercritical exponents and 
$\unicode[STIX]{x1D6FC}\neq 2$.
