Liouville theorems for an integral equation of Choquard type

Abstract

We establish sharp Liouville theorems for the integral equation \begin{document}$ u(x) = \int_{\mathbb{R}^n} \frac{u^{p-1}(y)}{|x-y|^{n-\alpha}} \int_{\mathbb{R}^n} \frac{u^p(z)}{|y-z|^{n-\beta}} dz dy, \quad x\in\mathbb{R}^n, $\end{document} where $ 0<\alpha, \beta<n $ and $ p>1 $. Our results hold true for positive solutions under appropriate assumptions on $ p $ and integrability of the solutions. As a consequence, we derive a Liouville theorem for positive $ H^{\frac{\alpha}{2}}(\mathbb{R}^n) $ solutions of the higher fractional order Choquard type equation \begin{document}$ (-\Delta)^{\frac{\alpha}{2}} u = \left(\frac{1}{|x|^{n-\beta}} * u^p\right) u^{p-1} \quad\text{ in } \mathbb{R}^n. $\end{document}